Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$? $A$ is a square matrix. All elements of $A$ depend on a parameter $t$, that is, $a_{ij}=a_{ij}(t)$. Let $S(A):=A^TA$, and take the derivative of $S$ w.r.t. $t$:
$\displaystyle \frac{dS}{dt}$
Now, pretty clearly
$\displaystyle \fr... | Short answer : no.
Think about the $ij$ entry of $A^T A$; it's
$$
s_{ij} = \sum_k a_{ki} a_{kj}
$$
Take the derivative with respect to $t$ (using primes to denote that) to get
$$
s'_{ij} = \sum_k a_{ki}' a_{kj} + \sum_k a_{ki} a_{kj}'
$$
The claim is that this is just $2 \sum_k a_{ki}' a_{kj}$, after some index-shuff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can a Mersenne number ever be a Carmichael number? Can a Mersenne number ever be a Carmichael number?
More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ (Fermat's Test)?
Cases potentially proved so far: (That are never Carmichael numb... | Let say $2^t-1$ has n prime factors like {$p_1,p_2,p_3,..,p_n$}.
If it is Carmichael number than $2^t-2$ should divisible by all n numbers which are like
{$num_1=p_1-1,num_2=p_2-1,num_3=p_3-1,..,num_n=p_n-1$} and all those numbers are even.
$2^t-2=2*(2^{t-1}-1))$ because of that equality n numbers all should just be ju... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880896",
"timestamp": "2023-03-29T00:00:00",
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Why these two series are convergent or divergent? I do not understand why $$\sum^{\infty}_{k=1} z_k = \sum^{\infty}_{k=1} \frac1k$$ is divergent but the other series $$\sum^{\infty}_{k=1} z_k = \sum^{\infty}_{k=1} \frac{(-1)^{k+1}}k$$ is convergent. For both cases the $\displaystyle\lim_{n \to +\infty} z_{n} = 0$. Coul... | The approach I would use would be to expand the sequences and see whether or not a lower or upper limit can be placed.
For $\frac1k$, consider that:
$$\frac12 + \frac13 + \frac14 + \frac15 + \frac16 + \frac17 + \frac18 + \frac19 + \frac1{10} + \frac1{11} + \frac1{12} + ...$$
...is certainly a larger sum than:
$$\frac12... | {
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"timestamp": "2023-03-29T00:00:00",
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If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$ I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms. So, is the following stat... | First, $\mathbb{Z}_n \oplus \mathbb{Z}_n$ is abelian, while there are many non-cyclic groups that are non-abelian (take $S_3$ for example), so the answer to your question as written is immediately no.
However, what if we only consider abelian non-cyclic groups? Then $\mathbb{Z}_2 \oplus \mathbb{Z}_6$ and $\mathbb{Z}_2 ... | {
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Solving recurrence relation: Product form Please help in finding the solution of this recursion.
$$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$
where $ f(1)=1$ and $f(2)=2$.
| As @Winther commented, letting $a_n=\log f(n)$ one has
$$a_n-a_{n-1}-a_{n-2}=-\log n.$$
We only need a particular solution. Let $F_n$ be the Fibonacci sequence $F_0=F_1=1, F_i=F_{i-1}+F_{i-2}$. And consider
$$b_n= \sum^n_{i=0}F_i\log (n-i).$$
It is easy to show that
$$b_n=b_{n-1}+b_{n-2}+\log n.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Probability in game of bridge In a game of bridge, find the probability that the North, East, South, and West hands will get respectively $a,b,c,d$ spades.
I tried like this. First I choose $a$ spades from the $52$ cards; then, from the remaining $39$ I choose $b$ spades; then, from the remaining $26$ cards I choose $c... | Imagine dealing in an unusual way, $13$ cards to South, then $13$ to East, and so on.
There are $\binom{52}{13}$ equally likely ways to choose the cards South gets. There are $\binom{13}{a}\binom{39}{13}$ ways to choose $a$ spades and $13-a$ non-spades. So the probability that South gets the right kind of hand can be c... | {
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Find 3rd side, given two sides and bearings The bearing from A to B is N $42^\circ$ E. The bearing from B to C is S $44^\circ$ E. A small plane traveling $65$ miles per hour, takes $1$ hour to go from A to B and $2$ hours to go from B to C. Find the distance from A to C.
| You have $<ABC = 42+44 = 86^\circ$, and $AB = 65$, $BC = 130$. Use law of cosine to get the answer.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Vitali set of outer measure 1 How to construct a Vitali set of outer measure 1. I couldn't understand the argument given here. Isn't there any easier way? I would also like if someone explains that to me.
Thank you in advance!
| Since $\mathbb{R}$ is a vector space over $\mathbb{R}$, there exists a
$\mathbb{Q}$ vector subspace $V$ of $\mathbb{R}$ such that $V\oplus \mathbb{Q} = \mathbb{R}$ ( it uses the existence of a basis of $\mathbb{R}$ extending $\{1\}$, so it uses the Zorn lemma).
Now, $V$ is a Vitali set, and moreover $\mathbb{Q}^{\time... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to interpret a discontinuity in 2D Pareto Frontier? I've solved a bi-objective optimization problem by means of NOMAD solver from OPTI Toolbox and as a result I've obtained a Pareto frontier:
How to interpret the visible "gap" in the Pareto frontier?
| I will try to answer myself.
Consider the Schaffer function no. 2:
$\begin{cases}
f_{1}\left(x\right) & = \begin{cases}
-x, & \text{if } x \le 1 \\
x-2, & \text{if } 1 < x \le 3 \\
4-x, & \text{if } 3 < x \le 4 \\
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Geometric meaning of reflexive and symmetric relations A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of those ordered pairs $(a,b)$ such that $ a \sim b$.
With this no... | Your description of reflexivity is correct.
For symmetry it means that the subset $R$ is "symmetric" around the line $y = x$, this means that for any point $(a, b)\in R$ its mirror point $(b, a)\in R$ (it's the point you get by doing reflection in the line $y=x$), i.e. either none of the two points $(a, b)$ and $(b, a)... | {
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Algebraic proof of $\tan x>x$ I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 x-1>0$ so $f$ is increasing, and $f(0)=0$.) $\tan x$ is defined to be $\frac{\sin x}{\cos x}$ where... | Here is a sketch of what you might be looking for:
Showing $\tan x > x$ is equivalent to showing $\sin x - x \cos x > 0$, since $\cos x > 0$ on $(0,\pi/2$).
The series for $\sin x - x \cos x$ is $\displaystyle\sum_{j=1}^{\infty} \dfrac{(2j)x^{2j+1}}{(2j+1)!} = x^3/3 - x^5/30 + x^7/840 - x^9/45360 \ldots$
Group the term... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proof of Descartes' theorem I came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any proof.Even Wikipedia also, just states the theorem!!I want to know the procedure to find the radius of the Soddy Circle??
I apologize if its duplicate and to ment... | Part I - Proof of Soddy-Gosset theorem (generalization of Descartes theorem).
For any integer $d \ge 2$, consider the problem of placing $n = d + 2$ hyper-spheres touching each other in $\mathbb{R}^d$. Let $\vec{x}_i \in \mathbb{R}^d$ and $R_i \in \mathbb{R}$ be the center and radius for the $i^{th}$ sphere. The condit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$
Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$.
I need to show that for each $\epsilon$ there exists an $n_0 \in \mathbb{N}$ such that $ \forall n \geq n_0: |2^{\frac{-1}{\sqrt{n}}}-1|\lt \epsilon$
I was simply trying to solve $2^{\frac{-1}{\sqrt{n}}}=1$ by tak... | No, it isn't good way, because from $2^{\frac{-1}{\sqrt{n}}}=1$ you get $\frac{-1}{\sqrt{n}}=0$-it's not possible.
What you can do:
1) If you know that for all $a>0$ $\lim_{n \to \infty} a^{\frac{1}{n}}=1$ you can use this.
2)If you don't know that for all $a>0$ $\lim_{n \to \infty} a^{\frac{1}{n}}=1$ you can prove thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/881860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the perimeter of the room If the length and breadth of a room are increased by $1$ $m$, the area is increased by $21$ $m^2$. If the length is increased by $1$ $m$ and breadth is decreased by $1$ $m$ the area is decreased by $5$ $m^2$. Find the perimeter of the room.
Let the length be $x$ and the breadth be $y$
... | $A=xy$
$(x+1)(y+1)=xy+21$
$(x+1)(y-1)=xy-5$
Foil out both equations to get:
$xy+x+y+1=xy+21 \quad \to \quad x+y=20 \quad \to \quad y=20-x$
$xy-x+y-1=xy-5 \quad \to \quad -x+y=-4 \quad \to \quad y=-4+x$
Set them equal to each other:
$20-x=-4+x$
$2x=24 \to x=12$
Since we know $x+y=20, y=8$.
You can verify this solution b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given two circles, find the length of a pulley belt that connects the two. So the problem is that there is one circle with radius of five and one circle with radius of 1. There centers are 8 units apart and there is a pulley belt that goes around the outside as shown in the image. It is given that the belt touches 2/3 ... | Hint: Let $A$ and $B$ be the centres of the bigger and smaller circles, respectively. Now let $C$ and $D$ be the endpoints of the upper part of the belt (so that $C$ is the point of tangency of the larger circle and $D$ is the point of tangency of the smaller circle). Now draw a line parallel to $CD$ that goes through ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/882067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find a conformal map from semi-disc onto unit disc This comes straight from Conway's Complex Analysis, VII.4, exercise 4.
Find an analytic function $f$ which maps $G:=$ {${z: |z| < 1, Re(z) > 0}$} onto $B(0; 1)$ in a one-one fashion.
$B(0;1)$ is the open unit disc.
My first intuition was to use $z^2$, which does the ... | The following trick works for any region bounded by two circular arcs (or a circular arc and a line).
Find the points of intersection of the arc and the line. (Here, they're $i$ and $-i$.) Now pick a Mobius transformation that takes one of those points to $0$ and the other to $\infty$; here $z \mapsto \frac{z-i}{z+i}$... | {
"language": "en",
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Contour Integration $\int_0^1\frac1{\sqrt[n]{1-x^n}}dx$ I want to compute:
$$\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$$
for natural $n>1$ using Residue Calculus.
I am thinking of using some kind of a keyhole or bone contour that could go around the $n$th roots of unity (singularities in this case). The problem is I beli... | In questions like this one, in order to avoid the problems of defining the right branch of the logarithm or the $n$th root, I suggest to, first start with a change of variables and to use Residue Theorem afterwards.
So, here how I do this. First the change of variables $x^n=\dfrac{e^t}{1+e^t}$ we get
$$
I_n~{\buildrel ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/882216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Find $\lim_{x\to0}\frac{\sin5x}{\sin4x}$ using $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$. I am trying to find $$\lim_{x\to0}\frac{\sin5x}{\sin4x}$$
My approach is to break up the numerator into $4x+x$. So,
$$\begin{equation*}
\lim_{x\to0}\frac{\sin(4x+x)}{\sin4x}=\lim_{x\to0}\frac{\sin4x\cos x+\cos4x\sin x}{\sin4x... | Hint:
$$\lim_{x\to 0} \frac{\sin 5x}{\sin 4x} = \frac{5}{4}\lim_{x\to 0} \frac{\sin 5x}{5x}·\frac{4x}{\sin 4x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/882392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral/infinite sum related to Bessels which pop up in optical coherence theory In propagating partially coherent optical fields, the following integral pops up:
$$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$
where $a$ and $b$ are real numbers. If we consider reducing the power on the cosine we fi... | In terms of the modified generalized Bessel functions introduced in [1] and [2],
$$
\int_0^{2\pi} \mathrm{d}\theta\; e^{i(a \cos\theta \,+\, b \cos 2\theta)} = 2\pi \sum_{m=-\infty}^\infty i^{-m} J_{2m}(a)\,J_m(b) = 2\pi J_0(a,b;-i) = 2\pi I_0(a,-ib)
$$
In [1], [2], and other publications of the authors, they discuss t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/882467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does anyone know of a non-trivial algebraic structure satisfying these four identities? Does anyone know of a non-trivial (i.e. cardinality $\geq 2)$ algebraic structure $(X,+,-)$ satisfying the following identities?
*
*$(x+a)-a=x$
*$(x-a)+a=x$
*$(x+y)+a = (x+a)+(y+a)$
*$(x-y)+a = (x+a)-(y+a)$
Remark. The Abelian... | Below I will demonstrate that, if $+$ is associative, then $x+a = x$ and $x-a = x$ for all $x$ and $a$. This is enough, I think, to qualify as a "trivial" algebraic structure, even though the underlying set can be as large as you like.
Beginning from (3):
\begin{align*}
(x+y)+a &= (x+a) + (y+a)\\
(x+y)+a &= ((x+a)+y) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Strict convexity and best approximations Let $V$ be a normed vector space.
It is said to be strictly convex if its unit sphere does not contain nontrivial segments.
A subset $A \subset V$ is said to have the unicity property if for any $x \in V$,
there is exactly one $x' \in A$ with $|x - x'| = \inf_{y \in A}|x - y|$.
... | Yes, the converse is also true. Suppose the space is not strictly convex. Let $[a,b]$ be a line segment contained in the unit sphere. The function $$t\mapsto \|(1-t)a+tb\|,\qquad t\in\mathbb R\tag1$$
is convex and is equal to $1$ on $[0,1]$. Therefore, it is greater than or equal to $1$ everywhere.
The distance from ... | {
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Integration with two unknowns I'm completely stumped with this one, I'm not sure how I should do this.
The equation of a parabola is $y=-3x(x-2)$. It intersects the $x$-axis at $0$ and $2$.
Given that the area of this parabola is $4\,{\rm units}^2$, there will be a straight line $y=mx$ which divides the area exactly in... | A good idea would be to integrate $\max(f(x)-mx,0)$ between 0 and 2, which indeed leads you to solve $f(x)=mx$, which then gives you $x_M=2-\frac{m}{3}$.
Now you simply have to integrate the following: $\int_0^{2-\frac{m}{3}} -3x(x-2) - mx dx = -\frac{1}{54}(m-6)^3$.
Making it equal to 2 gives you the result you're af... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the potential function of the field $\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$ The vector field is obviously conservative on every closed domain that doesn't encompass the point $(0,0)$, so there must be a potential function.
I've got $\arctan(\frac{x}{y})$ for $x$ unequal to zero and $\arctan(\frac{y}... | The field is $$ u = \frac{-y}{x^2+y^2}\mathbf{e_x}+\frac{x}{x^2+y^2}\mathbf{e_y}=\frac1{r} \mathbf{e_\theta}$$
In cylindrical coordinates
$$\nabla \phi = \frac{\partial \phi}{\partial r}\mathbf{e_r}+\frac1{r} \frac{\partial \phi}{\partial \theta}\mathbf{e_\theta}=\frac1{r} \mathbf{e_\theta}.$$
So $\phi = \theta$ has t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the Smallest Integer $N$ Where Reversing the Digits Makes $3N$? What is the smallest positive integer N such that the integer formed by reversing the digits of N is triple N? (Does such an integer even exist? If not, then for what multiplier for $N$ will such an integer exist?)
Here are my thoughts so far: (whe... | There is no such $N$ (besides $0$).
Now, say $N$'s first digit is $a$ and its last digit is $b$. Since $N$ and $3N$ have the same number of digits, $a$ can only be $1, 2$, or $3$. If $a=3$, then $b=9$ is the only choice, but this is impossible since then $3N$ would end in $7$, not $3$. $a=1$ also doesn't work, since t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integral of a function with two parts (piecewise defined) The function has 2 parts:
$$f(x) = \begin{cases} -\sin x & x \le 0 \\ 2x & x > 0\end{cases}$$
I need to calculate the integral between $-\pi$ and $2$.
So is the answer is an integral bewteen $-\pi$ and $0$ of $f(x)$ and then and $0$ to $2$.
but why the calculat... | Note that $$ \displaystyle\int_{-\pi}^{2} f(x) \, \mathrm{d}x = \displaystyle\int_{-\pi}^{0} f(x) \, \mathrm{d}x + \displaystyle\int_{0}^{2} f(x) \, \mathrm{d}x $$ because each integral is an area under the curve $f(x)$ under the given domain and we want to find the full area so we just add the two "components" of the ... | {
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"timestamp": "2023-03-29T00:00:00",
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Why does $\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}$ converge to $\log\left[\frac{1}{\epsilon}\right]$? while playing around with my equations, i found that the following has to hold for my universe to be consistent:
$$\lim_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{... | Your limit can be seen as a Riemann sum.
$$
\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}
=\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac1N\,\frac{1}{\frac{1}{1-\epsilon}-\frac iN}}
=\int_0^1\frac1{\frac1{1-\epsilon}-t}\,dt
=-\log\left.\left(\frac1{1-\epsilon}-t\right)\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/882982",
"timestamp": "2023-03-29T00:00:00",
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Best algebra text for Model Theory I'm looking for an algebra book that is tailored towards some of the ideas in Model Theory, I'm currently slogging through Hodges' Model Theory. I'm a bit rusty with my algebra and was curious if there are algebra texts aimed at Model Theory.
| I know this is a bit old, but two other references that may be worth looking into are Grätzer's Universal Algebra and Mal'cev's Algebraic Systems. They both contain material on model theory and are done "in the spirit", so to speak, of this discipline. I specially Mal'cev's book; although its notation is a bit non-stan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/883038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Quotient of Unit Quaternions by Subgoup (Lie Groups) Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ is a perfect group with one non-trivial normal subgroup of order $2$ (it... | Since the group G acts on the sphere by orientation preserving diffeo without fixed points, the quotient is an orientable manifold of dimension three, obviously compact. In particular, H_3 is Z The map from the sphere to M is clearly the universal covering space of M, so the fundamental group of M is G. One can check T... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The number of subgroups conjugate to a given subgroup of a finite group Let $H$ and $K$ are subgroups of $G$ conjugate to each other.
$A$ is defined as
$$A = \{a \in G \mid aHa^{-1} = H \}$$ for all $a\in G$.
Prove that $A$ is a subgroup of $G$ and prove that if $G$ is finite, then the number of subgroups that are c... | If $aHa^{-1}=bHb^{-1}$, $b^{-1}aH(b^{-1}a)^{-1}=H$, so $b^{-1}a\in A$, which is equivalent to $aA=bA$.
Thus ths number of different $aHa^{-1}$ is same as the number of different cosets of $A$, which is same as $|G|/|A|$ by Lagrange's theorem in group theory.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$
If $a$, $b$ and $c$ are positive real numbers, prove that:
$$\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$$
Additional info:We can use AM-GM and Cauchy inequaliti... | Another way to do this would be the following (I'm doing Liu Gang's suggested generalization):
We have to show
$$\frac{a^{n+1}}{b^n} + \frac{b^{n+1}}{c^n} + \frac{c^{n+1}}{a^n} - \frac{a^n}{b^{n-1}} - \frac{b^n}{c^{n-1}} - \frac{c^n}{a^{n-1}} \ge 0.$$
The left hand side equals
$$\frac{a^n(a - b)}{b^n} + \frac{b^n(b-c)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/883384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Simplification of expressions with radicals in Maple Having for example the expression $$\frac{abc\sqrt2}{d\sqrt{ab}}$$ (which results from a sequence of manipulations), can I force Maple to write it in the form $$\frac{c\sqrt{2ab}}{d}.$$ Many might find this as being the same thing, but I prefer the second to clarify ... | You wrote "for example". Does that mean that your example only has the form of your actual problem, and that say a and b are used by you here as placeholders for more involved expressions? If so, then do you know anything about their sign?
expr:=a*b*c*sqrt(2)/(d*sqrt(a*b));
1/2... | {
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"source": "stackexchange",
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Do two points determine a unique line in 4D space? I wish to generalize the notion of two points determining a unique line to four dimensions, but with the additional condition that all points on the line are a unit distance from the origin and the "line" is not straight, but forms a least-distance curve between the tw... | It sounds like you're just trying to carry out the picture of spherical geometry in 3-D to 4-D.
In the 3-D picture, the surface of the unit sphere is taken to be the set of points, and the "lines" are the great circles. Any two points which aren't antipodal determine a unique great circle.
The way to look at the grea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finitely many Supreme Primes? A challenge on codegolf.stackexchange is to find the highest "supreme" prime: https://codegolf.stackexchange.com/questions/35441/find-the-largest-prime-whose-length-sum-and-product-is-prime
A supreme prime has the following properties:
*
*the number itself is prime
*the number of digit... | This is not an answer, just a bit too long to be a comment. I didn't write the code for finding supreme primes, but I think it is simple.
All supreme primes $x$ are of the form:
$$x = \sum_{k=0}^n 10^k + 10^w\times(p-1) = \frac{10^{n+1} - 1}{9} + 10^w\times(p-1) \tag{1}$$
where $p$ is a prime number, and $0\le w \le n$... | {
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"timestamp": "2023-03-29T00:00:00",
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What's the difference between the different types of poles, zeroes and singularities in complex analysis? I am trying to get an understanding on the difference between the different types of poles, zeroes and singularities in complex analysis and how to identify them.
When is it a removable singularity, and why? When ... | This is how poles of different order looks like. If you are not sure, you can just plot it.
$\dfrac{z}{1-\cos z}$ is a good example. $1-\cos z$ has zero at $z=2πk$ but $z$ on the nominator removed the second order pole at $z=0$ by order 1. So, this function has a pole of order 1 at $z=0$ but of order 2 at other $z=2πk... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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1729, and related questions I just read this paragraph: (written by G. H. Hardy, on Ramanujan)
I remember once going to see him when he was lying ill at Putney. I
had ridden in taxi cab number 1729 and remarked that the number seemed
to me rather a dull one, and that I hoped it was not an unfavorable
omen. ‘No,’... | Very late for this party, but yes, there is an infinite number of taxicab numbers. The complete solution in positive integers to,
$$x_1^3+x_2^3 = x_3^3+x_4^3$$
was given by Choudhry's On Equal Sums of Cubes (1998). For positive integers $a,b,c$,
$$\begin{aligned}
d\,x_1 &= (a^2 + a b + b^2)^2 + (2a + b)c^3\\
d\,x_2 &= ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to describe $G/U$? Let $G=SL_2(\mathbb{C})$ and let $U = \{\left( \begin{matrix} 1 & x \\ 0 & 1 \end{matrix} \right): x \in \mathbb{C}\}$. We have an action of $U$ on $G$ by right multiplication. By definition, $G/U$ is the set of all $U$-orbits under this action. How to describe $G/U$ as a set?
Take $\left( \begin... | Consider the tautological action of $G:= SL_2(\mathbb C)$ on $\mathbb C^2$. It is transitive on the
points of $\mathbb C^2 \setminus \{0\}$, and the stabilizer of the vector $(1,0)$ is precisely $U$.
So the quotient $G/U$ is naturally identified with $\mathbb C^2\setminus \{0\}$.
As user165670 notes in their answer,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Two dice thrown, one comes up 6 If my friend throws two dice, and covers them up, but I see that one of them was a 6, what's the probability that they were both 6s given this knowledge?
I'm under the impression that the answer is 2/7, because the other die could be any of the other numbers, but if he really did roll do... | If the intuition is not yet clear, perhaps one can do a formal conditional probability calculation. Let $A$ be the event "at least one $6$" and $D$ the event "double $6$." We want $\Pr(D|A)$. By the definition of conditional probability this is $\frac{\Pr(A\cap D)}{\Pr(A)}$.
The event $A\cap D$ is just the event $D$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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Estimating the mean and variance of numbers assigned to each person in population of one billion Problem : Consider people of one billion, and each has one card containing one number. For instance first has card of number $7$. second has card of number $11$ and so on (simply if number means age or weight, it is fine). ... | You have described a particular type of bootstrap procedure that is called "$m$ out of $n$ bootstrapping".
In ordinary bootstrapping, we take a data set of $n$ observations and resample from it with replacement a large number of times, obtaining new samples that are also of size $n$. For each bootstrap sample, we com... | {
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p-adic cubic root Let $p$ be prime such that $p\equiv 2\bmod 3$. Show that for every $a\in \mathbb Z,p\nmid a$ there is a $x\in \mathbb Z_p$, where $\mathbb Z_p$ is the field of the p-adic integers, such that $x^3=a$.
| Hint (already given in comments): Hensel's lemma reduces to showing all elements of $\Bbb F_p^\times$ are cubes, which follows easily from $3\nmid(p-1)$. Can you see why?
Perhaps you'd get what's going on if I state it in a more general form: if $G$ is a finite group with order $n$ and $m$ is any number coprime to $n$,... | {
"language": "en",
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"source": "stackexchange",
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Help with Rudin rank theorem proof! I am struggling through Rudin's proof of the rank theorem (9.32) in the baby Rudin book. There is a part in the proof where he claims that for a finite-dimensional linear operator A, if the set V is open, then A(V) is an open subset of the range of A. I have seem things about the ope... | Pick any $x_0 \in V$. We will show that $Ax_0 $ is an interior point of $A(V)$.
By translating (i.e. consider $V - x_0$ instead of $V$), we can assume $x_0 = 0$.
Let $y_1, \dots, y_n$ be a basis of $\rm{Range}(A)$ and choose $x_1, \dots, x_n$ with $y_i = Ax_i$ for each $i$.
As $V$ is open with $0 \in V$, there is some ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show there exists a Cauchy subsequence Let $X$ be a separable reflexive real Banach space and $\{\psi_n\}$ be a dense sequence in
$$\{\psi\in X' : ||\psi||_{X'} \leq 1\}.$$
Consider in $X$ the scalar product defined by
$$(x | y)_0 = \sum_{n=1}^\infty 2^{-n} \langle \psi_n,x \rangle\langle \psi_n,y \rangle.$$
Show eve... | *
*It looks fine.
*I think the denseness is used implicitly when you want to show that $\|\cdot\|_0$ is a norm.
| {
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"source": "stackexchange",
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Extensions of degree $1$. My doubt is very simple:
Let $F|K$ be a field extension, if $[F:K]=1$, what can we say about $F$ and $K$? can I say $F=K$?
I'm trying to prove the equality without success.
Thanks in advance
| Suppose $\exists a \in K \setminus F$. Then $1, a$ are $F$-linearly independent. Proof: If $f_1, f_2 \in F$ such that $f_1 1 + f_2 a = 0$, it follows that $f_2 a = -f_1$. If $f_2 \neq 0$, then $a = - \frac{f_1}{f_2} \in F$, a contradiction. Otherwise, we have $f_1 = 0$ as well, i.e. the only solution is $f_1 = f_2 = 0$... | {
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Probability that a word contains at least 3 same consecutive letters? Assume we have a word of length $n$ and an alphabet of length $26$ (the small letters a through z, if you want so.
How likely is it that this word contains at least $k := 3$ consecutive letters of any type?
Examples that match:
aaabababab
aoeuuuuuuu... | The probability of no 3 consecutive letters in a word of length $n$ is $$\frac{(1-p)^2}{a-b}\,\left(\frac{a^{n-1}}{1-a}-\frac{b^{n-1}}{1-b}\right),$$ where $$a=\frac{p+\sqrt{p(4-3p)}}2,\quad b=\frac{p-\sqrt{p(4-3p)}}2,\quad p=1-\frac1{26}.$$ In particular, when $n\to\infty$, the probability of no 3 consecutive letters ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Selecting 180 days from 366: the probability of even distribution across months, or not having September among the first 30
In a draft lottery containing the 366 days of the year (including February 29). Select 180 days (draw 180 without replacement).
a) What is the probability that the 180 days drawn are evenly distr... | (a)
$$ \displaystyle \frac{{31 \choose 15}{29 \choose 15}...{31 \choose 15}}{366 \choose 180} $$
(b)
$$ {30 \choose 0}{336 \choose 180}\over {366 \choose 180} $$
Hypergeometric distribution. We divide the year up into different categories: 12 months in (a) and September versus the rest of the year in (b).
In both... | {
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I need help solving this indefinite integrals problem? I am doing indefinite integrals homework, and this problem popped up. I hate to post on here without any personal insight on the problem, but I really have no idea on how to approach this.I do not know what to do with the information given. Any insight on how to so... | Here's a hint:
The total time between dropping the stone and hearing the splash can be broken down into two parts:
*
*The time it takes for the stone to hit the water after being released from your hand ($t_1$)
*The time it takes for the sound of the splash to reach your ear ($t_2$)
You can write down an equation... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivation of Schrödinger's equation I recall a famous quote of the late physicist Richard Feynman:
Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger.
This quote was with reference to the derivation of Schrödinger's equation. I often found it str... | I think there is a post almost identical with yours at here:
https://physics.stackexchange.com/questions/30537/is-the-schr%C3%B6dinger-equation-derived-or-postulated
but there is a much better answer at here:
https://physics.stackexchange.com/questions/83450/is-it-possible-to-derive-schrodinger-equation-in-this-way/834... | {
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How long will it take to fill a water tank with two inlet pipes and one outlet? E11/40.
I can see here that I work the fill rate out as:
$\large\frac13 + \frac14 - \frac18 =$
Overall fill rate of $\large\frac{11}{24}$ tank per hour.
If I multiply $60$ minutes by $\large\frac{24}{11}$ I get the correct result of $131$ ... | Let S be the volume of the tank.
t is time
S is a number between 0 and 1 (empty and full)
$S(t) \in [0,1]$
*
*The first inlet pipe
liquid flows at speed $V_1$ (this is constant)
$S(0) = 0$ (empty)
$S(3) = 1$ (full)
$S(t) = V_1 * t + C$ (this comes from assuming constant water speed)
$0 = S(0) = V_1 * 0 + C$
so C =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why does the order of summation of the terms of an infinite series influence its value? I was looking through my lecture notes and got puzzled by the following fact: if we want to find the value of some infinite series we are allowed to rearrange only the finite number of its terms. To visualize this consider the alter... | It's quite easy to think up elementary counter-examples.
For example, consider the series
$$1-1+1-1+1-1+...=(1-1)+(1-1)+(1-1)+...\\
=0+0+0+...\\
=0.$$
If it is permissible to commute an infinite number of terms, you can rearrange the series into,
$$1-1+1-1+1-1+1-...=1+(-1+1)+(-1+1)+(-1+1)\\
=1+0+0+0+...\\
=1,$$
implyin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 1
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integral of $\frac{1}{(1+e^{-x})}$ I make the substitution $u=1+e^{-x}$ which gives $-\dfrac{e^x}{u}\ du$.
Integrating gives me $$-e^x\ln(1+e^{-x}) + C,$$
but the answer is $\ln(e^x +1) + C$.
What am I doing wrong?
| We have with $u=e^{-x}$ so $du=-e^{-x}dx\implies dx=-\frac{du}{u}$
$$\int \frac{dx}{1+e^{-x}}=-\int\frac{du}{u(1+u)}=\int\frac{du}{1+u}-\int\frac{du}{u}=\ln(1+u)-\ln u+C\\=\ln\left(1+e^x\right)+C$$
| {
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Prove that $f'(x_o) =0$ Let $f$ be a function defined on an interval $I$ differentiable at a point $x_o$ in the interior of $I$.
Prove that if $\exists a>0$ $ \ [x_o -a, x_o+a] \subset I$ and $ \ \forall x \in [x_o -a, x_o+a] \ \ f(x) \leq f(x_o)$, then $f'(x_o)=0$.
I did it as follows:
Let b>0.
Since $f$ is diff... | You proved that $$\forall b>0, |f'(x_0)|<b$$
and this means that $f'(x_0)=0$. So your proof is already finish.
| {
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#26, the Inversion of Sugar I'm trying to solve #26 from Chapter 7, Transcendental Functions (Thomas' Calculus 12th Edition) and I can't seem to figure out this problem:
The Processing of raw sugar has a step called "inversion" that changes the sugar's molecular structure. Once the process has begun, the rate of chang... | If, in 10 hours, 80% of the substance is remaining, then, since the rate of inversion is proportional to the amount of sugar left, 1 hour gives you $(0.8)^\frac{1}{10} \approx 0.977933$ or $97.7933 \%$ of the amount from the previous hour.
If you want to find this by solving a differential equation (which you don't nee... | {
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Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$? Any suggestions? I have tried using D'Alembert's test, but on the end I get 1. I can't think of any other series with which to compare it. In my textbook the give the following solution which I don't quite understand:
$\sum... | Squeezing if for the ultimate non-believers. Since $2\leq\left(1+\frac{1}{n}\right)^n\leq e$ and $n\leq\sqrt{n^2+1}\leq(n+1)$,
$$\sum_{n=1}^{N}\frac{1}{e(n+1)}\leq\sum_{n=1}^{N}a_n \leq \sum_{n=1}^{N}\frac{1}{2n},$$
but the LHS is greater than:
$$\frac{1}{e}\sum_{n=1}^{N}\log\left(1+\frac{1}{n+1}\right) = \frac{1}{e}\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/885680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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The blow up of of the plane and the Moebius band The (real) blow up of $\mathbb{R}^2$ is defined by $\tilde{\mathbb{R}}^2=\{(p,l)\in\mathbb{R}^2\times\mathbb{R}\mathbb{P}^1|p\in l\}$, with the projection $\pi:\tilde{\mathbb{R}}^2\to\mathbb{R}^2$, given by $(p,l)\mapsto p$.
Intuitively speaking, $\tilde{\mathbb{R}}^2$ l... | $\mathbb{RP}^1$ can be thought of as the line segment $[0,\pi]$ with its endpoints identified, so:
| {
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If $f$ satisfies $|f(x_1)-f(x_2)|\leqslant(x_1-x_2)^2$ on an interval, then it is constant
Prove that if $f$ is a function on an interval $[a,b]$ satisfying $$|f(x_1)-f(x_2)|\leqslant(x_1-x_2)^2 \ \text{ for all } \ x_1,x_2\in[a,b],$$ then $f$ is constant on $[a,b]$.
For any $x\in(a,b)$, we have $|f(x+h)-f(x)|/|h|\le... | I know an elementary answer which does not derivatives anywhere.
Taking $x_1=a$, $x_2=b$ then $|f(a)-f(b)|\leq |b-a|^2$.
If $x_1=a$, $x_2=(a+b)/2$ then $|f(a)-f((a+b)/2)|\leq |b-a|^2/4$.
If $x_1=(a+b)/2$, $x_2=b$ then $|f(b)-f((a+b/2))|\leq |b-a|^2/4$.
Therefore $|f(a)-f(b)|\leq |f(a)-f((a+b/2))|+|f(b)-f((a+b/2))|=|b... | {
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In calculus, which questions can the naive ask that the learned cannot answer? Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them.
Calculus is not known to be such a field, as far as I kno... | One result that may surprise most calculus students is that there is no algorithm for testing equality of real elementary expressions. This then implies undecidability of other problems, e.g. integration. These are classical results of Daniel Richardson. See below for precise formulations.
$\qquad\qquad$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/885934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "112",
"answer_count": 13,
"answer_id": 4
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How do I go from this $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$? So I am doing $\int\frac{x^2-3}{x^2+1}dx$ and on wolfram alpha it says the first step is to do "long division" and goes from $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$. That made the integral much easier, so how would I go about doing that in a clear... | Hint:
$$\frac{x^{2}-3}{x^{2}+1}=\frac{(x^{2}+1)-4}{x^{2}+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/885999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 0
} |
Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis I am not able to fill the gap in proof of following theorem which is stated as...
Let $U$ be an open set in a locally compact hausdorff space $X$, $K\subset U$ and K is compact. Then there exists an open set $V$ with compact closure such that : ... | I guess i got the solution and i thought it is not a better idea to edit the question..
So, I am writing this..
Suppose $\bigcap_{p\in C}(C\cap \overline{G}\cap \overline{W_p})$ is non empty...
Then i have $x\in \bigcap_{p\in C}(C\cap \overline{G}\cap \overline{W_p})$
In particular, $x\in C\cap \overline{G}\cap \overl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/886066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Help needed on convex optimization!!
Can some please help me in solving the below questions. I want to prove the below functions are convex, concave or neither.
| Here are some useful facts:
Any norm is convex. If $f$ is convex and $T$ is affine then $g(x) = f(T(x))$ is convex. A conic combination of convex functions is convex. A maximum of convex functions is convex.
These rules can be used to prove the functions in your question are convex.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/886132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to treat Dirac delta function of two variable? We can treat one variable delta function as
$$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$
Then how do we treat two variable delta function, such as $\delta(f(x,y))$?
for example, how calculate $\int\int \delta(x-y)$ ?
I first thought usi... | $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \mathrm{d}x\,\mathrm{d}y\, f(x)g(y)\delta(x,y) = \int_{-\infty}^{+\infty}\mathrm{d}y\, f(y)g(y),$$ or
$$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \mathrm{d}x\,\mathrm{d}y\, f(x)g(y)\delta(x,y) = \int_{-\infty}^{+\infty}\mathrm{d}x\, f(x)g(x),$$
which is the same... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/886253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Number of conjugates of a subgroup If $G$ is a simple non-abelian group and $H$ is a subgroup with $[G:H]=7$ then what is the number of conjugates of $H$ in $G$?
So far I found that the order of $H$ cannot be a prime number using Sylow theorems.
| The number of conjugates of $H$ is $[G:N_G(H)]$, where $N_G(H)$ is the normalizer of $H$ in $G$. Note that $H$ is normal iff $G=N_G(H)$. Since $H \subseteq N_G(H)$, and $H$ cannot be normal, $H=N_G(H)$, and $H$ has exactly 7 conjugates: $7=[G:H]=[G:N_G(H)] \cdot [N_G(H):H]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/886305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Generators for a finitely generated graded ring Given a Noetherian graded ring (commutative and with 1) $A=\bigoplus_{n=0}^\infty A_n$, that's generated as an $A_0$-algebra by $x_1,\ldots, x_s\in A$. I am having difficulties seeing why there is no loss in generality by assuming that the $x_i$ are homogeneous. Could som... | Write $x_i=\sum_{n\ge0}x_{in}$ with $x_{in}\in A_n$. Then $x_{in}$, $i=1,\dots,s$, $n\ge0$ is a homogeneous generating set for the $A_0$-algebra $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/886464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Properties of a set in $\ell^2$ space Let $\ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\}$
and $e_n \in \ell^2 $ be the sequence whose $n$-th element is $1$ and all other elements are $0$.
Equip the space with $\ell_2$ with the norm $$\|x\| = \left(\sum_{n=1}^\inft... | Every pair of orthonormal vectors $e_i\neq e_j$ has distance precisely $d(e_i,e_j)=\sqrt{2}$.
So $S$ is closed as it contains only isolated points, it is bounded as all its elements have norm one, it is not compact as it contains infinitely many isolated points, it contains a convergent subsequence the constant ones?!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/886557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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In Godel's first incompleteness theorem the Godel sentence G is true otherwise it contradicts itself, however its truth implies it is not provable . How can this be? I understand there are two basic definitions of truth in mathematics, one being the formalist definition which includes excluded middle and the second for... | Also if we stay with your awkward simplification, G's Incompleteness Th is no problem for intuitionism.
You are right in saying that for "the intuitionist [...] truth is based only on [...] provability", but this must not be read as "provability into a formal system".
G's proof is perfectly "sound" for an intuitionist ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
} |
Recurrence of the form $2f(n) = f(n+1)+f(n-1)+3$ Can anyone suggest a shortcut to solving recurrences of the form, for example:
$2f(n) = f(n+1)+f(n-1)+3$, with $f(1)=f(-1)=0$
Sure, the homogenous solution can be solved by looking at the characteristic polynomial $r^2-2x+1$, so that in general a solution for the homogen... | Let $f_p = A + Bn + Cn^2$.
$$
\begin{cases}
2f(n) = 2A +2Bn + 2Cn^2\\
-f(n+1) = -A - B(n+1) - C(n+1)^2\\
-f(n-1) = -A - B(n-1) - C(n-1)^2
\end{cases}
\quad \Rightarrow
$$
$$
3 = Cn^2 - C(2n^2 + 2) = -2C \quad \Rightarrow \quad C = -\frac{3}{2}
$$
Thus,
$$
f(n) = C_1 + C_2n -\dfrac{3n^2}{2}
$$
Now use the given initial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/886753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Find out whether two rectangles are intersecting in 3D space I've got two rectangles in 3D space, each given by the coordinates of their 4 corners. They are not axis aligned, meaning their edges are not necessarily parallel/perpendicular to the world axes. Each rectangle can have any orientation.
Is there an easy way t... | A point in each rectangle is individuated vectorially by
$$
\eqalign{
& {\bf P}_{\,1} = {\bf t}_{\,1} + a{\bf u}_{\,1} + b{\bf v}_{\,1} \quad \left| {\;0 \le a,b \le 1} \right. \cr
& {\bf P}_{\,2} = {\bf t}_{\,2} + c{\bf u}_{\,2} + d{\bf v}_{\,2} \quad \left| {\;0 \le c,d \le 1} \right. \cr}
$$
with an obv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$ Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers.
Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$.
I don't understand why that is. Any help would be greatly appreciated.
| Hint: $\bigl||x|-|y|\bigr|\leq |x-y|$ (the reverse triangle inequality) holds for complex numbers $x,y$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the sup-norm is not derived from an inner product I am trying to show that the norm $$\lVert{\cdot} \rVert _{\infty}=\sup_{t \in R}|x(t)|$$ does not come from an inner product (the norm is defined on all bounded and continuous real valued functions).
I tried to show that the inner product does not hold by us... | To simplify my answer, I'll ignore the "continuous" requirement and assume there is an appropriate inner product for that norm.
Let $b$ be a real number, $$f(x) = \left\{ {\begin{array}{*{20}{c}}
{1,}&{x = 0} \\
{0,}&{x \ne 0}
\end{array}} \right.$$
and $$g(x) = \left\{ {\begin{array}{*{20}{c}}
{1,}&{x = 2} \\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Exponential distribution - Using rate parameter $\lambda$ vs $\frac{1}{\lambda}$ Sometimes I see the exponential distribution defined as follows:
$$f(x) = \lambda e^{-\lambda x}$$ when $x > 0, 0$ otherwise
I have also seen it defined like so:
$$f(x) = \frac{1}{\lambda} e^{-\frac{x}\lambda}$$ when $x > 0, 0$ otherwise
S... | A matter of taste. If $\lambda$ is used as the rate parameter, which is the common usage, then we have $E[X] = \frac{1}{\lambda}$. However, it may sometimes be more intuitive to let the first moment parameterize the distribution (e.g. like the Poisson distribution), so if we have $E[X] = \mu = \frac{1}{\lambda}$, then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of Weierstrass' second theorem using the Fejér operator Weierstrass' second theorem states the following:
Let $f$ be a real continuous $2\pi$-periodic function (write $f\in C_{2\pi}$). Then for all $\epsilon>0$ there exists a trigonometric polynomial $p$ such that $\|f-p\|_{\infty}<\epsilon$
This theorem can be p... | $F_n(t)$ is a trigonometric polynomial of "degree" $n-1$, as exhibited by your second formula. Therefore the functions
$$g_t(\theta):= F_n(t-\theta)={1\over2}+\sum_{k=1}^{n-1}\left(1-{k\over n}\right)\bigl(\cos(kt)\cos(k\theta)+\sin(kt)\sin(k\theta)\bigr)$$
are trigonometric polynomials in $\theta$ for each fixed $t$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Computing the sum of an infinite series I am confused as to how to evaluate the infinite series $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}.$$
I tried splitting the fraction into two parts, i.e. $\frac{\sqrt{n+1}}{\sqrt{n^2+n}}$ and $\frac{\sqrt{n}}{\sqrt{n^+n}}$, but we know the two individual infinit... | Your problem may be converted to the following formula:
\begin{align}
& \lim_{N\to\infty}\sum_{n=1}^{N}\left({\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}}\right) =
\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)+\left(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}\right)+...+\left(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{N+1}}\right) \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Are there some functions that cannot be optimized using calculus? I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical function using calculus? I'd assume the reason genetic a... | Calculus methods are useful for functions which are differentiable. For functions which are not differentiable calculus won't help much. For instance the data set could be discrete. Some examples of this include:
*
*A set of binary variables like a set of yes/no decisions. Which set of filters and parameters shall w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Solutions for the given system with fractions I have to solve in $\Bbb{R}$ the following system :
$$
\ \left\{
\begin{array}{ll}
\frac{y}{x}+\frac{x}{y}=\frac{17}{4} \\
x^2-y^2=25
\end{array}
\right.$$
For this one I am stuck, I tried to use the fact that $x^2-y^2=(x-y)(x+y)$ and multiply by $x... | Hint:
$$x/y=t\Rightarrow y/x=1/t$$
from first equation
$$t+1/t=17/4\iff 4t^2-17t+4=0$$
$$t_{1,2}=\frac{17\pm15}{8}=4,1/4$$
$x=4y$ or $y=4x$
from second equation
$$(4y)^2-y^2=25,y^2=5/3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/887515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Canonical isomorphism between Cauchy sequence completion and inverse limit I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can see how these two are isomorphic as groups. H... | Outline of the proof.
Let $C$ be the group of Cauchy sequences (without the equivalence classes.)
If $\mathbf{a}=\{a_i\}$ is Cauchy, define $N_k(\mathbf {a})$ the be the least value so that for all $i,j\geq N_k(\mathbf a),\ a_i-a_j\in G_k$. This can be written as $a_i+G_k=a_j+G_k$ in $G/G_k$. Then define $\phi_k:C\to G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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A question from an engineering undergraduate My question primarily concerns the necessary transition from an undergraduate program in electrical engineering to graduate program in applied mathematics or pure mathematics.
I'm an electrical engineering student. During the first year in my university life, I found mys... | As an engineer interested in mathematics, you might want to look into the field of Continuum Thermomechanics. There are (applied) mathematics departments which offer such courses; yours might be such a school. Since you mentioned that you have done some self-study, books to look at as an introduction include:
1) The Me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
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Evaluating $ \int x \sqrt{\frac{1-x^2}{1+x^2}} \, dx$ I am trying to evaluate the indefinite integral of
$$\int x \sqrt{\frac{1-x^2}{1+x^2}} \, dx.$$
The first thing I did was the substitution rule:
$u=1+x^2$, so that $\displaystyle x \, dx=\frac{du}2$ and $1-x^2=2-u$. The integral then transforms to
$$\int \sqrt{\fr... | $\text {Let } x^{2}=2 \sin ^{2} \theta-1 \textrm{ for } \frac{\pi}{4} \leqslant \theta \leqslant \frac{\pi}{2}, \text {then } x d x=\sin \theta \cos \theta d \theta $.
\begin{aligned}\int x \sqrt{\frac{1-x^{2}}{1+x^{2}}} d x&=\int \sqrt{\frac{2-2 \sin ^{2} \theta}{2 \sin ^{2} \theta} }\sin \theta \cos \theta d \theta \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/887784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
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Using mean value theorem to show that $\cos (x)>1-x^2/2$ I have a question, by applying the mean value theorem to $f(x)=\frac{x^2}{2}+\cos (x)$, on the interval $[0,x]$, show that $\cos (x)>1-\frac{x^2}{2}$.
We know that $\frac{\text{df}(x)}{\text{dx}}=x-\sin (x)$, for $x>0$. By the MVT, if $x>0$, then $f(x)-f(0)=(x+0... | You started off well.
Notice that, by MVT:
$$f'(c) = \frac{f(x) - f(0)}{x - 0}$$
S0
$$xf'(c) = f(x) - f(0)$$
Notice that x is positive, and since $$f'(x) = x - sin(x)$$
Also, note that $x > \sin(x)$, so $f'(x) > 0$
Therefore,
We can conclude that
$$f(x) > f(0)$$
And
$$\cos(x) > 1- \frac{x^2}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/887954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Finding solution basis of $y^{(4)}-2y'''+5y''-8y'+4y=0$ Find a real-valued solution basis of $$y^{(4)}-2y'''+5y''-8y'+4y=0.$$
The corresponding characteristic equation is $$x^4-2x^3+5x^2-8x+4=0$$ $$\iff(x-1)^2(x^2+4)=0$$ which has the zeros $1, 2i, -2i$. How do I proceed from here? Please share a hint with me.
Thank yo... | Hint: the multiplicity of the root $x=1$ is $2$; this means that the functions
$$y_i(x)=x^i\exp(1\cdot x)$$
with $i=0,1$ are solutions of the ODE in the OP.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/888024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How do I show that f is strictly decreasing on (0, infinity)? I have been asked to define $f: (0, \infty) \to (0, \infty)$ by $f(x) = \frac 1 x$
a) How do I show that f is strictly decreasing on $(0, \infty)$? I realize that I have to show that $f'(x)<0$, but I'm not entirely sure how to go about this. Would anyone be ... | a)$$f'(x)=-\frac{1}{x^2}<0 \ \ \ \forall x \in (0, \infty)$$
Since the first derivative is negative at the whole interval $(0,\infty)$ the function $f$ is strictly decreasing on this interval.
b) You have to show that the function $f$ is injective.
Then to find $f^{-1}$, you have set $f(x)=y$ then you have to switch $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Solving the logarithimic inequality $\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$ I tried solving the logarithmic inequality:
$$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$
several times but keeping getting wrong answers.
| Here are the steps
\[
\log_2 \frac{x}{2}+\frac{\log_2 x^{2}}{\log_2 \frac{2}{x}} \le 1
\]
\[
\log_2 x -\log_2 2+\frac{2\log_2 x}{\log_2 2-\log_2 x} \le 1
\]
\[
\log_2 x -1+\frac{2\log_2 x}{1-\log_2 x} \le 1
\]
Let $\alpha= \log_2 x$, then
\[
\alpha -1+\frac{2\alpha}{1-\alpha} \le 1
\]
\[
\alpha -2+\frac{2\alpha}{1-\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Can someone explain in general what a central difference formula is and what it is used for? Topic- Numerical Approximations
| Even though I feel like this question needs some improvement, I'm going to give a short answer.
We use finite difference (such as central difference) methods to approximate derivatives, which in turn usually are used to solve differential equation (approximately).
Recall one definition of the derivative is
$$f'(x)=\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to graph $y=f(x^2)=\sin(x^2)$? How to graph $y=f(x^2)=\sin(x^2)$?
I have substituted as follows:
$$\begin{cases} y=f(a)=\sin a\\ a=x^2\end{cases}.$$
Then if I graph this with the coordinate axes $y$ and $a$ I get the ordinary sine function. But this doesn't solve my problem. Is it possible to graph my example $f(x^... | If you have the graphs of $y=f(x)$ and $y=g(x)$, you can create the graph of $y=g(f(x))$ from them easily in the following manner.
First, draw the graphs of $y=f(x)$ and $y=g(x)$ on the same set of axes, and additionally draw the line $y=x$ there as well.
To plot the point $(x,g(f(x)))$, start at the point $(x,0)$ on t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $f(x)=x^4$ is convex for $x\in (0,\infty)$ show $f(x)=x^4$ is convex.
I know it is convex since $f''(x)>0$
. How can we show by using definition?
do we have to use
Let L be linear space. $t\in[0,1],y\in L,f(xt+y(1-t))=(xt)^4+4(xt)^3((1-t)y)^1+6(xt)^2((1-t)y)^2+4(xt)(((1-t)y)^3+((1-t)y)^4$
edit:
$(xt)^4+4(x... | It is easy to show that $f(x) = x^2$ is convex and increasing on $\mathbb{R}_+$.
Hence $\forall x, y \in \mathbb{R}_+, t \in [0, 1]$ we have:
$$(tx + (1-t)y)^4 = ((tx + (1-t)y)^2)^2 \stackrel{(1)}\leqslant (tx^2 + (1-t)y^2)^2
\stackrel{(2)}\leqslant \\ t(x^2)^2 + (1-t)(y^2)^2 = tx^4 + (1-t)y^4.$$
$(1)$: using that $x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question about cardinals in ZF In Lévy's basic set theory refers a theorem of Bernstein as exercise problem:
Theorem. (Bernstein) Let $\mathfrak{a,b}$ be cardinals. If $\mathfrak{a+a=a+b}$, then $\mathfrak{b\le a}$.
I try to prove it using the following theorem:
Theorem. If $\mathfrak{a+c=b+c}$ then there are cardin... | Lets say you have a bijection $\mathfrak{b}\cup \mathfrak{a}\rightarrow \mathfrak{a}\times\{0,1\}$ where we assume that $\mathfrak{b}$ and $\mathfrak{a}$ are disjoint.
Consider the sequence
$$b\rightarrow (a_1,0)$$
$$a_1\rightarrow (a_2,0)$$
$$a_2\rightarrow (a_3,0)$$
$$\vdots$$
$$a_{n-1}\rightarrow (a_n,1)$$
Let u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Vector field ${\bf F}$ with $\int_S {\bf F}\cdot{\bf n}\ dS=c$ Find a vector
field ${\bf F}$ on $
{\bf R}^3$ with $$\int_S {\bf F}\cdot{\bf n}\ dS=c > 0 \tag{1} $$ where $S$ is any closed surface containing $0$ and ${\bf n}$ is
normal
Here there is a solution $\frac{k}{r^3} (x,y,z)$. Note that from divergence therem... | When they talk about "closed surfaces $S$ containing $0$" they tacitly mean that such $S$ should bound a compact body $B\subset{\mathbb R}^3$ which contains $0$ in its interior. Now we cannot have arbitrarily tiny such surfaces giving a fixed value $c>0$ for the integral in question unless something terrible happens a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/888810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Don't understand inequality in order to prove Algebraic Limit Theorem I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 2.3.3 on page 45, i.e., the Algebraic Limit Theorem.
In particular, letting $\lim a_n = a$ and $\lim b_n = b$, then I'm trying to follow the proof that... | Try using the triangle inequality as
$$|b| = |b - b_n + b_n| \leq |b - b_n| + |b_n|$$
Then use the assumption on $|b - b_n|$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/888883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The Mini-Max theorem for lattices I'm asking for help on an exercise in Davey and Priestleys's Introduction to Lattices and Orders. For those with the book, the exercise is specifically 2.9.
Let $A=(a_{ij})$ be an $m\times n$ matrix with entries in a lattice $L$. Show that
$$\bigvee_{j=1}^n\left(\bigwedge_{i=1}^m a_... | Hint: Try to use the same approach which is used when proving usual distributive inequality. Start with the following inequalities:
$$\bigwedge_{i = 1}^m a_{ij} \leqslant a_{kj} \leqslant \bigvee_{l = 1}^n a_{kl},\ k = \overline{1, m},\ j = \overline{1, n}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/888974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Why is the argument of a complex number measured anticlockwise (from the positive real axis), rather than clockwise? I was going through some basic examples of complex numbers (finding the argument and modulus) with my brother yesterday, and he asked
Why is the argument measured anticlockwise rather than clockwise [fr... | If you want Euler's formula to hold, then it follows from the sine and cosine functions going counterclockwise (otherwise you would have to shove an unnatural negative sign into an otherwise elegant formula). This fact for sine and cosine seems to follow from the convention to label the 'up' $y$ direction and 'right' $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Tough limit to evaluate I am trying to solve this limit problem
$$\lim_{x\to 1} {(1-x)(1-x^2)....(1-x^{2n})\over[(1-x)(1-x^2)....(1-x^n)]^2}$$
I am not able to figure how to to convert it to a compact form. Any tips?
| The $r(1\le r\le n)$th term is $$T_r=\lim_{x\to1}\frac{(1-x^{2r-1})(1-x^{2r})}{(1-x^r)^2}$$
$$=\lim_{x\to1}\frac{x^{2r-1}-1}{x-1}\cdot\lim_{x\to1}\frac{x^{2r}-1}{x-1}\frac1{\left(\lim_{x\to1}\dfrac{x^r-1}{x-1}\right)^2}$$
Now for integer $\displaystyle a>-1,\lim_{x\to1}\dfrac{x^a-1}{x-1}=\dfrac{d(x^a)}{dx}_{(\text{at }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How to do this integral $\int_{-\pi}^{\pi} x^n \cos^m(x) dx$? is there a way to explicitely evaluate this integral for natural numbers $n,m$:
$$\int_{-\pi}^{\pi} x^n \cos^m(x) dx.$$
Apparently, if $n$ is odd, this integral is zero due to symmetry.
| $$
\int_{-\pi}^\pi x^{10}\cos^{10}(x)\;dx =
{\frac {49408448066608271851}{16986931200000000}}\,\pi -{\frac {
13747940134011979}{7077888000000}}\,{\pi }^{3}
+{\frac {
3845425458091}{9830400000}}\,{\pi }^{5}
-{\frac {157029277}{
4096000}}\,{\pi }^{7}
+{\frac {49133}{20480}}\,{\pi }
^{9}
+{\frac {63}{1408}}\,{\pi }^{11}
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Conjecture for product of binomial coefficient Is it true that for any $n, k\in\mathbb N$
$$\frac{(kn)!}{k!(n!)^k} = \prod_{l=1}^k {{ln-1}\choose{n-1}} \quad?$$
I tested it for some small $k$ and $n$, but I don't know how to prove that it is true (or find example showing that it is not).
| To show that the theorem is true by algebraic manipulations, observe that
\begin{align}
\frac{(kn)!}{(n!)^k}
=\frac{(kn)!}{(kn-n)!n!}\cdot\frac{(kn-n)!}{(n!)^{k-1}}
&=\binom{kn}{n} \frac{(kn-n)!}{(n!)^{k-1}}\\
&=\binom{kn}{n}\binom{kn-n}{n}\cdot\frac{(kn-2n)!}{(n!)^{k-2}}\\
&\;\vdots\\
&=\binom{kn}{n}\binom{kn-n}{n}\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Transforming Limits of a Double Integral I´m working on trasnforming a double integral: $\int_0^\infty \int_0^x e^{-(x+y)} \,dy\,dx$ using the following identities identities. I need to get the limit to be 0 and1 in order to integrate it by using Monte Carlo´s integration
$$\theta=\int_0^\infty g(x)\,\mathrm dx,$$ we... | You don't need substitutions to solve this.
$$\begin{align}
\int_0^\infty \! \int_0^x e^{-(x+y)}\operatorname{d}y\operatorname{d}x
& = \int_0^\infty\!\int_0^x e^{-x}e^{-y}\operatorname{d}y\operatorname{d}x
\\[1ex]
& = \int_0^\infty e^{-x}\left(\int_0^x e^{-y}\operatorname{d}y\right)\operatorname{d}x
\\[1ex]
& = \ldo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is a bit-shifting standard C function for calculating $f(x) = \frac{(2^{16}- 1)}{(2^{32} - 1)}\cdot x$ I need to take 32-bit unsigned integers and scale them to 16-bit unsigned integers "evenly" so that $0 \mapsto 0$ and 0xFFFFFFFF $\mapsto$ 0xFFFF. I also want to do this without using a 64-bit unsigned integer b... | You're probably better off asking this on a programming forum.
That said, simply doing division is probably your best bet: not only does it make it obvious what you are doing, but the compiler is likely to automatically convert it to bit shifts for you anyways. And even if it doesn't, there's a good chance that it does... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Understanding trigonometric identities Can someone help me understand trigonometric identities? For example, it is known that $\cos(90-\theta)$ is equal to $\sin \theta$, and vice versa. But why? Is it something to do with the unit circle? Is it visual?
| Take a right triangle $ABC$ with $\angle B=90^\circ$ and $\angle A=\theta$. Then $\sin\theta = \frac{BC}{CA}=\cos \angle C=\cos (90^\circ-\theta)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/889549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Teaching the Concept of Infinity to Children. I was recently out with the family and we left it up to the children where we ate lunch (11 and 9 years old). They couldn't agree and were going back and forth calling each other names. This ultimately lead to the age old tradition of one kid saying to the other "You're stu... | To tell someone what infinity ... Take a sheet of $A4$ paper and divide it into two halves. Now take one of the halves and divide it again. Repeat this step indefinitely. Here ask the question 'Will this process finish?'.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/889618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
How many $3$ digit numbers with digits $a$,$b$ and $c$ have $a=b+c$ My question is simple to state but (seemingly) hard to answer. How many $3$ digit numbers exist such that $1$ digit is the sum of the other $2$. I have no idea how to calculate this number, but I hope there is a simple way to calculate it. Thank you in... | Assuming a digit is an element of $\{0,1,2,3,4,5,6,7,8,9,10\}$ we have three cases for $a,b,c$ to see:
*
*$a=b=c=0$. All easy here, yields $1$ combination.
*$b=c\ne 0$. $a=2b$, so $b<5$ giving us $4$ choices (digits $1$ to $4$). The position of $a$ uniquely determines the code, so multiply b $3$ to get $4\cdot 3 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Convergence in distribution/Distribution of X For each $n = 1, 2, ....$, suppose that $X_n$ is a discrete random variable with range $\{1/n, 2/n, ..., 1\}$ and
$\hspace{15mm}\mathrm{Pr}(X_n = j/n) = \frac{2j}{n(n+1)}$, $j = 1,...,n$.
Does $X_1, X_2, ...,$ converge in distribution to some random variable?
If it does, ... | Fix some $o\leq x\leq 1$. For any $n$, define $k(n)$ by $\frac{k(n)}{n}\leq x<\frac{k(n)+1}{n}$. We get
$F_n(x)=P(X_n\leq x)=\sum_{j=1}^{k(n)}\frac{2j}{n(n+1)}=\frac{2}{n(n+1)}\sum_{j=1}^{k(n)}j=\frac{2}{n(n+1)}\cdot\frac{k(n)(k(n)+1)}{2}=\frac{k(n)(k(n)+1)}{n(n+1)}$.
Calculate the limit as $n\to\infty$:
$$\lim_{n\to\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$ Let $f:I\rightarrow \Bbb{R}$, differetiable three times on the open interval $I$ which contains $[-1,1]$. Also: $f(0) = f(-1) = f'(0) = 0$ and $f(1)=1$.
Show that there's a point $c \in (-1, 1)$ such that $f^{(3)}(c) \ge 3$
I'd be glad to get a guidace here how to start.
| You can use argument contradiction to show. Suppose $f'''(x)<3$ for $\forall x\in(-1,1)$. In [-1,0], by MVT, we have
$$ f(0)-f(-1)=f'(c_1) $$
where $c_1\in(-1,0)$. So $f(c_1)=0$. Similarly in $[c_1,0]$, we have
$$ f'(0)-f'(c_1)=f''(c_2) $$
where $c_2\in(c_1,0)$. So $f''(c_2)=0$. In [0,1], we have
$$ f(1)-f(0)=f'(b_1) $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Implication related to carmichael function. If $g \in \Bbb Z_{n^2}^{*}$ and $x_1,x_2 \in \Bbb Z_n$ then help me in proving the following implication.
$g^{n \lambda(n)}\equiv 1 \mod{n^2} \implies g^{(x_1-x_2)\lambda(n)} \equiv 1 \mod{n^2}$
where $\lambda(n)$ is carmichael function.
I know how to prove the left side of ... | In the text there are additional conditions/restrictions on the $x_i,\;$
i.e. you have $r_1, r_2 \in \mathbb{Z}_n^{*}\;$ with
$$g^{x_1} r_1^n \equiv g^{x_2}r_2^n \pmod {n^2}.$$
Multiply both sides with $g^{-x_2},\;$ which exists because $g$ is invertible
$$g^{x_1-x_2} r_1^n \equiv r_2^n \pmod {n^2}$$
take powers with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/889981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A nice trignometric identity How to prove that:
$$\cos\dfrac{2\pi}{13}+\cos\dfrac{6\pi}{13}+\cos\dfrac{8\pi}{13}=\dfrac{\sqrt{13}-1}{4} $$
I have a solution but its quite lengthy, I would like to see some elegant solutions. Thanks!
| I add another answer although it is not different in principle from some of the others.
We have
$$t=\cos \frac{2\pi}{13}+\cos \frac{6\pi}{13}+\cos \frac{8\pi}{13}$$
it is natural to then consider the other even divisions,
$$s=\cos \frac{4\pi}{13}+\cos \frac{10\pi}{13}+\cos \frac{12\pi}{13}$$
Now
$$t+s=\cos \frac{2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$ I found a nice formula of the following integral here
$$\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$$
It states there that ... | Decomposing the integral into two, we have
$$
I= \int_0^{\infty} \frac{x^\alpha}{1+2 x \cos \beta+x^2} d x=\frac{1}{e^{\beta i}-e^{-\beta i}} \int_0^{\infty}\left(\frac{x^\alpha}{x+e^{-\beta i}}-\frac{x^\alpha}{x+e^{\beta i}}\right) d x
$$
Putting $u=\frac{x}{e^{\beta i}}$ transforms the second integral into
\begin{ali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/890210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 6,
"answer_id": 5
} |
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