Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Transforme $y'=-2x+2xy^2$ in an ODE of second oreder. I have the following Cauchy problem
$$(C):\begin{cases}y'=-2x+2xy^2\\ y(0)=\frac{1}{2}\end{cases}$$
1) Let $f$ a solution of $(C)$. Write an ODE of second order such that $f$ is solution.
2) Write the taylor expansion at second order of $f$.
My attempt
1) I have no ... | I think you are fine with
$$
f''=−2+2f^2+4xff'\text{ resp. } y''=−2+2y^2+4xyy'
$$
if there are no further requirement to the second order ODE. You could add multiples of the original ODE for variety, but that is not necessary.
Substituting the initial value for $f(0)=y(0)$ in the initial equation gives you $f'(0)=y'(0)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1503206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Using Euler-Lagrange equations to differentiate a Laplacian Assume I am given a functional of the form:
$$
I_0[u]:= - \int \nabla u \cdot \nabla u dx
$$
then, I know that by the Euler-Lagrange equations, I have:
$$
\frac{\delta I_0}{\delta u }= 2\Delta u
$$
But, what if I have higher-order derivatives inside the func... | The Euler-Lagrange equations implicitly assume that you have a functional which is of the form $I[y,y';x]=\int f(y,y',x)dx$. To do this, we use a Taylor expansion, incrementing $y$ by a small amount $\delta y$: $$\int_{a}^{b}f(y+\delta y,y'+\delta y',x)dx\\=\int_{a}^{b}f(y,y',x)dx+\int_{a}^{b}\left(\delta y \frac{\part... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1503563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Every Order Topology is Hausdorff I have the following proof:
Let $X$ be an ordered set having at least two elements with the order
topology, and let $x,y \in X$. Without loss of generality, let $x < y$. If there is a $z \in X$ with
$x < z < y$, then take $A = \{t \in X \mid t < z\}$ and $B = \{t \in X \mid t > z\... | Suppose $z \in A \cap B$. Then $z < y$ (from $z \in A$) and $x < z$ (from $z \in B$). So then $x < z < y$ but no such point existed by assumption, as we are in the second case. Contradiction, so $A \cap B = \emptyset$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1503693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there a rigorous definition of a line? In the sources I can find, line is considered a primitive concept (a one without definition). There is something, however, in such an intuitive definition that doesn't sit well with me.
Were there any attempts to rigorously define line?
And if there weren't, why? What makes lin... | In the plane geometry that bears David Hilbert's name, there are five undefined terms:
*
*Line
*Point
*Lies on (as in a point lies on a plane) (this is often called "incidence")
*Between (as in one point is between two other points)
*Congruent (as in two line segments or two angles are congruent)
However, just... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1503772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
A misunderstanding in elementary number theory Let $n$ be an even positive integer. $S_n={{n(n+1)}\over {2}}$. I know that the remainder of the euclidian division of $S_n$ by $n$ is $n/2$ since
$$S_n=n*{n\over 2}+{n\over 2};\;\;\; 0\leq {n\over 2}<n $$ but I want to know what is wrong about the following reasoning:
${... | One is the remainder divided by n. The other is the remainder divided by 2.
The remainders are not the same. The remainders divided by the divisor are the same. That is $remainder(\frac {S_n}{n})/n = remainder(\frac {n + 1}{2})/2$
Think of this.
12/8 has remainder 4. 6/4 has remainder 2. And 3/2 has remainder 1.
4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1503912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions? In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?
I tried shifting the second term to the rhs and squaring.Even after that i'm left with square ... | \begin{align}
&\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}} = 1\\ \implies &\sqrt{(x-1)-4\sqrt{x-1} + 4}+\sqrt{(x-1)-6\sqrt{x-1}+9}=1\\ \implies &\sqrt{(\sqrt{x-1}-2)^2}+\sqrt{(\sqrt{x-1}-3)^2}=1\\ \implies &|\sqrt{x-1}-2| + |\sqrt{x-1}-3| = 1\tag{1}
\end{align}
This calls for casework:
1. $\quad\sqrt{x-1}\geq 3$
$(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1503958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Is the zero vector always included in the orthogonal complement? This might be a really silly question, but I'm currently studying for my upcoming analysis exam, and I can't quite find an answer to this.
The reason I'm asking is actually, that I've been looking over sample questions for the exam, some of which are abou... | Orthogonal complement must have $\vec{0}$ in it. After all, if $A$ is some set, $$A^\perp = \{ x | x\cdot a = 0\ \forall a \in A\},$$
and so $\vec{0} \in A^\perp$ by definition.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible. Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible.
Here is what I have attempted:
We have $AB$ invertible so there exists an $m\times m$ matrix $C$ such that $C(AB)=(AB)C=I_{m}$. Since matrix mult... | Recall that a linear map $M$ can only lower the dimension of a subspace. By this I mean that if $V \subset \mathbb R ^m$ has dimension $n$ then $M(V)$ has dimension at most $n$.
Recall a matrix is invertible if and only if it has full rank, if and only if its null-space is empty, if and only if it is surjective, if an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Composition of a function and a scalar Are the following true?
*
*$\operatorname{scalar} \circ \operatorname{function} = \operatorname{scalar} \times \operatorname{function}$
*$\operatorname{function} \circ \operatorname{scalar} = $ the function evaluated at that scalar.
Background:
Let $f:E \to F, g: F \to G$, whe... | If $V$ is your pre-Hilbert space (which I am assuming is real), then let $$f:\ V \to \Bbb R\ :\ x \mapsto \langle x, x \rangle$$ Then for each $x \in V, Df|_x$ is a linear map from $V \to \Bbb R\ :\ v \mapsto 2\langle x, v\rangle$. Let $$g\ :\ \Bbb R \to \Bbb R\ :\ t \mapsto \sqrt t$$
Then $Dg|_t$ is a linear map from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
On Coprime Numbers and Age Differences Hypothetical Situation: Currently, I am $38$ and my wife is $33$, so this year our ages are coprime. Will our ages always be coprime? (If we happen to have the same birthday, so our age difference remains constant).
In general, if any two numbers $n$ and $n+\alpha$ are coprime, wi... | Your conjecture:
$n$ and $\alpha$ are coprime iff $n$ and $n + \alpha$ are coprime
Taking contrapositives for each conditional statement, we have the equivalent:
$n$ and $\alpha$ share a prime factor iff $n$ and $n + \alpha$ share a prime factor
Let us prove the latter, equivalent formulation.
Forward: If $n$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Asymptotic expansion of Bessel Function Hi I am interested in calculating an asymptotic expansion of the following function. Or, I would at least like to know how the function behaves for large values of x. I am having trouble simplifying the expression, it's not the nicest looking.
The function is
$$
f(x)=\sqrt{ \Re\... | I want to give some justification to the result given above.
Let's look at the quotient of two Bessel functions
$$
C(\alpha,\beta,x)=\frac{J_1(\alpha f(x) )}{J_1(\beta f(x) )}
$$
now assume that $f(x)\sim x^a+iO(x^{a-\delta})$ as $x\rightarrow \infty$ with $a>\delta>0$it is now tempting to just throw away the imagina... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to find $\int_0^1f(x)dx$ if $f(f(x))=1-x$? A friend of mine gave this question to me :
Find the value of $\int_0^1f(x)dx$ if $f$ is a real, non-constant, differentiable function satisfying $$f(f(x))=1-x \space\space\space\space\space\space\forall x\in(0,1)$$
The only thing that came to my mind was to form a diff... | I think $f(x)$ does not exist at all. $f(f(x))$ is decreasing on $(0,1)$.
$f(x)$ is either increasing or decreasing on $(0,1)$. (non-constant, real, differentiable)
If $f(x)$ is increasing, then $f(f(x))$ is increasing.
If $f(x)$ is decreasing, then $f(f(x))$ is increasing.
There is no chance for $f(f(x))$ to be a dec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Give an example of a ring $A$ such that $\dim(A) =0$ but $A$ is not a field.
Give an example of a ring $A$ such that $\dim(A) =0$ but $A$ is not a field.
One thing is sure, $A$ cannot be an integral domain.
| Let $R$ be a Boolean ring other than $\mathbb{Z}/2\mathbb{Z}$.
Lemma The only integral domain that is Boolean is $\mathbb{Z}/2\mathbb{Z}$, for otherwise, if $R$ an integral domain but $R \neq Z/2Z$, we may take $a \neq 0, 1$. But $a \cdot (a-1) = a^2 - a = a - a = 0$ a contradiction.
Hence, Every prime ideal $P$ of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding how many numbers are divisible by a prime number I'm trying to figure out how I can find out how many numbers are divisible by a certain prime (eg 3) in a certain range, eg 0-10000. I think it has something to do with permutations, but I'm not really sure and kinda stuck.
Could you please point me in the right ... | All you have to do is divide and throw away any remainder: $\textrm{floor}(10000/3)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
If a Borel measure is positive on every nonempty set, can it be finite? Let $\mu$ be a positive measure on $\mathscr{B}_R$, which is strictly positive on any non empty set. Can it be finite?
I thought about finding a very fine partition of $R$, and claim that the measure is infinity if you sum up all the measure on eac... | For every $x$ you have $a_x=\mu(\{x\})>0$. Since there are uncountably many $x$ this implies that $\mu$ is infinite.
Except you're worried that we only have countable additivity. Fine: Say $E_n$ is the set of $x$ with $a_x>1/n$. There exists $n$ so that $E_n$ is uncountable. In particular $E_n$ contains an infinite cou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
To show that group G is abelian if $(ab)^3 = a^3 b^3$ and the order of $G$ is not divisible by 3
Let $G$ be a finite group whose order is not divisible by $3$.
suppose $(ab)^3 = a^3 b^3$ for all $a,b \in G$. Prove that $G$ must be abelian.
Let$ $G be a finite group of order $n$. As $n$ is not divisible by $3$ ,$3$ ... | I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
This problem is the same problem as Problem 24 on p.48 in Herstein's book.
I solved this problem as follows:
$G\ni x\mapsto x^3\in G$ is a homomorphism since $(ab)^3=a^3b^3$ for all $a,b\in G$ by the assumption of this problem.
Assume that there exists $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Distributional derivative of absolute value function I'm tying to understand distributional derivatives. That's why I'm trying to calculate the distributional derivative of $|x|$, but I got a little confused.
I know that a weak derivative would be $\operatorname{sgn}(x)$, but not only I'm not finding that one in my cal... | Given a test function $\phi$, the goal is to rewrite $-\int |x|\phi'(x)\,dx$ so that it has $\phi$ in it instead of $\phi'$. Split into two integrals over positive and negative half-axes; then integrate by parts. The result:
$$
\int_{-\infty}^0 x\phi'(x)\,dx - \int_0^{\infty} x\phi'(x)\,dx
= -\int_{-\infty}^0 \phi(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove Null($A^t A$) = Null($A$) I couldn't find anything like this (though it may have been bad searching), so:
Given that $A$ is an m x n matrix, prove that:
Null($A^t A$)= Null($A$)
I'm not sure how to prove this or which properties to use.
| Assume $x\in\operatorname{Null}(A)$ so we have $Ax=0$ and therefore $A^TAx=A^T\cdot 0=0$ and $x\in \operatorname{Null}(A^TA)$ and we have proven that $\operatorname {Null}(A)\subset \operatorname{Null}(A^TA)$
Now assume that $A^TAx=0$ so we have $\langle A^TAx,x\rangle=0$ where $\langle , \rangle$ is the Euclidean inne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question regarding two properties of controllable subspace in control theory In Mathematical Control Theory II: Behavioral Systems and Robust Control
Two claims:
*
*$\langle A+BK | im B \rangle = \langle A | im B \rangle$
*$\langle A | im B \rangle$ is the smallest $A$-invariant subspace containing $B$
I am strug... | Take $x \in \operatorname{Im} B \subseteq V$. Since $V$ is $A$-invariant, $Ax \in V$. Since $x$ is arbitrary $A \operatorname{Im} B \subseteq V$. Remember that $V$ is a subspace, so $\operatorname{Im} B + A\operatorname{Im} B \subseteq V$. Similarly, since $Ax \in V$, then $A^2x \in V$ and so on. So
$$\operatorname{Im}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this a correct proof for a span? $V = P_2(\mathbb{R})$ with degree less than 3. Let
$$U = \{f \in V \mid f(2) = 0\} $$
So to prove that$ (t^2-4,t-2) .$ is a basis. Could I do this?
Let $a_1(t^2-4)$ +$ a_2(t-2) $ = $a^2t +bt +c$ and then make a,b and c subjects of $a_1$ and $a_2$ and therefore for any a,b and c you h... | First show that $t^2 - 4 \in U, t-2 \in U$, which are both clear.
Then show that these are linearly independent, which shouldn't be too hard.
Suppose that $f(t) = at^2 + bt + c$ is in $U$. So 2 is a root of $f$. This means $f(t)$ is divisible by $(t-2)$, so $f(t) = p(t)(t-2)$ where $p(t)$ is linear at most.
Write $p(t)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Can a pre-calculus student prove this?
a and b are rational numbers satisfying the equation $a^3 + 4a^2b = 4a^2 + b^4$
Prove $\sqrt a - 1$ is a rational square
So I saw this posted online somewhere, and I kind of understand what the question is saying. I'm interesting in doing higher order mathematics but don't quite... | We have
$$a^3+4a^2b=4a^2+b^4$$
$$a(a^2+4ab)=4a^2+b^4$$
$$a(a+2b)^2-4ab^2=4a^2+b^4$$
$$a(a+2b)^2=b^4+4ab^2+4a^2$$
$$a(a+2b)^2=(b^2+2a)^2$$
$$a=\frac{(2a+b^2)^2}{(a+2b)^2}$$
$$\sqrt{a}-1=\frac{2a+b^2}{a+2b}-1=\frac{a-2b+b^2}{a+2b}$$
I'm stuck there. I also have a PhD in math and I'm an algebraist. Ramanujan could have d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 8,
"answer_id": 6
} |
Is a vector field a closed map? A smooth vector field $V$ on a smooth manifold $M$ is a smooth section of the projection $\pi : TM \rightarrow M$. (This means that $V \colon M \rightarrow TM$ satisfies $\pi \circ V = \text{id}_M.$)
Does $V$ map closed sets to closed sets?
My attempt:
Let $C$ be closed in $M$. Then $C... | In general, $\pi^{-1}(\pi(V(X)))=\pi^{-1}(X)\ne V(X)$, since the preimage of $X$ contains all the fibers over $X$ and $V(X)$ just has one point per fiber.
Note that you just need to show that $V(M)$ is closed, since then you have $V(C)=\pi^{-1}(C)\cap V(M)$ because if $x\in V(C)$, then $\pi(x) \in C$ since $\pi\circ V=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Confused about basics of subsequences Hello I am a bit confused in regard to subsequences. The following image is taken from Introduction to Real Analysis, by Bartle and Sherbert.
For example, why is the following true?
My question mostly is , why can we say that $$n_{k} \ge k$$ ? for example what if $n_{2}=1$? what if... | I think that you have misunderstood what a sub-sequence is, which is what seems to be causing this confusion. Let the sequence be represented by the function f: N to X, where X is any non-empty set. Let g be a function (n(1), n(2), ...), an infinite subset of natural numbers, to N, such that g(k) = k. Then, a sub-seque... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Deductible and Policy limit I'm trying to figure out the solution to the following problem. I was working with the Adapt program for the p exam but I can't find the solution anywhere.
Problem:
Consider an insurance policy that reimburses collision damages for an insured individual. The probability that an individual h... | Just to add some details to what has already correctly explained in the previous answer. To solve this problem, we have to determine the pdf of reimbursement amount. Let us call this amount $y$. If I correctly interpret the data, the conditions stated in the OP imply that:
*
*in 20% of subjects, $y=0$ (these are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1505994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Boundary of a connected manifold is connected? I assume the answer to this question is simple, but I can't find any references:
Let $X$ be a topological space, and $M$ be a (path-)connected component of a manifold in $X$ with the same dimension as $X$. Then, is the boundary of $M$ also (path-)connected?
| The boundary need not be (path-)connected.
Consider $X = \mathbb{R}$ with its standard topology and $M = [0, 1]$. Note that $\partial M = \{0, 1\}$ which is not connected or path-connected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1506237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
For which $n ≥ 0$ is $2^n + 2 · 3^n$ divisible by $8$? Stuck on this problem for some time:
For which $n ≥ 0$ is $2^n + 2 · 3^n$ divisible by $8$?
I've reached the conclusion that $n = 1$ is the only solution to the question at hand, but I cant quite prove that/why this is the case.
The properties of the evaluation of... | If $n\geq 3$, $2^n+2\cdot 3^n$ is either $2$ or $6\pmod{8}$, since $3^2=9\equiv 1\pmod{8}$ and $8\mid 2^n$.
So we just have to check by hand the cases $n\in\{0,1,2\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1506327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Calculate double integral using Polar coordinate system Need to calculate $\int_{0}^{R}dx\int_{-\sqrt{{R}^{2}-{x}^{2}}}^{\sqrt{{R}^{2}-{x}^{2}}}cos({x}^{2}+{y}^{2})dy$
My steps:
*
*Domain of integration is the circle with center (0,0) and radius R;
*$x = \rho \cos \varphi ,\: y = \rho \sin \varphi,\: \rho \in \left... | In polar coordinates the integral is $\int_{-\pi /2}^{\pi /2}d \varphi \int_{0}^{R}cos({\rho}^{2} ) \rho d \rho =\pi \left[ \frac{\sin \left({\rho}^{2} \right)}{2}\right]_0^R=\frac{\pi }{2}\sin {R}^{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1506518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluate the Integral $\int^2_{\sqrt{2}}\frac{1}{t^3\sqrt{t^2-1}}dt$ $\int^2_{\sqrt{2}}\frac{1}{t^3\sqrt{t^2-1}}dt$
I believe I've done everything right; however, my answer does not resemble the answer in the book. I think it has something to do with my Algebra. Please tell me what I am doing wrong.
| Your integral is well calculated.
Computing the last part for a prominent answer, we get
$$\frac{1}{2}[1+\frac{1}{2}\sin 2\theta]^{\frac{\pi}{3}}_{\frac{\pi}{4}}$$
$$=\frac{1}{2}[(\frac{\pi}{3}-\frac{\pi}{4})+\frac{1}{2}(\sin (\frac{2\pi}{3})-\sin (\frac{2\pi}{4}))]$$
$$=\frac{1}{2}[\frac{\pi}{12}+\frac{1}{2}(\frac{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1506645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Basic Counting -- Selecting groups of people from a group of 10 I'm reading a series of basic counting problems regarding a ten person club. I just want to make sure I'm sufficiently grasping the material and my problem solving methods are correct.
$1.$ In how many ways may a ten person club select a president and a s... | It is correct since you understand that the first answer requires a permutation since it is in a certain order, second requires combination because it is not chosen in any order, and the third is permutation and combination since one person is chosen in a specific order and the next two are chosen at random.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1506723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
The meaning of "subdirect sum"? In paper "unique subdirect sums of prime rings" L. S. Levy define the notion of "subdirect sum". I can not understand this definition! In this definition the map $h$ is isomorphism and i think that it is wrong. Is it wrong? Can you help me? thanks.
| Here's the definition from the paper you mention.
Let a set of rings $\{R_\alpha:\alpha \in A\}$ be given.
A ring $R$ is a subdirect sum of $\{R_\alpha\}$ if there is an isomorphism $h$ of $R$ into the (complete) direct product $\prod_{\alpha} R_\alpha$ such that each of the induced projections $R \to h(r)_\alpha$ m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1506814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What mathematical object accepts a sequence as input? A function inputs a single number and outputs a single number. (e.g. $y = f(x)$)
What mathematical object inputs a sequence, or even just a vector or tuple?
Edit:
I understand that a function doesn't have to take a single number and output a single number. But what... | Well, actually... the real numbers are equivalent to rational sequence classes which converge to that number so any real number function is equivalent to a function taking in a sequence as an input. (Although the domain the domain of one representative sequence from an infinite class of sequences, all converging to th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1506890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Calculate the area under $f(x) = \sqrt x$ on $[0,4]$ by computing the lower Riemann sum for $f$ with the given partition Where $x_i = \dfrac{4i^2}{n^2}$ and letting $n \rightarrow \infty$
I don't know how and where to begin.
| $$
\sum_{i=0}^{n-1} f(x_i) \, \Delta x_i = \sum_{i=0}^{n-1} \sqrt{\frac{4i^2}{n^2}} \left( \frac{4(i+1)^2}{n^2} - \frac{4i^2}{n^2} \right).
$$
Etc. Why don't you know where to begin? Are you unaware of what Riemann sums are? If so, maybe you could ask about that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability of $ \sum_{n=1}^\infty \frac{x_n}{2^n} \leq p$ for Bernoulli sequence Probability of $ \sum_{n=1}^\infty \frac{x_n}{2^n} \leq p$ for a Bernoulli(1/2) sequence $(x_n)$ and $p \in [0,1]$. I know that the answer should be just $p$ but how can you prove it?
Say for $p= 1/2$ I can look at it as 1-( Probability... | Write $p$ in base $2$.
$$\sum_{i = 1}^{\infty} \frac{p_i}{2^i}$$
We don't know these coefficients explicitly but we won't need to in the end. Then you have to look at what are the possible values for the $x_i$.
To get something smaller than $p$, you need to have $x_i = p_i$ up to a certain point where $p_k = 1$ and $x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Best practice for naming variables to distinguish chosen upper bound from computed maximum. Say I have a vector of a matrix's singular values $[s_1, s_2, \cdots, s_N]$, and would like to define two variables, one which holds the maximum value in the vector, and another which holds an arbitrarily chosen upper limit whic... | I might use "cap" rather than "max" to briefly describe the second quantity.
Alternatively, and perhaps more conventionally, give the cap a different name like $M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the maximum number of prime numbers by which N can be divided?
MyApproach
$$(a+1)(b+1)(c+1)\cdots=45$$
To see maximum number of prime numbers by which $N$ can be divided.
$a,b,c$ are the powers of prime numbers.
I took factors of $45$. I got $3^2 \cdot 5$. From this, I think maximum $3$ factors could be there ... | Suppose $p,q,r$ are prime numbers, and consider $\{ p^i q^j r^k : 0\le i\le 2,\ 0\le j\le 2,\ 0\le k\le 4 \}$. This is a set of $45$ numbers, all factors of $N = p^2 q^2 r^4$.
The highest power of $p$ that divides a factor of $N$ is either $1$ or $p$ or $p^2$.
The highest power of $q$ that divides a factor of $N$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
In a triangle prove that $\sin^2({\frac{A}{2}})+\sin^2(\frac{B}{2})+\sin^2(\frac{C}{2})+2\sin(\frac{A}{2})\sin(\frac{B}{2})\sin(\frac{C}{2})= 1$ Let ABC be a triangle. Thus prove that
$$\sin^2\left({\frac{A}{2}}\right)+\sin^2\left(\frac{B}{2}\right)+\sin^2\left(\frac{C}{2}\right)+2\sin\left(\frac{A}{2}\right)\sin\left... | Let $x=\frac{A}{2}$, $y=\frac{B}{2}$, $z=\frac{C}{2}$, $x+y+z=\pi/2$
$$\sin^2(x)+\sin^2(y)+\sin^2(z)+2\sin(x)\sin(y)\sin(z)\\=\sin^2(x)+\sin^2(y)+\sin^2(\pi/2-x-y)+2\sin(x)\sin(y)\sin(\pi/2-x-y)\\=\sin^2(x)+\sin^2(y)+\cos^2(x+y)+2\sin(x)\sin(y)\cos(x+y)\\=\sin^2(x)+\sin^2(y)+\cos^2(x+y)+(\cos(x-y)-\cos(x+y))\cos(x+y)\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluate the Integral $\int \sqrt{1-4x^2}\ dx$ $\int \sqrt{1-4x^2}\ dx$
I am confused as I get to the end. Why would I use a half angle formula? And why is it necessary to use inverses?
| When you're doing the trigonometric substitution, you write $x=a\sin\theta$, which is good; you should also remember how to get back from $\theta$ to $x$, that is,
$$
\theta=\arcsin\frac{x}{a}=\arcsin\frac{x}{1/2}=\arcsin(2x)
$$
which actually should be the starting point, because it guarantees the angle $\theta$ is be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
How do we find integral closure?
Find the integral closure of $\mathbb C[x^2,x^2-1] $ in $\mathbb C(x)$
I don't know much about integral closure,I've just learned about it.How do we find integral closure in practice ? Thanks for your help.
| $\mathbb C[x^2]\subset\mathbb C[x]$ is integral ($x$ is a root of $T^2-x^2\in\mathbb C[x^2][T]$), so the integral closure of $\mathbb C[x^2]$ in $\mathbb C(x)$ is $\mathbb C[x]$ since the last one is integrally closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
General linear group over integers I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold?
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a ... | Integers with multiplication do not form a group. For example the 1x1 matrix (2) has an inverse (1/2) which is not integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
Determining row space of a $n$ by $n$ matrix (Linear Algebra) Prove the following statements or provide a counterexample if it is false
For two $n$ by $n$ matrices $A$ and $U$ the row space of $UA$ is contained in the row space of $A$
What i tried
I started by trying out some simple 2 by 2 matrices to determine whethe... | The elements of the row space of the $m\times n$ matrix $B$ can be obtained by computing all products
$$
\begin{bmatrix}x_1 & x_2 & \dots & x_m\end{bmatrix}B
$$
where $x=\begin{bmatrix}x_1 & x_2 & \dots & x_m\end{bmatrix}$ is a row with $m$ columns.
So, if $r$ is an element of the row space of $UA$, there is a row $x$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing that the solution of the in-homogeneous ODE: $u''(x)-Cu(x)=-f(x)$ is unique Given the inhomogeneous ODE on the form: $u''(x)-Cu(x)=-f(x)$
where $0 < x < 1, C > 0$ and $f(x)$ is cont. on the interval for x.
With IC: $u(0) = u(1) = 0$
Solving that yields the general solution for the homogeneous part:
$u_c = Ae^\s... | It should of course be
$$
u_c(x) = Ae^{\sqrt{C}x} + Be^{-\sqrt{C}x}
$$
giving you the conditions
$$
A+B=0\\
Ae^{\sqrt{C}} + Be^{-\sqrt{C}}=0
$$
which indeed tells you that there are no solution except the trivial solution $u_c\equiv 0$.
For the inhomogeneous equation you could either use variation of constants or use... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If A is invertible and $||B-A|| < ||A^{-1}||^{-1}$ prove $B$ is invertible. Just having really hard time trying to proof :
If $A$ is invertible and $||B-A|| < ||A^{-1}||^{-1}$ prove $B$ is invertible.
It is related to Neumann Series but i don't understand how to proof with math.
Thanks for your help and time.
Brian Ign... | This is an old question, but I think it deserves a simple solution which does not need a convergence argument. So here we go.
By assumption, we have $\| A-B \| \| A^{-1}\| < 1$. Moreover because the induced $2$-norm is consistent, we have $\| (A-B)A^{-1}\|\leq \| A-B \| \| A^{-1}\|$. Therefore, $\| (A-B)A^{-1}\|=\|I-BA... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Integrating $\sqrt{x^2+a^2}$ I'm trying to integrate this function wrt $x$, substituting $x = a \tan \theta$
$$ \int \sqrt{x^2+a^2} dx = a^2 \int \frac {d\theta}{\cos^3\theta} = $$
$$= a^2 \cdot \frac 12 \left( \tan\theta \sec\theta + \ln\lvert \tan\theta + \sec\theta \rvert \right) = \frac 12 \left( x \sqrt{a^2 + x^2... | $$\frac 12 \left( x \sqrt{a^2 + x^2} + a^2 \ln \left| \frac x a + \frac{\sqrt{a^2+x^2}}{a} \right| \right)\\=\frac 12 \left( x \sqrt{a^2 + x^2} + a^2 \ln \left| x + \sqrt{a^2+x^2} \right| \right)-\frac{a^2\ln|a|}{2} $$
and
$$\frac 12 \left( x \sqrt{a^2 + x^2} + a^2 \ln \left| x + \sqrt{a^2+x^2} \right| \right)$$
are di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that this Set contains all multiples of a whole number Let S be a nonempty subset of $\mathbb{Z}$. Suppose S satisfies the following constraints
If $x,y \in S$ then $x+y \in S $
If $x \in S$ and $y \in \mathbb{Z}$ then $xy \in S$
Show that S is the set of all integer multiples of m for some $m \in \mathbb{N}\cup{\... | Let $m$ be the minimal positive in $S$. Then by the second rule $km\in S$ for each $k\in\Bbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that $a^{m} a^{n} = a^{m+n}$ Prove that if $G$ is a group and $a\in G$, then we have $\forall m,n\in\mathbb{Z} $ that
$$a^{m} a^{n} = a^{m+n}.$$
I've proved the case when $m,n>0$ but I'm stuck on how to prove the case when one or both are negative without assuming that $(a^{-1})^n = a^{-n} $.
| I assume that you would like to use $(a^{-1})^n = a^{-n}$ by saying that if $n > 0$ then
$$
a^m a^{-n} = a^m (a^{-1})^n = a^{m-n}
$$
by leveraging the case for $m,n > 0$. The problem is that you can't really do this, because in that case you had two powers with the same basis, but usually $a \neq a^{-1}$.
Instead, wha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Group Theory Cyclic Generators Proof
Suppose $a$ is a power of $b$, say $a=b^k$, then $b$ is equal to a power of $a$ if and only if $\langle a\rangle=\langle b\rangle$.
I am not even sure how to really start this. I want to say something about how $b$ has order $k$ meaning it has $k$ elements. But I am not sure if th... | If $b$ is equal to a power of $a$ then $b\in\langle a \rangle$. Hence, $\langle b\rangle\subset\langle a\rangle$. Since $a$ is a power of $b$ the reverse inclusion is true.
Finally, $\langle a\rangle=\langle b\rangle$.
If $\langle b\rangle=\langle a\rangle$, since $b\in \langle b\rangle$, $b\in \langle a\rangle$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Closed Form Generating Function for sum of natural numbers I need to find a Closed Form Generating Function for a sequence whose $n$-th term is the sum of the first $n$ natural numbers, i.e:
$$f(x) = \sum_{i=1}^{n}\frac{n(n+1)}{2}x^n$$
and am having difficulties. I am trying to approach it through differentiation since... | Ok I have it figured out.
The answer is : $$f(x) = \frac{x}{(1-x)^3}$$
We get this by starting with $\frac{1}{1-x} = 1 + x + x^2 + ...$, taking the second derivative yields $2 + 6x + 12x^2 + ... = \frac{2}{(1-x)^3}$ then multiplying by $x\over2$ gives the desired series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Using trig to integrate $x^2/\sqrt{16-x^2}$ I'm trying to integrate: $\int\frac{x^2}{\sqrt{16-x^2}}\ dx$
My try was to convert $x$ to $4\sin(u)$ and $dx$ to $4\cos(u)du$, but I'm not sure. Thanks.
| Notice, let $x=4\sin u\implies dx=4\cos u$
$$\begin{align}
\int \frac{x^2}{\sqrt{16-x^2}}\ dx&=\int \frac{16\sin^2u}{\sqrt{16-16\sin^2u}}(4\cos u\ du)\\\\
&=\int \frac{16\sin^2u}{\cos u}(4\cos u\ du)\\\\
&=\int \sin^2 u\ du\\\\
&=\int \frac{1-\cos (2u)}{2}\ du\\\\
&=\frac{1}{2}\int \ du-\frac{1}{2}\int \cos (2u)\ du\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$2\times 2$ matrix with $2$ Distinct Eigenvalues is Diagonalizable In a class of mine, my teacher said that any $2\times 2$ matrix with $2$ distinct eigenvalues is diagonalizable. I've been trying to find a proof of this or a theorem to support it but cannot find anything. Is this statement true?
| We can use the fact that eigenvectors corresponding to distinct eigenvalues are linearly independent. Now recall that an $n\times n$ matrix is diagonalizable if and only if it has $n$ linearly independent eigenvectors.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Solving degree three inequality
$$\frac{2}{(x^2+1)}\geq x.$$
I thought to do this:$$2-x^3-x\geq 0$$
But we haven't learnt how to solve equations of third grade. Could you help me maybe by factorizing sth?
| Hint : Try factorizing the LHS. Let $f(x) = x^3 + x - 2$, $f(1) = 0$ thus $x - 1$ is a factor of $f$.
$$ f(x) = x^3 + x - 2 = (x - 1)\cdot P(x)$$
where $P(x)$ is a quadratic polynomial of $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
If $A=\left(\begin{smallmatrix} 1 & \tan x\\ -\tan x & 1 \end{smallmatrix}\right)$ and $f(x)=\det(A^TA^{-1})$, then what is $f(f(f(f \cdots f(x))))$?
Let $$A := \begin{bmatrix} 1 & \tan x\\ -\tan x & 1\end{bmatrix}$$ and $$f(x) := \det(A^TA^{-1})$$ Which of the following can not be the value of $f(f(f(f \cdots f(x))))... | As you figured out that $f(x) = 1$
1.$f^n(x) = 1$ as $f(x) = 1 \Rightarrow 1^n = 1$
2.Use the same argument as $(1)$
3.That is indeed right.
4.$n f(x) = n (1) = n$, which can't be the value of $f(f(f(f(f(....f(x)))))))$
Hence, option $(4)$ is the answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\sum\limits_{n=1}^\infty(-1)^n\frac{x^2+n}{n^2}$ converges uniformly, but not absolutely My Work:
i) Uniform Convergence (By Weierstrass M-Test):
I am attempting to show that the series converges uniformly on the interval $I=[-a,a]$, in which:$$\sum_{n=1}^\infty(-1)^n\frac{x^2+n}{n^2}\le\sum_{n=1}^\infty(-1... | Your argument is correct for the non-absolute convergence. Indeed, the series $\sum_{n\geqslant 1}x^2/n^2$ is convergent while $\sum_{n\geqslant 1}1/n$ is divergent, hence $\sum_{n\geqslant 1}\left(x^2/n^2+1/n\right)$ is divergent.
For the uniform convergence, define $s_n(x):=\sum_{j=1}^n(-1)^j(x^2+j)/j^2$. Defining $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to interpret the text of this exercise: find formulas for the following implicitly functions I have some issues in the interpretation of this exercise:
Find formulas for the following implicitly defined functions. What are their domains and ranges?
$y = f(x)$ is the solution of equation $x^{3}y + 2y = 5$.
I have p... | Yes, you are supposed to do algebraic manipulations to have the left side $y$ and the right side not contain $y$. The reason it is called "implicitly defined" is that $x$ and $y$ are both on the same side of the equation, so you cannot (easily) plug in a value for $x$ and find $y$. This will also make it easier to re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Problem proving Cartan's identity There is a famous identity stating that, if $X$ is a field and $\omega$ a form, then:
$$\mathcal{L}_X\omega=\iota_Xd\omega+d\iota_X\omega.$$
I'm trying to prove it. Thanks to Anthony Carapetis, I know that:
$$\iota_X(\alpha\wedge\beta)=(k+\ell)(\iota_X\alpha\wedge\beta+(-1)^k\alpha\wed... | As Anthony noted in his comment, correcting the normalization coefficient by eliminating the denominator $(k+\ell)!$ removes the problem and yields:
$$\iota_X(\alpha\wedge\beta)=\iota_X\alpha\wedge\beta+(-1)^k\alpha\wedge\iota_X\beta,$$
which allows for Cartan to be proved my way.
Another way is this, which John Ma pos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Conversion from euler angles to versors I am attempting to create a script to convert between the output of one long program and the input of another, neither of which I can edit. The output of the first gives euler angles for rotation and the input of the other requires a 3 float versor. All the online information I h... | From this ref. we see how determine a unitary quaternion $z=a+b\mathbf{i}+c\mathbf{j}+c\mathbf{k}=a+\mathbf{v}$ that corresponds to three rotation around the coordinate axis (note that the quaternion depends on the order of the rotations not only from the angles).
We can write such unitary quaternion $z$ in polar form... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
die roll probability with four sided die How many times must you roll a four sided die so that the probability of getting at least $3$ twos is $0.3214$.
$$P(n \ge 3)=0.3214$$
$$P(x)=(1/2)^3=1/8$$
answer given is $8$
| The probability of getting at least $3$ twos out of $n$ rolls is $\sum\limits_{k=3}^{n}\dfrac{\binom{n}{k}\cdot3^{n-k}}{4^n}$
Using trial & error, we get $\sum\limits_{k=3}^{n}\dfrac{\binom{n}{k}\cdot3^{n-k}}{4^n}=0.3214 \implies n=8$
You can also calculate $1$ minus the probability of the complementary event:
$1-\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Triangulation for a 1-manifold I'm taking a course on Algebraic Topology and I had a question while studying. I know the answer is affirmative but I don't know why. What I want to prove is
There exist a triangulation for every 1-manifold
I tried to use that the manifold is II Axiom of countability, so there exist a c... | Do you know the classification of 1-manifolds? Every compact one-manifold is a finite union of circles. Now you should be able to explicitly construct triangulations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding the equivalence class of a relation |a| = |b| For the following relation $R$ on the set $X$ determine whether it is $(i)$ reflexive, $(ii)$ symmetric and $(iii)$ transitive. Give proofs or counter examples. In the case where $R$ is an equivalence relation, describe the equivalence classes of $R$.
$X=\mathbb C,\... | If I've understood correctly, you need to characterize the equivalence classes for the above equivalence relation on the set $X=\mathbb{C}$.
The definition of complex absolute value is very geometric. Given some complex number $a+bi$, $\lvert a+bi\rvert=\sqrt{a^2+b^2}$. It follows from here that an equivalence class o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Remark about $\sup(f+g)$ and $\sup (A+B)$ 1. Let $A$ and $B$ are some nonempty subsets of $\mathbb{R}$. Define $A+B=\{a+b: a\in A, b\in B\}$ it's easy to prove that $$\sup(A+B)=\sup A+\sup B \qquad(1)$$
2. Let $f(x)$ and $g(x)$ are some real-valued functions on $[a,b]$. Then $$\sup\limits_{[a,b]} (f(x)+g(x))\leqslant\s... | In one you are taking the sum over fixed $x\in[a,b]$, whereas in the case of $A+B$ you are summing over all combinations of $a$ and $b$. You would achieve an equality in the second case if you instead had
$$\sup_{x,y\in[a,b]}(f(x)+g(y))$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1509969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Random Walk Upper-Bound. For any symmetric Random Walk, absolute value of partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}$? Is it possible to prove that for any symmetric Random Walk, the absolute value of its partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}\quad\large?$
I run some... | No
For example you might with positive probability get nine $+1$s in a row.
But $\sqrt{18 \pi}+\sqrt{\pi / 2} \lt 9$.
More substantially, the law of the iterated logarithm says $\displaystyle \limsup_{n \to \infty} \frac{S_n}{\sqrt{2n \log\log n}} = 1$ almost surely. For big enough $n$ you will have $\dfrac{\sqrt{2n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$. List (with proof) all field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$.
So I can see that the field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$ are
*
*$p+q\sqrt{5}$
*$p-q\sqrt{5}$
But how do I formally proof this?
| Such monomorphism is equal to the identity on $\mathbb Q$. You can verify that the two maps that you have described are indeed field monomorphisms.
Finally there are no others as a field monomorphism sends a root of $X^2-5$ to a root of the same polynomial. And a field monomorphism $\sigma$ of $\mathbb Q(\sqrt{5})$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Likelihood of a correct diagnosis of a disease A certain cancer is found in one person in 5000. If a person does have the disease, in
92% of the cases the diagnostic procedure will show that he or she actually has it. If
a person does not have the disease, the diagnostic procedure in one out of 500 cases
gives a false ... | $P(H_1)=2/10000$
$P(u=1|H_1)=92/100$
$P(u=1|H_0)=2/1000$
Using the Bayes rule:
$$P(H_1|u=1)=\frac{P(u=1|H_1)P(H_1)}{P(u=1)}$$
where
$$p(u=1)=P(u=1|H_1)P(H_1)+P(u=1|H_0)P(H_0),$$
and
$$P(H_1)+P(H_0)=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Understanding a necessary step in a solution in variational calculus I'm reviewing calculus of variations using a pdf that I found online (link) and in the example about the minimal surface of revolution, the writer simplified an equation tagged $(3.16)$ as follows:
$$\sqrt{1+(u')^2} - \frac{d}{dx} \, \frac{uu'}{\sqrt{... | Let $v = \sqrt{1+(u')^2}$. Then $vv' = u'u''$, and the LHS is
$$\begin{align} v - \frac{d}{dx}\frac{uu'}{v} &= v - \frac{({u'}^2 + uu'')v - uu'v'}{v^2}
\\&=\frac{v^4}{v^3} - \frac{({u'}^2 + uu'')v^2 - uu'vv'}{v^3}
\\&=\frac{v^4 - (v^2 - 1 + uu'')v^2 - u{u'}^2u''}{v^3}
\\&=\frac{v^2 - uu''(1+{u'}^2) + u{u'}^2u''}{v^3}
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A nice property of $L_1$ function in $[0,1]$. Let $f$ is in $L^1([0,1],m)$, $m$ is a Lebesgue measure and suppose that $f(x)>0$ for all $x$. Show that for any $ 0<\epsilon<1 $ there exists $\delta$ >0 so that $\int_{E} f(x)dx\ge \delta$ for any set $E\subset[0,1]$ with $m(E)=\epsilon$
I have done a similar problem some... | Hint: take $\eta > 0$ so that $m\left(\{x: 0 < f(x) < \eta \}\right) \le \epsilon/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Computing an indefinite integral(2) $$\int \sqrt{1- \sin(2x)}dx =?$$
My attempt:
$$\int \sqrt{1- \sin(2x)}dx = \int \sqrt{(\cos x - \sin x)^{2}} dx = \int|\cos x- \sin x| dx = ??$$
| Notice, $$\cos x-\sin x=\sqrt 2\left(\frac{1}{\sqrt 2}\cos x-\frac{1}{\sqrt 2}\sin x\right)=\sqrt 2\sin\left(\frac{\pi}{4}-x\right)$$
but $\sin \theta\ge 0\iff 2k\pi\le \theta\le 2k\pi+\pi$
hence, solving the inequality
*
*$$2k\pi\le \left(\frac{\pi}{4}-x\right)\le 2k\pi+\pi$$
$$ 2k\pi\le \left(\frac{\pi}{4}-x\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why does a compact Rieman surface not admit a Greens function? Here is my definition of a Greens function:
Definition: Let $X$ be a Riemann surface and $x_0$ a point in $X$.
(1) The Greens function of a relatively compact domain $G \subseteq X$ with regular boundary in $x_0$ is the unique continuous function
$$
g \colo... | Assume that if $g : X \setminus\{x_0\} \to \mathbb R$ is a harmonic function so that around $x_0$, $g(z) + \log |z|$ has a harmonic extension. Thus there is a small open disk $D$ around $x_0$ so that $g(z) \ge M > \inf g$ for all $z\in \overline D\setminus\{x_0\}$. In particular, minimum of $g$ when restricted to the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a mistake in this proof?
Proposition 1: The number of occurrences of an event within a unit of
time has a Poisson distribution with parameter $\lambda $ if and only
if the time elapsed between two successive occurrences of the event
has an exponential distribution with parameter $\lambda $ and it is
... | The $\text{“ if ''}$ assertion is correct; the $\text{“ only if ''}$ is not.
If the interarrival times are independent and exponentially distributed with rate $\lambda$ (and therefore with expected value $1/\lambda$), then the number of arrivals between time $0$ and time $1$ has a Poisson distribution with expected val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1510948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $u_1u_2\cdots u_n=1$ in a commutative ring, then all of $u_i$ are units If $u_1u_2\cdots u_n=1$ in a commutative ring, then all of $u_i$ are units.
Does the proof follow some logic like the following:
$u_1(u_2\cdots u_n)=1\implies u_1,u_2\cdots u_n$ are both units, so $u_1$ is a unit,
$u_2(u_1u_3u_4\cdots u_n\implie... | Here is a proof by induction.
The cases $n=1$ and $n=2$ are clear.
If $n>2$, let $v=u_1u_2$. Then $vu_3\cdots u_n=1$ and by induction $v, u_3, \ldots, u_n$ are units. Use the case $n=2$ on $v$ to conclude that $u_1, u_2$ are units.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How many sets contains 6 or its multiple given the following conditions?
MyApproach
I created
@Edit
S1={1,2,3,4,5} ...B)
S2={2,3,4,5,6}
S3={3,4,5,6,7}
S4={4,5,6,7,8}
S5={5,6,7,8,9}
S6={6,7,8,9,10}
S7={7,8,9,10,11} ....A)
S8={8,9,10,11,12}
From this information I analyzed that these $8$ sets have $6$ sets that have 6... | Hints:
*
*The sets have $5$ consecutive elements so can at most contain $1$ multiple of $6$.
*For a multiple of $6$ find out of how many of these sets it will be an element.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Examples for $ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$ I saw the answers here about how to prove:
$$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrat... | If $x>1$ then $0< \frac 1x < 1$ so $\lfloor \frac 1x \rfloor = 0$ and the equation reduces to
$$x + \frac{1}{x} = 1 + \lfloor x \rfloor$$
Any number $x>1$ can be written on the form $x = n + r$ where $n$ is an integer and $0<r<1$. Take this form for $x$ in the equation above. Since $\lfloor n + r\rfloor = n$ the equati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Show how $\lim\limits_{n\to\infty}\left(\frac{n-1}{n+1}\right)^n = \frac{1}{e^2}$ How does one evaluate this limit?
$$\lim\limits_{n\to\infty}\left(\frac{n-1}{n+1}\right)^n$$
I got to $$\lim_{n\to\infty}\exp\left(n\cdot\ln\left(\frac{n-1}{n+1}\right)\right)$$ but I'm not sure where to go from there.
| $$\lim_{n\to\infty} \left(\frac{n-1}{n+1}\right)^n=$$
$$\lim_{n\to\infty} \exp\left(\ln\left(\left(\frac{n-1}{n+1}\right)^n\right)\right)=$$
$$\lim_{n\to\infty} \exp\left(n\ln\left(\frac{n-1}{n+1}\right)\right)=$$
$$\exp\left(\lim_{n\to\infty}n\ln\left(\frac{n-1}{n+1}\right)\right)=$$
$$\exp\left(\lim_{n\to\infty}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 5
} |
Simplification ideas for an integration Does anyone see any shortcuts for calculating the following integral analytically?
$$
-\int_{-\infty}^{+\infty} \frac{1}{(\pi x)^2}\sin{(2\pi x)}\sin{(2\pi n x)} e^{2\pi i k x} dx \tag{1}
$$
*
*Where $n\in \mathbb{N}.$
*Alternate writing of the two sin terms: $\sin{(2\pi x)}... | You are looking for the Fourier transform of a product, that is the convolution between the Fourier transform of $\frac{\sin(2\pi x)}{\pi x}$ and the Fourier transform of $\frac{\sin(2\pi m x)}{\pi x}$. That is quite easy to compute, since the Fourier transform of $\frac{\sin x}{x}$ is just the indicator function of a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Contraction operator In a proof of Picard's theorem using the contraction mapping theorem, we define an operator $T$ which is applied to a function $y$. I don't really see below how $Ty$ is any different from $y$ as the RHS for both are the same. Could someone please explain how they are different?
| Perhaps the author wrote it in a slightly confusing way. The first equation
$$y(x) = b + \int_{a}^{x}f(s,y(s))ds$$
is true of the solution $y(x)$ to the differential equation $$\frac{d}{dx}y(x) = f(x,y(x))$$ with initial condition $y(a) = b$. I would think of 1.21 as an equation that is true when the variable $y$ takes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Two numbers are independently picked from [0,1], what's the probability that the two picks differ by less than 0.5? So using a graphical method, mapping the two sample spaces along X and Y axes in a unit square, I graphically calculate the area relevant to this probability to be $\frac{3}{4}$
But, using the Random Vari... | If I understand your post correctly, your PDF is incorrect. The PDF of the (absolute value) difference $Z$ between two variables $X$ and $Y$, both uniformly drawn from the interval $[0, 1]$, is in fact
$$
f_Z(z) = \begin{cases}
2-2z & 0 \leq z \leq 1 \\
0 & \text{otherwise}
\end{cases}
$$
This can be obtained as follo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do finite fields work? I know that a field must satisfy a set of axioms, the one that is causing me most discomfort is closure under addition. All the roots of $X^q-X$ are all the elements of a finite field of order $q$. However, I tested this with $X^5-X$, its roots are ${1,-1,0,i,-i}$ however, if I add $i+i=2i$ a... | You're not taking the roots in the complex numbers, you're taking the roots in the finite field itself. In the finite field $\mathbb{F}_5$ (the integers $\bmod 5$), there is in fact an element that deserves to be called $i$ (in the sense that it squares to $-1$), namely $2$, and $2 + 2 = 4$ is also in the field (if you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to evaluate $1234^{1234} \pmod{5379}$? Note: $5379 = 3 \times 11 \times 163$.
I tried Chinese Remainder Theorem and Fermat's Little Theorem, got as far as:
$$
1234^{1234} = 1 \pmod{3} \\
1234^{1234} = 5 \pmod{11}
$$
With a bit more work:
$$1234^{1234} = 93^{100} \pmod{163}$$
But $93^{100}$ doesn't really help?
Wolf... | Well this is far from perfect,but it works if you have enough time or a calculator.
$$93\equiv -70\pmod{163}$$
$$\begin{align}
93^{100}&\equiv(-70)^{100}\\
&= 490^{50}\cdot10^{50}\\
&\equiv 10^{50}\\
&= 2^{50}\cdot 5^{50}\\
&= 1024^5\cdot 3125^{10}\\
&\equiv 46^5\cdot 28^{10}\\
&=2^{25}\cdot 23^5\cdot 7^{10}\\
&= 2^{25... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Einstein Tensor Notation: Addition inside a function Main Question
Can I represent addition of multi-dimensional variables in this linear function in Einstein Summation Convention?
$$
f(\mathbf{x} + \mathbf{v})
$$
This didn't seem right $f(x_{\mu}+v_{\mu})$ and nor did $f(x+v)_{\mu}$
More Information...
I am doing an u... | After discussion with a few course mates we came to the conclusion that it should generally be left as the following,
$$
f(\mathbf{x} + \mathbf{v}) = \cdots
$$
with index notation on other relevant terms.
On the topic of taylor series for tensors there is a really interesting treatment of Taylor's Theorem in Tensor Cal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1511985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Convergence/Divergence of $\sum_{n=1}^\infty \left(\frac {1+\cos(n)}3 \right)^n$ I need to see if this series $\sum_{n=1}^\infty\limits \left(\frac {1+\cos(n)}3 \right)^n$ either converges or diverges. I was thinking that because the inside terms are going to fluctuate between $(0,\frac 23)$, the inside is never negat... | Hint:
$$
\sum_{n=1}^\infty \left|\frac{1+\cos(n)}3 \right|^n
\leq\sum_{n=1}^\infty \left(\frac23\right)^n<\infty,
$$
hence the series converges absolutely, hence converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove $|h(x)|\leq (Mx^2)/2$ I repeated this question because I would really appreciate a hint.
Let $h:[0,a]\rightarrow \mathbb{R}$ be twice differentiable, $h'(0)=h(0)=0$, and $|h''(x)|\leq M$ for all $x\in [0,a]$.
I proceed with the mean value theorem, but I do not have the factor of $1/2$ in the result above (I get$|... | Show that $h(x) = \int_0^x (x-t) h''(t) dt$, then proceed by using the esgtimate for $h''$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Let $f: A \to B$ and let $\{D_{α} : α\in Δ\}$ Prove that $f\left( \bigcup_{α\in Δ} D_{α}\right ) = \bigcup_{α\in Δ} f( D_{α})$ So I know that I need to show that each is a subset of the other but other than that, I don't know where to start. Any help is appreciated.
| You can use the fact that
$$
A\subset B\implies f(A)\subset f(B)
$$
Since for any $\alpha$
$D_{α}\subset \bigcup_{α\in Δ} D_{α}$, there is
$$
f(D_{α})\subset f(\bigcup_{α\in Δ} D_{α})\quad\text{and so}\quad \bigcup_{α\in Δ}f(D_{α})\subset f(\bigcup_{α\in Δ} D_{α})
$$
On the other hand, for any $y\in f(\bigcup_{α\in Δ}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Analytical closed solution for an integral I came across the following integral:
$$\int e^{-t^2}\cdot \text{erf}(a+b\cdot t)dt$$
Does anyone know whether it has a closed form? I have seen related solutions for the interval t=[0,inf] or for the case with a=0 with lower integration limit at t=0. A similar specific soluti... | It definitely does.
I am going to call $I_{a,b}$ the integral you defined, and assume that the error function is defined as usual (like here for instance). The first thing to notice is that $I_{0,1}=0$ because the error function is odd. Let us consider $I$ as a function of $a$ and try to differentiate. Namely, set
$$f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Rewriting summation formula I just watched a tutorial on recurrence by substitution. In the tutorial, it mentioned about rewriting
$\sum\limits_{i=1}^\mathbb{k}{2^i}$
as (2k+1 - 2). My question is can I generalize it as xlimit + 1 - x where x is the base.
| Nice observation and almost true:
$$
\sum\limits_{i=1}^\mathbb{n}{a^i} = \frac{a^{n+1}-a}{a-1}
$$
It's a special case of a geometric progression.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
An identity for the factorial function A friend of mine was doodling with numbers arranged somewhat reminiscent of Pascal's Triangle, where the first row was $ 1^{n-1} \ \ 2^{n-1} \ \cdots \ n^{n-1} $ and subsequent rows were computed by taking the difference of adjacent terms. He conjectured that the number we get at... | Oh, here's another way to look at it that I rather like. It's specific to the particular polynomial we're working with here.
So, let $a_0(n) = n^k$ and then $a_{d+1}(n)=a_d(n)-a_d(n-1)$; this question is about $a_k$. Here's the key observation: $a_d(n)$ counts $k$-tuples of numbers from $\{1,\dots,n\}$ that include at ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 8,
"answer_id": 6
} |
If $\phi$ is an endomorphism of an elliptic curve, and $\phi = \hat{\phi}$ then $\phi = [m]$? I heard a reference to this fact, but I cannot find a reference. (I can find the converse in Silverman, namely that $\hat{[m]} = [m]$.)
Notation: $[m]$ is multiplication by $m$ in the group law, and $\hat{\phi}$ is the dual en... | I think that if $E$ has CM this is false: take an endomorphism $\phi$ such that $\phi^2=[d]$, where $d$ is square-free. Then $\phi\circ\widehat{\phi}=[\deg\phi]=[d]$. Thus $\phi^2=\phi\circ\widehat{\phi}$, which implies $\phi=\widehat{\phi}$, but $\phi$ cannot be multiplication by an integer since it has non-square deg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $R$ is an integral domain, then $(R[x])^\times=R^\times$ If $R$ is an integral domain, then $(R[x])^\times=R^\times$
So since $R$ is an integral domain, it follows that $R[x]$ is an integral domain. We have $f(x)g(x)=1$ then we know that $\deg(f(x)g(x))= \deg(f(x))+\deg(g(x))=0$
Since neither can be equal to zero, ... | Your proof seems fine.
Using your result try to prove the following general result:
Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.
Try to prove this re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1512926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$d(x,y) = \sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2 }.$ $(\mathbb R^2,d)$ is a metric space . I am facing problem while solving its triangular inequality. In case of collinear points equality holds but what happen if points are not collinear.
| For three point $x,y,z \in \mathbb{R}^2$, you need to show that
$$d(x,y) \le d(x,z)+d(y,z)$$
Applying your metric, gives us
$$\sqrt{(x_1-x_2)^2+(y_1-y_2)^2} \le \sqrt{(x_1-x_2)^2+(z_1-z_2)^2} + \sqrt{(y_1-y_2)^2+(z_1-z_2)^2}$$
Let $\Delta x := x_1-x_2$, $\Delta y := y_1-y_2$ and $\Delta z := z_1-z_2$. We need to prove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Are there numbers $x, y \in \mathbb{Q}$ such that $\tan(x) + \tan(y) \in \mathbb{Q}$? My question is related to the following one: Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?
It was shown that the above set is not closed under addition using the Lindemann-Weierstrass theorem. That tells us that $\tan(x) ... | Let $x=\dfrac{p}{q}$ and $y=\dfrac{r}{s}$ and $z=\dfrac{1}{qs}$.
We observe that tan x and tan y are polynomials in tan z; further tan x + tan y is a non-constant polynomial if $x+y \neq 0$. Here some calculation should show that tan $A\theta$ + tan $B\theta$ is a non-constant polynomial for $A + B \neq 0$.
Hence we ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Probability of a machine with two switches failing A certain type of switch has a probability p of working when it is turned on. Machine A, which has two of these switches, needs both such switches to be switched on to work, and machine B (also with two of the same switch) only needs at least one of them switched on to... | Let $ q=1-p $ be the probability of a switch failing and let $ P(n) $ be the probability that, for a given machine, n switches succeed. For machine A to fail, you need to find $ P_a( \leq 1)=P_a(1)+P_a(0) $ because as long as one switches fails, the whole thing does. For machine B to fail, you need $ P_b(0) $ as both s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Fisher Information for Exponential RV Let $X \sim exp(\lambda_0)$; i.e, an exponential random variable with true parameter $\lambda_0 > 0$. The density is then $f(x;\lambda_0) = \lambda_0 e^{-\lambda_0 x}$. For a given $\lambda > 0$, the Fisher information is defined as
\begin{align*}
I(\lambda) & := E\left( \left(\f... | You're right to say that the actual realization of the random variable $X$ does not affect the (true and unknown since it does depend on the true parameter) Fisher information since in the definition we integrate over the density of $X$.
The Fisher information is the 2nd moment of the MLE score. Intuitively, it gives a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to show uniform convergence of this series of functions I was working the following question from a previous qualifying exam, the solution below seems to have a minor snag in it below
Let $f_{1}: [a,b] \rightarrow \mathbb{R}$ be Riemann integrable function. Define the sequence of functions $f_{n}: [a,b] \rightarro... | Your original work has been correct, but as you noticed, obtaining a global bound for $f_{n+1}$ from a global bound for $f_n$ via
$$\lvert f_{n+1}(x)\rvert = \biggl\lvert \int_a^x f_n(t)\,dt\biggr\rvert \leqslant \int_a^x \lVert f_n\rVert_{\infty}\,dt = \lVert f_n\rVert_{\infty}\cdot (x-a) \leqslant \lVert f_n\rVert_{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Choose $3n$ points on a circle, show that there are two diametrically opposite point
On a circle of length $6n$, we choose $3n$ points such that they split the circle into $n$ arcs of length $1$, $n$ arcs of length $2$, $n$ arcs of length $3$. Show that there exists two chosen points which are diametrically opposite.
... | Considering the $6n$ evenly spaces points on the circle, call a point exterior if it is one of the chosen points which splits the circle and interior if it is in the interior of one of the arcs. Let the dual splitting of the circle be the one generated by changing every interior point to exterior and vice versa. If no ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Exchangeability of union and intersection of open balls around all rational numbers in $[0,1]$ Let $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For $n,k \ge1$ set $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $.
Is it true that $$ \bigcup_{n\ge1} \bigcap_{k\ge1} I_{n,k} = \bigcap_{k\ge1} \bigcup_{n\ge1}... | You're wrong, the left side is not included in the right side.
Since $2^{-(n+k)}$ is decreasing in $k$, $$\bigcup_{k\ge 1} I_{n,k} = I_{n,1} = X \cap (v_n - 2^{-n-1},v_n + 2^{-n-1})$$
Take some $v_n$ that is near $0$ and $v_m$ that is near $1$ and you'll
have $I_{n,1} \cap I_{m,1} = \emptyset$. So the right side is e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is the natural exponential function defined as being its own derivative? Is $e^x$ actually defined as being the function $f$ for which $\dfrac{d}{dx}f=f$?
By which I mean not "does the identity hold", of course I know it does and that this definition is sufficient for $e$, but did Euler actually sit down and think "gee... | Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa around 1690 studied the question of how to compute areas under the hyperbola $xy=1$, which led to the notion of how to compute the area under the curve $y=1/x$. While the case $y=1/x^n$, $n > 1$ was simpler and solved by Cavalieri earlier, a new function had to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
What to do with the $y$ term when solving $yu_{y} + uu_{x} =u-y$ I have an initial value problem that looks like this
$$yu_{y} + uu_{x} =u-y; \qquad u(x,1)=x$$
I think I can just use the method of characteristics to solve it, which I would be perfectly capable of doing, except I am not sure how to treat the $-y$
Perha... | The equation of the characteristics is
$$
\frac{dy}{y}=\frac{dx}{u}=\frac{du}{u-y}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1513869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $ \int_0^{\pi}f(x)\cos(nx)\,dx=0$ for all $n$ then prove that $f\equiv 0$
If $f:[0,\pi]\to \mathbb R$ is continuous and $f(0)=0$ such that $\displaystyle \int_0^{\pi}f(x)\cos(nx)\,dx=0$ for all $n=0,1,2,\cdots$ then prove that $f\equiv 0$ in $[0,\pi]$.
I want to apply Weierstrass approximation theorem. As $f$ is c... | (Complete revamp of my previous answer.)
Extend $f$ by $f(x) = f(-x)$ for $x \in [-\pi,0)$, then $f$ is $2\pi$ periodic
and so for any $\epsilon>0$ there is a trigonometric polynomial $q$ such that $\|f-q\|_\infty = \sup_{|x|\le \pi} |f(x)-q(x)| < \epsilon$.
(Recall that a trigonometric polynomial is of the form
$q(t) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
Solve the equation $1-\tan x + \tan^2 x - \tan^3 x + ... = \frac{\tan 2x}{1+\tan2x}$ How to solve this? Any advice?
$$1-\tan x + \tan^2 x - \tan^3 x + ... = \frac{\tan 2x}{1+\tan2x}$$
Next step I do this
$\sum\limits_{n=0}^\mathbb{\infty}(-1)^n \tan^nx = \frac{\tan 2x}{1+\tan2x} $
But I don't know next step. I am cule... | Notice, for the sum of infinite series on LHS, $|\tan x|<1$
$$1-\tan x+\tan^2 x-\tan^3 x+\ldots =\frac{\tan 2x}{1+\tan 2x}$$
$$\frac{1}{1-(-\tan x)}=\frac{\frac{2\tan x}{1-\tan^ 2x}}{1+\frac{2\tan x}{1-\tan^ 2x}}$$
$$\frac{1}{1+\tan x}=\frac{2\tan x}{1-\tan^2 x+2\tan x}$$
$$1-\tan^2 x+2\tan x=2\tan x+2\tan^2 x$$
$$3\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Intuition behind alternate expression of impulse train So it's known that $\sum_n \delta(x-nT) = \frac{1}{T}\sum_m e^{2\pi imx/T}$. This can be proven by expressing the left hand as a Fourier series and finding $c_m$. But it's just mindboggling that this is the case, is it not? Why would one expect this to happen?
If I... | just play with this interactive Desmos Graph and see.
The point is constructive and destructive interferences : every cos contributes around the Dirac comb support or progressively cancel each other at other places, resulting into the Dirac comb.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Complex line integral - where the length of the complex number is given I'm trying to study ahead of class so I alleviate some upcoming stress in the semester. I have currently read the chapters on line integrals of complex numbers and am currently trying to deepen the content with some problems given in the workbook.
... | I'll consider the integral $\oint_{|z|=1}\frac{dz}{z(z+2)}$. The second one can be done in the similar way (but you need some trick there). I will use Cauchy Integral Theorem (CIT) then.
From CIT we know that if some function $f$ is holomorphic on an open subset $U$ containing a closed disk $D$, then
$$f(a)=\frac{1}{2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$
Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$
I start with the median $AM$. Since $\angle BAM, \angle CAM$ are known I can construct them. So I have point $A$, line segment $AM$ an... | The symmetric of $B$ with respect to $M$ is $C$, hence in order to find our triangle it is enough to intersect the $AC$ line with the symmetric of the $AB$ line with respect to $M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding the derivative of $(\frac{a+x}{a-x})^{\frac{3}{2}}$ This is a very simple problem, but I am stuck on one step:
Differentiate $(\frac{a+x}{a-x})^{\frac{3}{2}}$
Now, this is what I have done:
$$
(\frac{a+x}{a-x})^{\frac{3}{2}} \\
\implies \frac{\delta}{\delta y}\frac{f}{g} \\
\implies gf' = (a-x)^{\frac{3}{2}} \t... | i think the right answer is this here
$$\frac{3}{2} \sqrt{\frac{a+x}{a-x}}
\left(\frac{a+x}{(a-x)^2}+\frac{1}{a-x}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.