Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solutions to $z^3 - z^2- z =15 $ Find in the form $a+bi$, all the solutions to the equation
$$z^3 - z^2- z =15 $$
I have no idea what to do - am I meant to factor out z to get $z(z^2-z-1)=15$ or should I plug in $a+bi$ to z?
Please help!!!!!!!
| Hint: First find the one real root, say $r$ (it's not hard). Then you can factor out $z-r$ out of $z^3-z^2-z-15=0$, and find the remaining roots using the quadratic formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
The digit at the hundred's place of $33^{33}$ I would want to know how to start with the question. And if you get hung up somewhere there's the answer it's $5$.
Any help is appreciated thanks,
My approach was to look at the factors to somehow crack the nut. But still in vain. Any help or tip or approach is alright as I... | Here goes, to see how quickly it can be done. Jack's extraction of a factor $8$ from the modulus is helpful.
Let's work with "the last three digits", using $\equiv$ when the thousands are dropped, but allow negative numbers if that simplifies things.
$$33^2=1089$$
$$33^4\equiv89^2=(90-1)^2=7921\equiv-79$$
$$33^8\equiv ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What vector field property means “is the curl of another vector field?” I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under suitable smoothness conditions on the component fun... | The concept of a pseudovector is quite similar, being associated with the curl of a vector. An apparent generalization is to the concept of a pseudovector field, for which I found a reference here.
There is also the related notion of a vorticity arising as the curl of a vector field but this is not generic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 1
} |
Product of a character with an irreducible character a non-negative integer Why is the inner product of a character with an irreducible character a non-negative integer?
I can see that by properties of the inner product it will be non-negative but I cannot see why it would be an integer?
| I assume we're talking about finite groups and complex scalars (so, the semisimple situation).
Note that if $U$ and $V$ are reps of $G$, then $\hom(U,V)$ is also a rep: $g$ acts as $T\mapsto gTg^{-1}$ for all linear maps $T:U\to V$. The invariant subspace $\hom(U,V)^G$ is precisely the space of equivariant maps (aka in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What I have to do to write and have published an article? Let's suppose that I have proved a theorem myself and I want to write an article about it, I have a few questions:
1) How do I have to write it ? I mean, what character should I use, what conventions I should follow, what format should I use and so on...
2) How ... | I'd say: Write a good introduction to your article and answers to these questions will come naturally.
Since you are supposed to place your result in the already existing literature on the problem, you will:
1) See what notations/conventions/format are used by others to deal with the problem. Use them! People like what... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
arccos and arcsin integral contradiction: I am shown:
$$f(x) = \arcsin x \implies f'(x) = \frac{1}{\sqrt{1-x^2}}$$
$$f(x) = \arccos x \implies f'(x) = -\frac{1}{\sqrt{1-x^2}}$$
These two derivatives can be very readily derived by a bit of implicit differentiation. However... I want to undo my differentiation for both e... | Two functions defined over an interval that have the same derivative differ by a constant.
Since the derivatives of $\arcsin x$ and $-\arccos x$ are both equal to $\dfrac{1}{\sqrt{1-x^2}}$ on the interval $(-1,1)$, you can say that there exists $c$ such that
$$
\arcsin x=c-\arccos x
$$
If you evaluate at $0$, you get
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1278997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Number of digits function Let $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ be the $10$ digits of the base $10$ system. The set of all finite sequences of those digits that don't start with a leading zero, together with the single one-element sequence $(0)$, can be bijectively mapped onto the natural numbers starting with 0, in t... | The number of digits in zero is $1$, since we have associated zero with the one-element sequence $(0)$, and the number of digits function is defined as the length of that sequence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to know if equation can be solved by factorising before trying? So, I have core 1 test tomorrow and there is a lot of solving of quadratic equations without calculator and my weakest point is the time I waste in trying to factorise and equation but then it ends up it doesn't factorise or the opposite way in which I... | For any quadratic equation $ax^2+bx+c=0$, $a,b,c\in \mathbb{R}$, there exist solution(s) (i.e., you can factorise) if and only if the discriminant
$$D=b^2-4ac$$
is higher than or equal to zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
What's the name for this mathematical device used by programmers? So a friend is trying to figure out what this is called so we can read more about it.
The concept is/was used by database designers, who needed a compact way to store a list of selected options as a single numerical value, where that number would be uni... | This is normally called a "Bit field."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field? Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field?
I'm confused because the polynomial $x^2+1$ in $F_2[x]$ is inseparable ($f(x)$ and $f'(x)$ a... | The definition of a field $E$ being perfect is that every irreducible polynomial $p\in E[x]$ is separable (Wikipedia link). But the polynomial $x^2+1$ is equal to $(x+1)^2$ in $\mathbf{F}_2[x]$, so it's not irreducible.
Similarly, the field $\mathbf{Q}$ being perfect doesn't conflict with $x^2+2x+1=(x+1)^2\in\mathbf{Q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find wave equation initial condition such that solution consists of right-going waves only Let $u(x,t)$ solve the wave equation $u_{tt}=c^2u_{xx}$ and let $u(x,0)=A(x)$ for some function $A(x)$. Find the function $B(x)=u_t(x,0)$ such that the solution consists of right going waves only.
My attempt:
By d'Alembert's form... | The general solution to the homogeneous wave function $u_{tt}-c^2u_{xx}=0$ is
$$u(x,t)=f(x-ct)+g(x+ct)$$
Applying the initial condition $u(x,0)=A(x)$ reveals that
$$f(x)+g(x)=A(x) \tag 1$$
Applying the initial condition $u_t(x,0)=B(x)$ reveals that
$$-cf'(x)+cg'(x)=B(x) \tag 2$$
Now, we differentiate $(1)$ to obtain... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Unique factorization in fields Suppose $A$ is a commutative $R$-algebra and that is also a field. Define:
*
*For $x,y \in A$, say that $x$ divides $y$ iff $xr = y$ for some $r \in R$.
*Call $x,y \in A$ associates iff each divides the other.
*Say that $I \subseteq A$ is an ideal iff $A$ is an additive subgroup of $... | There is a very nice book by Alfred Geroldinger and Franz Halter-Koch which studies factorization from the monoid standpoint. There is a survey article here: http://www.uni-graz.at/~geroldin/54-non-unique-fact-survey.pdf By these two, which I assume turned into the book (which is a bit pricey, but your library might ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How can $\frac{\mathrm{d}}{\mathrm{d}x}\left[y(u(x))\right] = \frac{\mathrm{d}y}{\mathrm{d}x}$? I just saw a video on the chain rule, and it stated that
$$\frac{\mathrm{d}}{\mathrm{d}x}\left[y(u(x))\right] = \frac{\mathrm{d}y}{\mathrm{d}x}$$
I don't understand this; if I let $y(x) = x^2$ and $u(x) = \sqrt x$ then
$$\fr... | Your confusion is reading the abbreviation of $\frac{\mathrm d y}{\mathrm d x}$ as $\frac{\mathrm d y(x)}{\mathrm d x}$ instead of $\frac{\mathrm d y(u(x))}{\mathrm d x}$ as you ought.
$$\begin{align}
u(x) & \mathop{:=} \surd x
\\ y(u) & \mathop{:=} u^2
\\ \therefore y(u(x)) & = y(\surd x)
\\ & = (\surd x)^2
\\ & = x
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Finding $k$-clique in a graph with running time of $|V|^{k-1}$ This is a homework problem. Let's say I have a graph $G$, how can I find a $k$-clique (i.e. a complete graph with $k$ vertices) inside $G$? So far I can think of a naive solution where I check if each node in k is connected to other nodes, and the running t... | Nešetřil and Poljak give an algorithm running in time $O(n^{(\omega/3)k})$, for $k$ divisible by 3, which seems to be the state-of-the-art. This generalizes the well-known $O(n^\omega)$ algorithm for the case $k = 3$. Here $\omega$ is the matrix multiplication constant, currently known to be at most $2.373$, but conjec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that collection of finite dimensional cylinder sets is an algebra but not $\sigma$-algebra I am trying to prove that collection of all finite dimensional cylinder sets is an algebra but not $\sigma$-algebra. Cylinder sets are defined as:
$\mathcal{B}_n$ is defined as the smallest $\sigma-$ algebra containing the r... | I presume that by a cylinder set you mean a set of the form $B \times \mathbb{R} \times \mathbb{R} \times \cdots$, with $B \in {\cal B_n}$ for some $n$. Let ${\cal C}_n$ denote the $n$ dimensional cylinder sets and ${\cal C} = \cup_n {\cal C}_n$.
Note that if $C \in {\cal C}_n$ then $C \in {\cal C}_m$ for all $m \ge n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\lim_{x\rightarrow0}\frac{1-\left(\cos x\right)^{\ln\left(x+1\right)}}{x^{4}}$ Could you please check if I derive the limit correctly?
$$\lim_{x\rightarrow0}\frac{1-\left(\cos x\right)^{\ln\left(x+1\right)}}{x^{4}}=\lim_{x\rightarrow0}\frac{1-\left(O\left(1\right)\right)^{\left(O\left(1\right)\right)}}{x^{4}}=\lim_{x\... | By your logic:
$$\lim_{x\to 0}\frac{1 - \cos x}{x^2} = \lim_{x\to 0}\frac{1-O(1)}{x^2} = \lim_{x\to 0}\frac{0}{x^2} = 0$$
but in actual fact, the limit should be equal to $\frac12$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Angle of rotation of $w = \frac{1}{z}$ at the points $1$ and $i$ What angle of rotation is produced by the transformation $w=\frac{1}{z}$ at the point
(a) $z_{0} = 1$
(b) $z_{0} = i$
I'm not sure what to do. At first I though about representing $w=\frac{1}{z}$ in polar coordinates and then plugging in the point in ques... | Here , $f(z)=\frac{1}{z}$. $f'(z)=-\frac{1}{z^2}$.
Angle of rotation at the point $z_0=1$ is $\arg f'(1)=\arg(-1)=\pi$.
Angle of rotation at the point $z_0=i$ is $\arg f'(i)=\arg(1)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1279957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\lim_{x\rightarrow0}\frac{\ln\cos2x}{\left(2^{x}-1\right)\left(\left(x+1\right)^{5}-\left(x-1\right)^{5}\right)}$ Could you help me to find the limit:
$$\lim_{x\rightarrow0}\frac{\ln\cos2x}{\left(2^{x}-1\right)\left(\left(x+1\right)^{5}-\left(x-1\right)^{5}\right)}$$
| In a neighbourhood of the origin:
$$\log(\cos 2x)=\log(1-2\sin^2 x)=-2x^2+o(x^3),$$
$$2^x-1 = e^{x\log 2}-1 = x\log 2+o(x),$$
$$ (x+1)^5-(x-1)^5 = 2+o(x),$$
hence the limit is zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Analyticity of $\dfrac{1}{z}$ vs. $\dfrac{1}{z^2}$ I am learning complex analysis on my own. I am familiar with the theorems, and I am able to compute by hand and get correct results. But there is something that escapes me.
What is the criteria for analyticity?
For example:
derivative if $\dfrac{1}{z}=-\dfrac{1}{z^2}$
... | If you change variables to $w=\frac{1}{z}$ then you end up with $\int z^{-n}dz$ becoming $-\int w^n \frac{dw}{w^2}=-\int w^{n-2}dw$ and hence, for each $n\geq 2$ this is analytic again.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Having a difficult time figuring out if linear equations is true or false. I'm having a very difficult time figuring out if these linear equations are true or false:
$$
x + 2y > 6\\
x - y < 3
$$
How do you identify if these equations are true or false? At first I thought I was supposed to write out the solution, then ... | Hint: It would be helpful to sketch the lines
$$
l_1\colon \quad y=\frac{6-x}{2}\\
l_2\colon \quad y=x-3
$$
and check when the following two conditions hold true jointly:
$$
y>\frac{6-x}{2}\\
y>x-3
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Group of Order $5$ Let $G$ be a group of order $5$ with elements $a, b, c, d, 1$ where $1$ is the identity element. This is the definition of the group.
We all know that this can't be a group because any group of order $5$ is abelian but according to my definition this group is not abelian. But my question is why can'... | $$(ab)d=a \neq d= a(bd)$$
Associativity fails.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Equivalence of norms: $\|x\|$ and $\|x\|_1=\|x\|+\lvert\,f(x)\rvert$ Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous.
I've tested some examples and get the answer is positive... | Hint. It suffices to prove the following:
$$
\|x_n-x\|\to 0 \quad\text{iff}\quad \|x_n-x\|_1\to 0.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Book recommendation on infinitesimals I'm a sophomore mechanical engineering student and I'm looking for a book which completely describes the concept of "infinitesimals" (used in thermodynamics or dynamics or...), In fact, I was taught that dy/dx is not quotient ,but in physics it seems they are treated and being man... | Since differential geometry seems to be the flavour of this thread, my recommendations are as follows.
Personally, I think the best text for differential forms is set out in R.W.R. Darling's Differential Forms and Connections: see link here.
For a physics slant, try Nakahara's Geometry, Topology and Physics, the text c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Counting binary strings of length n with no two adjacent 1's I need to calculate the total number of possible binary strings of length $n$ with no two adjacent 1's.
Eg.
for n = 3
f(n) = 5
000,001,010,100,101
How do I solve it?
| Hint:
Use a recurrence relation. What if the string with $k$ bits starts with a 1, how many possibilities gives this for strings with $k+1$ bits? If the string with $k$ bits starts with a 0, how many possibilities gives this for strings with $k+1$ bits?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find $x$ and $y$ If $\frac{\tan 8°}{1-3\tan^{2}8°}+\frac{3\tan 24°}{1-3\tan^{2}24°}+\frac{9\tan 72°}{1-3\tan^{2}72°}+\frac{27\tan 216°}{1-3\tan^{2}216°}=x\tan 108°+y\tan 8°$, find x and y. I am unable to simplify the first and third terms. I am getting power 4 expressions. Thanks.
| HINT:
$$\tan(3\cdot8^\circ)=\dfrac{3\tan 8^\circ-\tan^38^\circ}{1-3\tan^28^\circ}$$
Now, $$\frac{\tan 8^\circ}{1-3\tan^28^\circ}-y\tan8^\circ=\dfrac{(1-y)\tan 8^\circ-(-3y)\tan^38^\circ}{1-3\tan^28^\circ}$$ which will be a multiple of $\tan(3\cdot8^\circ)$ if
$$\dfrac{1-y}{-3y}=\dfrac31\iff y=-\dfrac18$$
$$\implies\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Inductive proof that every term is a sequence is divisible by 16 I have this question:
The $n$th member $a_n$ of a sequence is defined by $a_n = 5^n + 12n -1$. By considering $a_{k+1} - 5a_k$ prove that all terms of the sequence are divisible by 16.
I can do the induction and have managed to rearrange the expression... | A simple proof with some arithmetic modulo $16$:
$$\begin{array}{r|ccc}
n\equiv\pmod 4\quad & 5^n & 12n-1& a_n \\
\hline
0 & 1 & -1 & 0 \\
1 & 5 & 11 & 0 \\
2 & 9 & 7 & 0 \\
3 & 13 & 3 & 0
\end{array}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
"Intermediate Value Theorem" for curves Let $f$ be a continuous real function defined on the unit square $[0,1]\times[0,1]$, such that $f(x,0)=-1$ and $f(x,1)=1$.
If we walk from $y=0$ to $y=1$ on any path, we must cross some point where $f=0$. Intuitively, there is a connected "wall" (a curve) that separates $y=0$ fro... | Converting my comment to a loose idea of proof:
Assume it's not true - ie, the set of zeros of f is not connected between x=0 and x=1. Then there are 2 disjoint open sets, A and B, one containing the x=0 side, and the other one containing the x=1 side, which cover the entire set of zeros.
Now let's prove that in the re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How do I compute $ \sum\limits_{n=0}^{\infty} \frac{1}{(n+2)7^n} $? I'm trying to evaluate the following sum:
$$ \sum\limits_{n=0}^{\infty} \frac{1}{(n+2)7^n} $$
Probably the best way to do that is to try and find a closed formula for
$$ \sum\limits_{n=0}^{\infty} \frac{1}{(n+2)x^n} $$
Probably it requires some integ... | $$\sum_{n\geq 0}\frac{1}{(n+2)7^n}=49\int_{0}^{1/7}\sum_{n\geq 0}x^{n+1}\,dx = 49\int_{0}^{1/7}\frac{x}{1-x}\,dx = \color{red}{-7+49\log\frac{7}{6}}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1280913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Intersection of Hilbert spaces Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ respectively. I am trying to understand when $H = H_1 \cap H_2$ becomes a Hilbert space with t... | Asking questions about the intersection $H_1 \cap H_2$ of two arbitrary Hilbert spaces $H_1$ and $H_2$ is "evil" in the mathematical sense of the word. Unless the two Hilbert spaces are linear subspaces of some larger Hilbert space, then this question is ill-posed.
In general, it is "evil" to ask questions about struc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to integrate when integration by parts never ends? So I have the following integration: $\int{e^{-x}\sin{x}}$
I let u be the $e^{-x}$ and $dv = \sin{x} dx$
After doing the integration by parts I get:$(e^{-x})(-\cos{x})-\int{-e^{-x}*(-\cos{x})}$
From what it looks like, if I continue integration by parts, will it ju... | If you continue (do integration by part for $\int e^{-x}\cos x\,dx$) you will get $\int e^{-x}\sin x \,dx= f(x) - \int e^{-x}\sin x\, dx$, therefore $\int e^{-x}\sin x\, dx= \frac12 f(x)$.
Every time you integrate by part you will get an extra minus, but you integrating $\sin x$ twice get one minus, that's why in this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
If I wanted to randomly find someone in an amusement park, would I be faster roaming around or standing still? Assumptions:
*
*The other person is constantly and randomly roaming
*Foot traffic concentration is the same at all points of the park
*Field of vision is always the same and unobstructed
*Same walking sp... | Suppose your friend moves around taking a step in a uniformly random direction each time step and suppose you can either stand still or move by the same rule as your friend.
Then, if the step size is small compared to the visibility radius and the park is large (so boundary effects are to be ignored), I claim you'll fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Among the various subgraphs of $K_5$, how many are cycles? Among the various subgraphs of $K_5$, how many are cycles?
I know the answer is $37$ because the number of $3$-cycles is $10$, the number of $4$-cycles is $15$, and $5$-cycles is $12$. Could anyone explain in detail how these numbers are reached?
| First notice that any $n \in \{3,4,5\}$ vertices you choose from $K_5$, you can make a cycle out of them. When you are counting the $n$-cycles, you need to count the number of ways you can choose $n$ vertices out the given $5$ vertices, and for each of those choices you need to count how many ways those $n$ vertices ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
First order differential equation integrating factor is $e^{\int\frac{2}{x^2-1}}$ So i got the first order ode
$$(x^2-1)\frac{dy}{dx}+2xy=x$$
I divided both sides by $x^2-1$
$$\frac{dy}{dx}+\frac{2}{x^2-1}xy=\frac{x}{x^2-1}$$
in the form $y' + p(x)y = q(x)$
So that means the integrand is...
$$e^{\int\frac{2}{x^2-1}}$$
... | your equation $$(x^2 - 1) \frac{dy}{dx} + 2x y = x $$ is an exact differential equation. the reason is it can be written as $$\frac d{dx}\left((x^2 - 1) y\right) = x $$ on integration gives you $$(x^2 - 1)y = \frac12 x^2 + c \to y = \frac{x^2}{2(x^2 - 1)} + \frac c{x^2 - 1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to simplify this summation, or express as integral? $\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$ How to simplify this summation, or express as integral?
$$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$$
$a$ is a constant, say, 24.
$$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+24x)^2+4}}$$
| Just to make an answer from sos440's comment: From the integral test you get:
$$\sum_{t=1}^\infty\frac1{\sqrt{(t+24x)^2+4}} > \int_1^\infty \frac1{\sqrt{(t+24x)^2+4}} dt = \infty$$
because $\frac1{\sqrt{(t+24x)^2+4}} \approx \frac 1x$ for $x\rightarrow\infty$ and $\int_1^\infty \frac 1x dx$ diverges. Because each sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does one of these conditions on a sequence imply the other one? Let ${(r_n)}_{n \geq 0}$ be a sequence of integers $\geq 2$. Set $q_n=\prod_{i=0}^{n-1} r_i$ (agreeing with $q_0=1$). I want to know whether one of these two conditions implies the other one (I think the answer is no but I don't find counter-examples). The... | @Adayah's answer provides the proof of this lemma:
Lemma: Let ${(u_n)}_{n \geq 0}$ and ${(v_n)}_{n \geq 0}$ be two sequences of positive numbers such that $v_n \searrow 0$. If $\sum u_n v_n < \infty$ then $\epsilon \sum_{i=0}^{n(\epsilon)} u_i \to 0$ when $\epsilon \to 0^+$, where $n(\epsilon)=\min\{n \mid v_{n+1} < \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\left\lceil \frac{n}{m} \right\rceil =\left \lfloor \frac{n+m-1}{m} \right\rfloor$ On a discrete mathematics past paper, I must prove that
$$\left\lceil \frac{n}{m} \right\rceil = \left\lfloor \frac{n+m-1}{m} \right\rfloor$$
for all integers $n$ and all positive integers $m$.
I have started a proof by in... | We have
$$
\lfloor \frac{n+m-1}{m}\rfloor=\lfloor \frac{n-1}m \rfloor+1
$$
Now you have two cases $m|n$ or not...
If $m|n$, then $\lceil \frac nm\rceil=\frac nm$ and $\lfloor\frac{n-1}m\rfloor=\frac nm-1$
Now if $m$ does not devide $n$, then $\lceil \frac nm\rceil=\lfloor\frac nm\rfloor+1$ and $\lfloor\frac nm\rfloor=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
What is a branch point? I am really struggling with the concept of a "branch point". I understand that, for example, if we take the $\log$ function, by going around $2\pi$ we arrive at a different value, so therefore it is a multivalued function. However, surely this argument holds for all points in the complex plane, ... | A branch point of a "multi-valued function" $f$ is a point $z$ with this property: there does not exist an open neighbourhood $U$ of $z$ on which $f$ has a single-valued branch.
In the case of $\log$, the only branch point is $0$: indeed, if $z \ne 0$, we could take
$$U = \{ w \in \mathbb{C} : \lvert w - z \rvert < \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Difference between domain and range for relations and functions? What is the difference between the definition of domain and range for a relation, and that for a function?
| There is no difference. Note, that each function can be seen as a special relation. So $f:A \rightarrow B$ can be seen as a relation $R_f \subseteq A\times B$ such as
$$\forall x \in A: \exists ! y \in B: xR_fy$$
The domain and the range is so defined, such that the relation $R_f$ has the same domain and the range as $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1281931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Representation of the Orthogonal Projection We want to show that orthogonal projection can be written as
\begin{align}
C\cdot C^+=\left(I_N-\frac{1}{N}\iota\iota'\right),
\end{align}
where $C=\left(\omega\iota'-I_N\right)$, $I_N$ is the identity matrix of dimension $N$, $\omega$ is an $N\times1$ vector, $\iota'\omega=1... | We find finally that $C^+$ has the following form:
\begin{align}
C^{+}= w \iota' - I \frac{1}{\omega'\omega} \omega \omega' - \left( \frac{1}{N} \frac{1}{\omega'\omega} + 1\right) \omega \iota' + \frac{1}{N} \iota \iota'
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$\sum a_n$ converges $\implies\ \sum a_n^2$ converges? If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ always convergent? Either prove it or give a counter example.
Im trying in this way, Suppose $a_n \in [0,1] \ \forall\ n.\ $ Then ${a_n}^2\leq a_n\ \forall\ n.$ Therefore by comparison test $\sum {a... | If $\sum_{n=1}^\infty a_n$ is convergent, then $\lim_{n\to\infty}a_n=0$, hence from somewhere upward we have $0<a_n<1$, now use comparison test...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 2
} |
Derivative of an Expectation over an Indicator function I have a small question on how to compute the derivative of an expectation when an indicator function gets in the way. Let $x$ be a random variable. We are interested in computing the derivative wrt A of
$E_x [(A-x)1_{(x \leq A)}]$
where $1_n$ is the indicator fu... | $$E_x [(A-x)1_{(x \leq A)}]$$
$$=\int_{-\infty}^\infty(A-x)1_{(x\leq A)}p_x(x)dx$$
$$=\int_{-\infty}^A(A-x)p_x(x)dx$$
$$=\int_{-\infty}^AAp_x(x)dx-\int_{-\infty}^Axp_x(x)dx$$
$$=A\int_{-\infty}^Ap_x(x)dx-\int_{-\infty}^Axp_x(x)dx$$
$$=AF(A)-\int_{-\infty}^Axp_x(x)dx$$
So taking the derivative with respect to $A$ we get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that if a Mobius transformation has 3 fixed points then it is the identity map. I have that any non trivial Mobius transformation has at most 2 fixed points since f(z)-z=0 has at most 2 roots. But I cannot deduce why it must then be the identity.
| Suppose $Tz=(az+b)/(cz+d)$. The fixed points of $T$ must satisfy the equation:
$$\begin{align}
\frac{az+b}{cz+d}=z & \iff az+b=cz^2+dz\\
& \iff cz^2+(d-a)z+b=0
\end{align}$$
If this quadratic equation has more than 3 solutions, the polynomial must be identically zero. So, $c=0,a=d,b=0$.
Therefore $Tz=az/a=z$.
This arg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Multivariable limit which should be simple ! How to calculate the following limit WITHOUT using spherical coordinates?
$$
\lim _{(x,y,z)\to (0,0,0) } \frac{x^3+y^3+z^3}{x^2+y^2+z^2}
$$
?
Thanks in advance
| Let $\epsilon \gt 0$. If $(x,y,z)$ is close enough to $(0,0,0)$ but not equal to it, then $|x^3|\le \epsilon x^2$, with similar inequalities for $|y^3|$ and $|z^3|$. It follows that
$$\frac{|x^3+y^3+z^3|}{x^2+y^2+z^2}\le \frac{|x^3|+|y^3|+|z^3|}{x^2+y^2+z^2}\le \frac{\epsilon (x^2+y^2+z^2)}{x^2+y^2+z^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving UNIT INTERSECTION NP-complete I am working on some review problems right now and am extremely stuck on how to solve problem - any help would be so appreciated. We are told to consider the following combinatorial problem:
Unit Intersection: Let X = {1, 2,...,n}. Given a family of subsets $S_1,...,S_m$ of X, det... | The reduction is straightforward. As an example, I'll reduce an instance of Exactly-One-in-3SAT to an instance of Unit Intersection.
$$ (x_1 \lor x_4 \lor x_3) \land (\overline{x_4} \lor \overline{x_2} \lor x_3) \land (x_2 \lor x_1 \lor \overline{x_3}) $$
Let $n$ be the number of distinct (positive and negative) liter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The box has minimum surface area Show that a rectangular prism (box) of given volume has minimum surface area if the box is a cube.
Could you give me some hints what we are supposed to do??
$$$$
EDIT:
Having found that for $z=\frac{V}{xy}$ the function $A_{\star}(x, y)=A(x, y, \frac{V}{xy})$ has its minimum at $(\s... | we will keep the volume at $1.$ let the base have length $x$ and width $y.$ then the volume constraint makes the height of the box $\frac1{xy}.$ you need to minimize the surface area $$A = 2\left(xy+\frac 1x + \frac 1y\right), x > 0, y > 0 $$ now you can use the am-gm inequality $\frac{a+b+c}3\ge (abc)^{1/3}$ to show... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 2
} |
A reference for the Tannaka-Krein theorem I am looking for a reference for the Tannaka-Krein theorem on compact groups. By the Tannaka-Krein theorem which is also called (classic) Tannaka duality (because of the quantum theory), I mean the theory which is to reconstruct a compact (Lie) group from its representations.
| I found it myself. It is in
Hewitt, Edwin; Ross, Kenneth A. Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups. Die Grundlehren der mathematischen Wissenschaften, Band 152 Springer-Verlag, New York-Berlin 1970 ix+771 pp.
Section 30. It is written... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove that $\lambda(s)=(1-2^{-s})\zeta(s)$? The Lambda function is defined as:
$$\lambda(s)=\sum_{n=0}^{\infty} \frac{1}{(2n+1)^s},\; \mathfrak{Re}(s)>1$$
How to prove that $\lambda(s)=(1-2^{-s})\zeta(s)$?
Basically, I was dealing with the functional equation of $\eta$ and while I was trying to prove its functi... | I give the solution according to Lucian's comment.
We start things off by using the $\zeta$ Riemann's function, defined as:
$$\zeta(s)= \sum_{n=1}^{\infty}\frac{1}{n^s}, \; \mathfrak{Re}(s)>1$$
which converges absolutely since all terms are positive. That the series converges if $\mathfrak{Re}(s)>1$ is an immediate con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Solving non-linear second order differential equation: radius of curvature $= k \theta$ I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates
radius of curvature $= k \theta$
$$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + 2r'^2 - rr''} = k \theta$$
where k ... | I understand you are looking for any curve satisfying the property $$\rho=k\theta$$
Therefore choose a curve which has the property that the radius vector makes a constant angle $\alpha$ with the tangent vector. In intrinsic form, we have $$\frac {dy}{dx}=\tan\psi$$
The radius of curvature is $$\rho=\frac{ds}{d\psi}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1282901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Let $n$ be an odd natural number , to find a continuous real valued function on $\mathbb R$ which takes every value exactly $n$ times Let $n$ be an odd natural number . We know $\mathbb R = \cup_{k \in \mathbb Z} [nk\pi , n(k+1)\pi]$ . So for every $k \in \mathbb Z$ , define
$h(x):=2k+1-(-1)^k \cos x , \forall x \in [... | Yes, your function appears to meet the requirements. Here is a sketch for $n=5$
Another approach without the need for intervals might be to take a sinusoidal curve and add a linear element to shift successive peaks and troughs up, perhaps something like
$$f(x)= \frac{2}{n\pi}x +\cos(x)$$
which for $n=5$ looks like
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sum of variances of multinomial distribution. I've k fair coins, and I would like to know the number of heads obtained in $n$ trials. But that is simple binomial distribution. But if I want to find out how much it varies from binomial expectation, them it becomes a multinomial distribution for the number of heads from ... | There is nothing wrong with the derivation, only problem with experimental setup. The experiment is calculating expected value of standard deviation and not of variance.
Change
RMSExptAvg(S, n) = mean([RMSExpt(S, n) for i in 1:10000]))
to this:
RMSExptAvg(S, n) = sqrt(mean([RMSExpt(S, n)^2 for i in 1:10000]))
Then e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Existence of a metric space where each open ball is closed and has a limit point Show that there exists a metric space in which every open ball is closed and contains a limit point.
I think that the space $\{\frac{1}{n}\mid n\in\mathbb{N},n>0\}\cup \{0\}$ with the standard Euclidean metric is an answer, but it is not t... | $\mathbb{Z}$ with the $p$-adic metric $d(m,n) = p^{-\nu_p (m-n)}$ has the property that every open ball is closed (to prove this, note that the set of positive distances forms a discrete space), and every integer $n$ is the limit point of the sequence $\{n+p^k\}_{k\geq 1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving vectors as a basis in $E^{m}$ Show that if the vectors $a_{1}$, $a_2$, $\cdots$, $a_m$, are a basis in $E^{m}$, the vectors $a_{1}$, $a_2$, $\cdots$, $a_{p-1}$, $a_{q}, a_{p+1}, \cdots,a_{m}$, also are a basis if and only if $y_{p,q} \neq 0$, where $y_{p,q}$ is defined by the following tableau:
\begin{matrix}
1... | Remark: Your problem has nothing to be with the rightmost column
$(y_{1,0},y_{2,0},\dots,y_{m,0})^T$. I'll omit that column from the matrix.
Settings
Let $A = \begin{bmatrix}\mathbf{a}_1&\mathbf{a}_2&\cdots&\mathbf{a}_m&|\mathbf{a}_{m+1} & \cdots&\mathbf{a}_n\end{bmatrix}$ be the original coefficient matrix in $E^{m \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why can't I get this with Gram-Schmidt Can someone assist me with a very simple problem. I cannot get these two vectors to be orthogonal using Gram-Schmidt:
$\{(1,-1,1),(2,1,1)\}$
What am I doing wrong?
Let $v_1=(1,-1,1)$.
$v_2 =
> (2,1,1)-\frac{\langle(2,1,1),(1,-1,1)\rangle}{\lvert (1,-1,1) \rvert^2}(1,-1,1)$ =$... | The numerator of the fraction should be
$$
\langle(2,1,1),(1,-1,1)\rangle
$$
Edit
After your correction, I can see that there is still an error, given that
$$
\frac{\langle(2,1,1),(1,-1,1)\rangle}{|(1,-1,1)|^2}=\frac{2}{3}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that the set $A \cap B = \emptyset$ Let $A$ and $B$ be two sets for which the following applies: $A \cup B = (A \cap B^{C}) \cup (A^{C} \cap B)$. Show that $A \cap B = \emptyset$.
How?! I am seriously stuck. One thought I had is to distribute the right part, so that: $(A \cap B^{C}) \cup (A^{C} \cap B) = (A \cap ... | I'll add an answer expanding your idea, since that works too!
Only, you made a few mistakes.
It should be:
$(A \cap B^{C}) \cup (A^{C} \cap B) = ((A \cap B^{C}) \cup A^{C}) \cap ((A \cap B^{C}) \cup B)$
Then,
$=((A\cup A^{C}) \cap (B^{C}\cup A^{C})) \cap ((A\cup B) \cap (B^{C}\cup B))=$
$=(B^{C}\cup A^{C}) \cap (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact under the following metrics:
$1. d(f,g)=\sup_{x\in [0,1]}|f(x)-g(x)|$
$2.d(f,g)=\int _0^1 |f(x)-g(x)| dx$
My try:
In order to show not compact if we can find a sequence which has no conv... | As mentioned in previous answers, an animation much aids comprehension. Here's an open cover for the closed unit ball with respect to sup norm metric. Let,
\begin{align}
f^n_{\mathrm{upper}}(x) &= \text{the continuous function obtained by connecting the points } \begin{cases} (0,0),\\ (1/n,2),\\(1-1/n,2),\\(1,0) \end{c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Fourier series of cotangent I have found the Fourier series of $\cot(ax)$ and i get:
$$\cot(ax) \sim \frac{ \sin(a \pi)}{a\pi}\left[ \left(\frac{1}{2a^2}\right)- \sum_1^\infty \frac{(-1)^n \cos(nx)}{n^2-a^2}\right]$$
How can I deduce the Fourier series of $\cot(x)$ where $x$ isn't multiple of $\pi$? Any help please...
| Expand the exponential form of $\cot x$:
\begin{align}\cot x=\frac{i(e^{ix}+e^{-ix})}{e^{ix}-e^{-ix}}&=i+\frac{2ie^{-ix}}{e^{ix}-e^{-ix}}=i+2ie^{-2ix}(1+e^{-2ix}+e^{-4ix}+\cdots)\\[1ex]
&=i+2i\sum_{k\ge1}(\cos2kx-i\sin2kx)
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Application of Taylor's Theorem in Number Theory I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote:
It is easily seen that no polynomial $f(n)$ with integer coefficients can be prime for all $n$ in $\mathbb{N}$, o... | By Tayor's Theorem
$$f(x) = f(n) + f'(n)(x-n) + \frac{f''(n)}{2!}(x-n)^2 + \cdots + \frac{f^{(k)}(n)}{k!}(x-n)^k + h_k(x)(x-n)^k$$
Now take $k$ big enough so that $h_k(x)=0$ (possible since $f$ is a polynomial).
So
$$f(x) = f(n) + f'(n)(x-n) + \frac{f''(n)}{2!}(x-n)^2 + \cdots + \frac{f^{(k)}(n)}{k!}(x-n)^k$$
Now plug... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
} |
Formula to get mean and standard deviation of this multi-variable equation $$
\binom n x \times\left(\frac1r\right)^x\times\left(\frac{r-1}r\right)^{n-x}
$$
If you have $n$ boxes and have a $\frac1r$ chance to fill each one, this equation returns the chance that you fill exactly $x$ boxes. When $n$ and $r$ are given, t... | Let $X_i$ be the random variable which equals $1$ when box $i$ is filled, and $0$ otherwise. Then the variable you want to know about is $X = \sum_{i = 1}^{n}X_i$.
We have:
$$E(X) = E(\sum_{i = 1}^{n}X_i) = \sum_{i = 1}^{n}E(X_i)$$
And, since the $X_i$ are independent:
$$\sigma^{2} = Var(X) = Var(\sum_{i = 1}^{n}X_i) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Let $A$ be a subset of a topological space. Prove that $Cl(A) = Int(A) \cup Bd(A)$ Let $A$ be a subset of a topological space. Prove that $Cl(A) = Int(A) \cup Bd(A)$
Here are my defintions:
Closure: Let $(X,\mathfrak T)$ be a topological space and let $ A \subseteq X$ . The closure of $A$ is $Cl(A) = \bigcap \{U \... | The closure of $A$ is the smallest closed subset of $X$ containing $A$. The interior of $A$ is the largest open subset contained in $A$. So we have that $Int(A) \subseteq A \subset Cl(A)$. Let $x \in Bd(A)$. Suppose $ x \notin Cl(A)$. $X \setminus Cl(A)$ is open, and $x \in X \setminus Cl(A)$, so by your definition of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1283975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Which of the following collections are topologies for $\mathbb R$? Am I correct? Which of the following collections are topologies for $\mathbb R$?
I think I have these correct I just want to double check my answers.
(a) $\{\mathbb R, \emptyset, (-\infty, 0), (0,\infty)\}$- Yes, empty set and whole line are included, i... | You are correct in thinking that (b) is not a topology, for the reason that you gave. However, (a) is also not a topology: it doesn’t contain $(-\infty,0)\cup(0,\infty)=\Bbb R\setminus\{0\}$. Finally, (c) also fails to be a topology:
$$\bigcup_{b<0}(-\infty,b]=(-\infty,0)\;,$$
which is not in the family.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proof of Cauchy-Schwarz inequality from Terry Tao's notes, meaning of "cancelling the phase" I was reading Tao's proof of the Cauchy-Schwarz inequality by exploiting certain inherent symmetries and making some transformations.
He says that first we use the fact that the norm is positive, i.e.
$$\|u-v\|^2>0$$ to conclud... | Cancelling the phase means to choose $\theta$ so that $e^{i\theta}\langle u,v\rangle$ is real. A priori all we know about $\langle u,v\rangle$ is that it is some complex number $\langle u,v\rangle = re^{i\varphi}$, so choosing $\theta = -\varphi$ (i.e. "cancelling the phase" on this complex number) will make $e^{i\thet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Why 9 mod -7 = -5? Quotient and remainder with negative integers. Forgive me if this question does not belong on this site for it is simplistic and this is my first post, however I do not seem to understand the modulo function when it comes to negative numbers.
I'd assume the process for calculating modulo would be the... | Recall that $a \equiv b \bmod n$ if and only if $n$ divides $a - b$. This is the definition.
So is it true that $-5 \equiv 9 \bmod -7$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
Proving an identity Given $a,b\in\mathbb{R}$ with $a < b$ and defining $F(z):=\int_0^z f(s) \, ds$ with $z \in \mathbb{R}$, how can one establish that $$F(a+b)=F(a)+f(a)b+ b^2\int_0^1 (1-s)f'(a+sb) \, ds,$$ which is part of a larger proof on page 508 of PDE Evans (2nd edition)?
Should I perform a change of variable to ... | Taylor's formula with integral remainder:
$$
F(a+b)=F(a) + F'(a) b + \int^{a+b}_a (a+b-t) F''(t) \ dt
$$
Make a substitution $t=a+bs,$ and we get
$$
F(a+b)=F(a) + F'(a) b + b^2\int^{1}_0 (1-s) F''(t) \ dt.
$$
Using $F(x)=\int^x_a f(w) \ dw,$ we get exactly the required result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove a set of connectives is functionally complete What is the right way of proving that a set of logical connectives is or not functionally complete?
For example, if I have
{→,∨}
how can I show it is or not functionally complete?
Any ideas?
| To prove that a set of connectives is functionally complete, you simply need to show that you can derive any other logical connective using only this restricted set.
In your case, you want to show that you can obtain definitions for $\left\{\land,\leftrightarrow,\neg\right\}$ from $\left\{\to,\lor\right\}$, i.e. the qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Prove every group of order less or equal to five is abelian Is it possible to prove that every group of order less or equal to five is abelian?
We know that groups of prime order are cyclic and therefore commutative. As the number $4$ is the only composite number $\le5$, it basically remains to show this for groups ... | Group of order 1 is trivial, groups of order 2,3,5 are cyclic by lagrange theorem so they are abelian.
For a group of order 4, if it has an element of order 4, it is abelian since it is cyclic(isomorphic to Z4).
If orders of every element are 2, then the inverse of an element is the element itself so you can verify ev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 1
} |
$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $
$$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $$
(can this be duplicate? I think not)
I tried it using many methods
$1.$ Solve this conventionally taking $1^\infty$ form in no luck
$2.$ Did this, expand $ {(1+x)^{\frac1x}}$ using binomial th... | Use Taylor expansion of $(1+x)^{1/x}$.
$$(1+x)^{1/x}=e(1- \frac{x}{2} + \frac{11x^2}{24} + ..........)$$
The other terms wont be required in this limit.
The numerator simplifies and the $x$ terms get cancelled out.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Maclurin Series. (Approximation) Given that $y=\ln \cos x$, show that the first non-zero terms of Maclurin's series for $y=-\frac{x^2}{2}-\frac{x^4}{12}$. Use this series to find the approximation in terms of $\pi$ for $\ln 2$.
My question is how to determine value of $x$ which is suitable?
| We have
\begin{align}
\ln(\cos(x)) & = \dfrac{\ln(\cos^2(x))}2 = \dfrac12 \cdot \ln(1-\sin^2(x)) = - \dfrac12 \sum_{k=1}^{\infty} \dfrac{\sin^{2k}(x)}k\\
& = -\dfrac12 \sin^2(x) - \dfrac14 \sin^4(x) - \cdots\\
& = -\dfrac12 \left(x-\dfrac{x^3}{3!} + \mathcal{O}(x^5)\right)^2-\dfrac14 \left(x + \mathcal{O}(x^3)\right)^4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why is it so hard to prove a number is transcendental? While reading on Wikipedia about transcendental numbers, i asked myself:
Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e,
\frac{\pi}{e}$ etc. are transcendental numbers?
Answer by @hardmath:
It is generally more difficult to prove a number ... | For example, $\pi$ and $1-\pi$ are transcendental, but $\pi+(1-\pi)=1$ is not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1284948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 3,
"answer_id": 1
} |
How to calculate a Modulo? I really can't get my head around this "modulo" thing.
Can someone show me a general step-by-step procedure on how I would be able to find out the 5 modulo 10, or 10 modulo 5.
Also, what does this mean: 1/17 = 113 modulo 120 ?
Because when I calculate(using a calculator) 113 modulo 120, th... | Modulo is counting when knowing only a limited amount of numbers. E.g. modulo three, instead of counting
$$0,1,2,3,4,5,6,7,8,9,10,11,...$$
you count
$$
\underset{\color{lightgray}0}0,
\underset{\color{lightgray}1}1,
\underset{\color{lightgray}2}2,\;\;
\underset{\color{lightgray}3}0,
\underset{\color{lightgray}4}1,
\un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 2
} |
Prove that if $G$ is a finite group and $H$ is a proper normal subgroup of largest order, then $G/H$ is simple. We define the group action of $G$ on the set of left cosets of $H$ in $G$ by left multiplication. Let $\Phi$ be the associated permutation representation.
Then by Generalised Cayley's Theorem, $G/ \text{Ker}(... | A normal subgroup of $G/H$ is of the form $K/H$ where $K \le G$ is a normal subgroup containing $H$. By maximality of $H$ you have $K=H$. Hence $G/H$ has no proper normal subgroups.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Does the Laplace operator include the second derivative with respect to time variable? Does the Laplacian of a function $f(x,z,t)$ equal $f_{xx} + f_{yy} + f_{tt}$?
We aren't sure whether or not time is included in it or not.
| It depends... the Laplacian is always defined with respect to some metric: $\Delta f = \nabla \cdot \nabla f$ and divergence requires a metric. Alternatively, you can define $\Delta$ as the gradient, in the sense of the calculus of variations, of the Dirichlet energy $\int \langle \nabla f, \nabla f\rangle\,dV$ and her... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If the sum of two squares is divisible by $7$, both numbers are divisible by $7$
How do I prove that if $7\mid a^2+b^2$, then $7\mid a$ and $7\mid b$?
I am not allowed to use modular arithmetic.
Assuming $7$ divides $a^2+b^2$, how do I prove that the sum of the squares of the residuals is 0?
| Since you said that you are not allowed to use modular arithmetic, argue by the contrapositive. If 7 does not divide $a$ or $b$. Then,
$a = 7k \pm 1, 7k \pm 2, 7k \pm 3 \implies$
$a^2 = 7(7k^2 \pm 2k) + 1, 7(7k^2 \pm 4k) +4, 7(7k^2 +8k + 1)+2$, $k \in \Bbb Z$
Now, look at any linear combination of $a^2$ (with integer w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How to find cotangent? Need to find a $3\cot(x+y)$ if $\tan(x)$ and $\tan(y)$ are the solutions of $x^2-3\sqrt{5}\,x +2 = 0$.
I tried to solve this and got $3\sqrt{5}\cdot1/2$, but the answer is $-\sqrt{5}/5$
| Since by Vieta's formulas one has
$$\tan x+\tan y=-\frac{-3\sqrt 5}{1}=3\sqrt 5,\ \ \ \tan x\tan y=\frac{2}{1}=2,$$
one has
$$3\cot(x+y)=3\cdot\frac{1}{\tan(x+y)}=3\cdot\frac{1-\tan x\tan y}{\tan x+\tan y}=\frac{3(1-2)}{3\sqrt 5}=-\frac{\sqrt 5}{5}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Evaluate the complex integral $\int_{C_R}\frac{z^3}{(z-1)(z-4)^2}$
Let $C_R$ be the positively oriented circle with centre $3i$ and radius $R > 0$.
Use the Cauchy Residue Theorem to evaluate the integral $$\int_{C_R}\frac{z^3}{(z-1)(z-4)^2}$$
Your answer should state any values of R for which the integral cannot... | Sketch 3i , 1, 4 on a paper and use pythagorean theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving an Equation for y in terms of x How is it possible to solve the equation:
$$ x = \frac{2-y}{y} \sqrt{1-y^2} $$
for $ y $ in terms of $ x $?
I know that it is possible to do it, since WolframAlpha gives a complicated answer but entirely composed of elementary functions. However, I am stumped on how to even start... | The typical method would be to square both sides (possibly picking up extraneous solutions) and then arrange the terms into a polynomial equation in $y$:
$$x = \frac{2-y}{y} \sqrt{1-y^2}$$
$$x^2 = \frac{(2-y)^2}{y^2} (1-y^2)$$
$$y^2 x^2 = (4-4y+y^2)(1-y^2) = (4-4y+y^2) - 4y^2 + 4y^3 - y^4$$
Which yields:
$$y^4 - 4y^3 +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Some questions about notation in "$[T]_\alpha^\beta$" I just have a few questions about the general meaning of the notation "$[T]_\alpha^\beta$". I would really appreciate if someone would dumb it WAY down to the most basic level (no assumptions, no leaps of logic) because most of the literature I have read on this not... | It means the matrix associated to $T$ respect the basis $V\subset\mathbb{R}^n$ and the basis $W\subset\mathbb{R}^m$
For example, if we have $T\colon\mathbb{R^2}\to\mathbb{R^3}$ such $T(x,y)=(x,x+y,y)$, and $V=\{e_1=(1,0);e_2=(0,1)\}$ a ordered basis for $\mathbb{R^2}$ and $W=\{f_1=(1,0,0);f_2=(0,1,0);f_3=(0,0,1)\}$ a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Q: Nowhere dense sets.
Given $X$ a metric space, $A\subset X$ a nowhere dense set. Show that every open ball $B$ contains another open ball $B_1 \subset B$ such that $B_1 \cap A = \emptyset$.
EDIT: I modify my proof with the help of a great hint!
Given $B(x,\epsilon)$ lets suppose that for every open ball $B_1 \subse... | That step is not legitimate. Consider the set $A=\left\{\frac1n:n\in\Bbb Z^+\right\}$ in $\Bbb R$. For each $\epsilon>0$ we have $B(0,\epsilon)\cap A\ne\varnothing$, but we can’t take the limit as $\epsilon\to 0^+$ to conclude that $\{0\}\cap A\ne\varnothing$: this is clearly false.
Try this instead. Suppose that every... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
integrating something from a partial derivative $v=\int \frac{2x}{x^2+y^2}\,dy$ i am trying to learn harmonic analysis, and i have$$\frac{\partial u}{\partial x}=\frac{2x}{x^2+y^2}=\frac{\partial v}{\partial y}$$
and i want to get $v$. so what i do is:
$$v=\int \frac{2x}{x^2+y^2}\,dy$$
$$=\int\frac{2x}{2x}\frac{1}{a}\,... | Remember that you can treat $x$ as a constant because we are integrating with respect to $y$. This motivates the following:
$$\begin{aligned}
v &= \int \frac{2x}{x^{2} + y^{2}} dy \\
&= \int \frac{2}{x \left( 1 + \left(\frac{y}{x} \right)^{2}\right)} dy \\
&= \frac{2}{x} \int \frac{dy}{\left( 1 + \left( \frac{y}{x} \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Topics in Algebra I.N.Herstein Problem 7 Given that if $A$ and $B$ are cyclic of orders m and n and $\gcd(m,n)=1$ then $A\times B$ is cyclic.
Using this prove that if $u,v\in \mathbb Z$ then $\exists x$ such that $x\equiv u(\mod m);x\equiv v(\mod n)$
My try:
Now $(u,v)\in \mathbb Z\times \mathbb Z\implies (u\mod m,v\mo... | Consider the element $(1,1)$. What is the order of this element? It is the smallest $N$ that is both a multiple of $m$ and $n$. Since $m$ and $n$ are relatively prime, $N=mn$. This the group has $mn$ elements and an element of order $mn$, hence it is cyclic.
To show $x$ exists, note that $u,v$ determine an element in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1285961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Proving simple trigonometric identity I need help with this one
$$
\frac{\sin^2 \alpha}{\sin\alpha - \cos\alpha} + \frac{\sin\alpha + \cos \alpha}{1- \mathrm{tan}^2\alpha} - \cos\alpha = \sin \alpha
$$
I tried moving sin a on the other side of the eqation
$$
\frac{\sin^2 \alpha}{\sin\alpha - \cos\alpha} + \frac{\sin\... | $\left(\dfrac{\sin^2 \alpha}{\sin \alpha+\cos\alpha}-(\sin\alpha+\cos\alpha)\right)+\dfrac{\sin \alpha+\cos\alpha}{1-\tan^2\alpha}\\=\left(\dfrac{\sin^2 \alpha-\sin^2 \alpha+\cos^2\alpha}{\sin \alpha-\cos\alpha}\right)+\dfrac{\sin \alpha+\cos\alpha}{1-\tan^2\alpha}\\=\dfrac{\cos^2\alpha}{\sin \alpha-\cos\alpha}+\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Show that the comb space has fixed-point property By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment.
How to prove that this space has the fixed-point property?
I've tried to embed $X$ as a retrac... | Let $f:X\to X$ be an arbitrary continuous map. We shall prove that $f$ has a fixed point.
Let $I=[0,1]\times\{0\}$ be the bottom interval and let $\pi:X\to I$ be the projection $\pi(x,y)=(x,0)$. The map $\pi\circ f$ maps $I$ to $I$ so it has a fixed point in $I$. Therefore, there exist $x_0$ and $y_0$ such that $$f(x_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
comparing MSE of estimations of binomial random variables $X$ is a binomial random variable defined over 12 Bernoulli trials with a success probability of $p$ in each (i.e. $X\sim\operatorname{Bin}(12,p)$. Consider $\hat p=\frac X{10}$
Determine the range for which the mean squared error of $\hat p =\frac X{10}$ is w... | I will write $\tilde p = \frac1{10}X$ and $\hat p=\frac1{12}X$ to avoid confusion. Recall that the mean-squared error of an estimator $\hat\theta$ of a parameter $\theta$ is
$$\operatorname{MSE}\left(\hat\theta,\theta\right) = \mathbb E\left[(\hat\theta, \theta)^2\right] = \operatorname{Var}\left(\hat\theta\right) + \o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Dual Bundles' Local Trivializations Confusion $$\newcommand{\R}{\mathbb R}$$
Let $\pi:E\to M$ be a rank $k$ smooth vector bundle over a smooth manifold $M$. I will below describe how to form the dual bundle, wherein lies my question.
Let $E_p=\pi^{-1}(p)$ for each $p\in M$, and define $E^*=\bigsqcup_{p\in M}E_p^*$. Fur... | The identification of $V$ with $V^*$ via a scalar product is done for vector spaces in which there is no preferred basis. This is not the case of $\mathbb R ^n$ which has the usual basis (in terms of which it is constructed, after all) that allows you to define the identification $e_i \mapsto e^i$ where $e^i (v) = e^i ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Show that $f(x) = 0$ for all $x \in \mathbb{R}$
$f: \mathbb{R} \to \mathbb{R}$ is continuous with $f(x)=0$ for all $x \in \mathbb{Q}$. Show that $f(x) = 0$ for all $x \in \mathbb{R}$.
Can anyone please point me in the right direction as to how I can go about answering this question?
| Assume that
$$ \exists q \in \mathbb{R} \setminus \mathbb{Q} : f(q) \not = 0$$
Now by trichotomy we have either $f(q) > 0$ or $f(q) < 0$. If we consider the first case, by continuity of $f$ we have
$$ \exists \delta > 0 : f(x) > 0 : x \in (q - \delta, q + \delta)$$
(Proving this is a very useful exercise in itself). Ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Inf is not a stopping time in general If ${\tau_n}$ , $n=1,2,3...$ are stopping times to a given filtration $F_t$, why in general it's not true to claim that $\inf_n {\tau_n}$ is a stopping time also?
Thanks
| I guess we are talking about continuous time: stochastic basis $(\mathcal F_t)_{t \in (0,\infty)}$, say.
Write $\sigma = \inf_n \tau_n$. Can we verify that $\{\sigma \le 5\} \in \mathcal F_5$ ???
Well, suppose $B \in \bigcap_{t>5}\mathcal F_t$ but $B \not\in \mathcal F_5$. Then for each $n$,
$$
\tau_n(\omega) = \be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Another combined limit I've tried to get rid of those logarithms, but still, no result has came.
$$\lim_{x\to 0 x \gt 0} \frac{\ln(x+ \sqrt{x^2+1})}{\ln{(\cos{x})}}$$
Please help
| It can be computed this way
$$
\frac{\ln (x+\sqrt{x^{2}+1})}{\ln {(\cos {x})}}=\frac{\ln (1+(x-1)+\sqrt{%
x^{2}+1})}{(x-1)+\sqrt{x^{2}+1}}\cdot \frac{(x-1)+\sqrt{x^{2}+1}}{x^{2}}%
\cdot \frac{x^{2}}{(\cos x)-1}\cdot \frac{(\cos x)-1}{\ln (1+\cos x-1)}
$$
$$
\lim_{x\rightarrow 0^{+}}\frac{\ln (1+(x-1)+\sqrt{x^{2}+1})}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Find the integer x: $x \equiv 8^{38} \pmod {210}$ Find the integer x: $x \equiv 8^{38} \pmod {210}$
I broke the top into prime mods:
$$x \equiv 8^{38} \pmod 3$$
$$x \equiv 8^{38} \pmod {70}$$
But $x \equiv 8^{38} \pmod {70}$ can be broken up more:
$$x \equiv 8^{38} \pmod 7$$
$$x \equiv 8^{38} \pmod {10}$$
But $x \equiv... | You made a small slip up when working $\bmod 2$, this is because $0^n$ is always congruent to $0$ no matter what congruence you are working with . I repeat this step once again.
$8^{38}\equiv 3^{38}\equiv3^{9\cdot4}3^{2}\equiv(3^{4})^{9}3^2\equiv1^9\cdot3^2\equiv9\equiv 4\bmod 5$
$8^{38}\equiv0 \bmod 2$ since it is cle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 8,
"answer_id": 1
} |
Electric field off axis inside a charged ring. If I have a charged ring of radius $a$ what I'm trying to find the electric field of a point $r$ from the axis in the plane for $r\ll a$. The example sheet hints to use a gaussian surfaces of radius $r$ but I'm not sure what purpose a gaussian surface that encloses no char... | So if we take a very small gaussian pillbox centred on the origin of height $2z$ and radius $r$ in the limit the field out of the top and bottom surfaces is: $$\frac {Qz\pi r^2}{4\pi\epsilon_0(a^2+z^2)^{3/2}}$$
therefore as the total charge enclosed is zero and we know the field through the sides of our pillbox is radi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
any good way to approximate this non-convex function with convex function? There is a non-convex constraint in my optimization problem, which is given by
$\displaystyle -xy\log\left(1+\frac{z}{xy}\right)$. Obviously, it is neither convex or concave. Is there any good convex approximation method to approximate this fun... | I don't know if this is particularly helpful in your situation, but you might consider the 'double' Legendre transform.
What I mean is the following: Let $f$ represent your function. I assume $f: D \to \mathbb{R}$, where $D \subset \mathbb{R}^3$ is some suitable domain.
The Legendre transform of $f$, denoted by $f^{*}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Discrete metric, countable basis?
Give an example of a metric space which does not have a countable basis.
I was thinking of some uncountable set, with a metric which results in an uncountable number of open subsets. Which resulted in this:
$$(\mathbb{R}, d_{\text{discrete}})$$
Where $$d_\text{discrete}(x,y) =
\begi... | Your example is fine, but your argument is not: $\Bbb R$ with the usual topology, for instance, is second countable but has uncountably many open sets. Let $\mathscr{B}$ be a base for the discrete topology on $\Bbb R$. For each $x\in\Bbb R$ the set $\{x\}$ is open, so for each $x\in\Bbb R$ there is a $B_x\in\mathscr{B}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Sum of the digits of a perfect square
Prove that the sum of the digits of a perfect square can't be 2, 3, 5
, 6, or 8.
I'm completely stumped on this one, how would I go about proving it?
| If your square number had the sum of digits equal to 3 or 6, then it must be a multiple of 3. But a square that is a multiple of 3 must be a multiple of 9. Therefore the sum of its digits must be a multiple of 9. 3 or 6 are not multiples of 9.
Therefore 3 and 6 are impossible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
clarification of a logic proof I am a bit confused on what this question is asking me to prove:
Prove $$
\exists z\forall x\in\mathbb{R}^{+}[\exists y(y - x = y/x)\leftrightarrow x \neq z]
$$
Am I asked to prove that there exists a z where the bi-conditional statement is true? Or is z a given and I should just prove t... | It is conventional that quantifiers bind from left to right. Therefore, to prove this statement, you must first choose a $z$. Then, your adversary picks an $x$, and regardless of this choice, you must now prove that the biconditional holds.
To do so you must prove that under certain circumstances $y$ exists, and unde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The maximality of a family of pairwise disjoint meager open sets implies the denseness of its union Consider the following theorem from A. Kechris's Classical Descriptive Set Theory:
(8.29) Theorem. Let $X$ be a topological space and $A \subseteq X$. Put $$U(A) = \bigcup \{ U\text{ open} : U \Vdash A \}.$$ Then $U(A) ... | If $W$ is not dense in $U(\varnothing)$, then $U(\varnothing) \not\subseteq \overline{W}$, and so $U(\varnothing) \setminus \overline{W}$ is a nonempty open set. Picking any $x \in U(\varnothing) \setminus \overline{W}$ by definition of $U(\varnothing)$ there must be an open meager $U$ containing $x$. In particular, $U... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Matrices such that $M^2+M^T=I_n$ are invertible
Let $M$ be an $n\times n$ real matrix such that $M^2+M^T=I_n$. Prove that $M$ is invertible
Here is my progress:
*
*Playing with determinant:
one has $\det(M^2)=\det(I_n-M^T)$ hence $\det(M)^2=\det(I_n-M)$
and $\det(M^T)=\det(I_n-M^2)$, hence $\det(M)=\det(I_n-M)\... | as a footnote, just to look at the simplest cases:
in dimension 1 $m^2 + m - 1=0$ so $m=\frac12(-1 \pm \sqrt{5})$
in 2 dimensions, set $M=\begin{pmatrix} a & b \\c & d \end{pmatrix}$ so
$$
M^{\mathrm{T}}=\begin{pmatrix} a & c \\b & d \end{pmatrix}
$$
and
$$
M^2 = \begin{pmatrix} a^2+bc & b(a+d) \\c(a+d) & d^2+bc \end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 0
} |
Rolling two dice... Let $A_n$ be the number of fives, $B_n$ the number of sixes and $C_n$ the number of eights in $n$ rolls of two dices. For which n do we have:
$E(A_n) < E(min(B_n,C_n))$ ?
| I run a program with PARI/GP :
? n=0;gef=0;while(gef==0,s=0;su=0;n=n+1;for(a=0,n,for(b=0,n,for(c=0,n,for(d=0,n,
if(a+b+c+d==n,su=su+n!/a!/b!/c!/d!*p1^a*p2^b*p3^c*p4^d*min(b,c);s=s+n!/a!/b!/c!/
d!*p1^a*p2^b*p3^c*p4^d)))));if(su>n/9,gef=1);print(n," ",s," ",su*1.0," "
,n/9*1.0," ",(su-n/9)*1.0))
The end of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$ Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$)
show that
$$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$
if we use
Closed-form expression
$$F_{n}=\dfrac{1}{\sqrt{5}}\left(\left(\dfrac{1+\sqrt{5}}{2}\right... | Here's one rough approach which works for $n$ sufficiently large. The idea is that for large $n$, we have that
$$
F_n \approx \phi^n/\sqrt{5}
$$
since the conjugate root is less than one, and so it tends to zero.
So consider now the generating function
$$
G(q) = \sum_{n=0}^\infty \frac{q^n}{F_n}
$$
We note that your su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 1
} |
Convergence of $\sum_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$ I have encountered the following problem:
Determine whether $$\sum \limits_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$$ converges or diverges.
What I have tried so far:
Assume that $a_n = \sqrt[3]{n^3+1} - n$.
*
*$n$th term test: $\lim \lim... | Hint We can apply a cubic analogue of the technique usually called multiplying by the conjugate, which itself is probably familiar from working with limits involving square roots. In this case, write $$u = \sqrt[3]{n^3 + 1},$$ so that $$u^3 - n^3 = (n^3 + 1) - (n^3) = 1.$$
On the other hand, factoring gives (in analogy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Absolute Value Equation $$x^3+|x| = 0$$
One solution is $0.$ We have to find the other solution (i.e, $-1$)
$$Solution:$$
CASE $1$:
If $x<0,~|x| = -x$, we can write $x^3+|x| = 0$ as $-x^3-x=0$
$$x^3+x=0$$
$$x(x^2+1)=0$$
$$\Longrightarrow x=0, or, x=\sqrt{-1}$$
Please tell me where I've gone wrong.
| the function $$f(x) = \begin{cases} x^3 - x & if \, x \le 0,\\x^3+x &if \, 0 \le x. \end{cases} $$
has one negative zero at $x = -1$ and a zero at $x = 0.$ you can see from the function definition that $f(x) > 0$ for $x > 0.$ the graph of $y = f(x)$ has a cusp at $(0,0)$ which is also a local minimum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the value of undefinite integral Find
$$\int \frac{dx}{(x+1)^{1/2}+(x+1)^{1/3}}$$
I have tried with let $u=(x+1)^{1/2}+(x+1)^{1/3}$ but I have nothing to solve that undefinite integral. please give me a clue for solve it.
| $$\begin{align*}
\int \frac{dx}{\left ( x+1 \right )^{1/2}+ \left ( x+1 \right )^{1/3}} &\overset{u^6=x+1}{=\! =\! =\! =\!}\int \frac{6u^5}{u^3+ u^2}\, du \\
&= \int \left ( 6u^2 - 6u - \frac{6}{u+1} + 6 \right )\, du\\
&= u\left ( 2u^2-3u+6 \right )-6\log (u+1)
\end{align*}$$
Now substite $u$ with $\sqrt[6]{x+1}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solve $|1 + x| < 1$ I'm trying to solve $|1 + x| < 1$. The answer should be $ -2 < x < 0$ which wolframalpha.com agrees with.
My approach is to devide the equation to:
$1+x < 1$ and $1-x < 1$ and then solve those two:
$ 1+x < 1 $
$ x < 0 $
$ 1 - x < 1$
$ -x < 0$
$x > 0$
And this gives me $ x = 0$. Where am I going wron... | Every exercise dealing with $| \cdot |$ can be solved as follows:
\begin{align}|1+&x| < 1 \\
(\iff) -1 < 1&+x < 1 \\
(\iff) -2 < &x < 0
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1288140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.