Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How can I solve $(xy^3 + y)dx + 2(x^2y^2 + x + y^4)dy = 0$?
Solve this differential equation
$$(xy^3 + y)dx + 2(x^2y^2 + x + y^4)dy = 0$$
I tried converting it to the form: $\frac{dy}{dx} + yp(x) = q(x)$ but couldn't. The equation is also not homogeneous. Keeping $\frac{dy}{dx}$ on one side will not render the nume... | By multiplying both sides by $y$ (see Moo's comment) we have that
$$0=y(xy^3 + y)dx + 2y(x^2y^2 + x + y^4)dy=d\left(\frac{y^2(2y^4+3x^2y^2+6x)}{6}\right).$$
Hence
$$y^2(2y^4+3x^2y^2+6x)=C$$
where $C$ is an arbitrary constant. Have you any initial condition?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Transformation that rotates eigenvalues Let $A \in \mathbb{R}^{n \times n}$ be a square matrix.
Is there a transformation $T_{\theta}: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$, not necessarily linear, that rotates the eigenvalues by an angle $\theta$ on the complex plane?
In other words, for each e... | In general, you can't. The reason is that you want to keep real entries, hence a characteristic polynomial with real coefficients, and the complex roots of such a polynomial have to come in conjugate pairs. If you rotate the eigenvalues of $A$ arbitrarily, you lose this symmetry property and the set that you obtain is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proof to go broke almost surely Suppose you play a game where you start with capital $K_0 = 1$. In each turn $i = 1,2, \dots, n$ you throw a fair coin independently of the history. If you get "heads", you get back $3/2$ of your capital, if you get "tails" you only get back $1/2$ of your capital.
Prove that $K_n \to 0$ ... | So, firstly I will explain the error in your approach via the strong law, and the correct way in which the Strong Law does suggest this to be true.
However, I think you still need to be careful with this approach, so then present an argument via Martingales.
Intuition via the strong law
Your intuition to take logarithm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A subgroup $H$ of $G$ is normal iff for all $a,b \in G$, $ab \in H \iff ba \in H$. I need to show the following:
Let $H$ be a subgroup of $G$. $H$ is normal iff it has the following
property: for all $a,b \in G$, $ab \in H$ iff $ba \in H$.
I have to use the following definition of a normal subgroup:
Let $H$ be a s... | A different way to see it: $ab\in H$ if and only if $a$ and $b^{-1}$ are in the same right coset of $H$. $ba\in H$ if and only if $b$ and $a^{-1}$ are in the same right coset of $H$, which is equivalent to saying that $a$ and $b^{-1}$ are in the same left coset of $H$.
Thus the property under consideration means exactl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
How many ways can a number $k$ number be written as a sum of $1$s and $2$s? The order of the numbers does matter, which means that for $k=4$ $4= 1+1+1+1,1+1+2,1+2+1,2+1+1,2+2$ I have calculated it until $k=6$ and I get that the number of the valid calculations are:$1,2,3,5,8,13$. I assume that this is about the Fibonac... | We can prove this using direct proof. First, we let $Q(k)$ be the number of ways can a number $k$ be written as a sum of 1s and 2s.
Hypothesis: For Fibbonaci number $F_1=1, F_2=1, F_{n}=F_{n-1}+F_{n-2}$, $Q(k)=F_{k+1}$
We say that we have constructed every order of $Q(n)$. Then, we construct $Q(n+1)$ this way:
*
*Fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $k_i = \{x \in \mathbb N, m\in \mathbb N: x=im\}$ for $i \in \mathbb N$ and $i>1$. Is the $\cap_{i=1}^{\infty} k_i$ empty? I want to give a counter example to the statement saying that:
Given a collection of closed (not necessarily bounded) sets where the finite intersection of any of these sets is nonempty, the i... | There is no number that is a multiple of every integer. For example, $n$ can not be a multiple of $n+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $V$ is isomorphic to $W \times V/W$. I have been asked to prove the following:
Let $V$ be a finite dimensional vector space of a field $K$ and $W$ be a subspace of $V$. Prove that $V$ is isomorphic to $W \times V/W$ (the direct product of $W$ and $V/W$).
Here is my proof thus far:
Define $\pi: V \rightarrow ... | Hint: consider a basis $\{w_1,\dots,w_m\}$ of $W$ and extend it to a basis
$$
\{w_1,\dots,w_m,v_1,\dots,v_n\}
$$
of $V$. Now note that $\{[v_1],\dots,[v_n]\}$ is a basis for $V/W$.
Some more details. A vector $v\in V$ can be uniquely written as
$$
v=a_1w_1+\dots+a_mw_m+b_1v_1+\dots+b_nv_n
$$
Define $f(v)= a_1w_1+\dots+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding first return time in an infinite markov chain
I have an infinite markov chain that looks like this. I need to show that the chain is recurrent by computing the first return time to 0 for the chain that started at 0. Intuitively to me, this makes sense because any state will eventually return to state 0. Howeve... | To figure out how recurrent this Markov chain isn't, you'll probably want to know two things:
*
*The probability that, starting at $0$, you'll ever return to $0$.
*The expected number of steps it takes to return to $0$.
In this Markov chain, it's very clear what the path has to be if we never return to $0$, and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Example of a multivariable nondifferentiable function with directional derivatives. Please give an example of a continuous function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ all of whose directional derivatives exist, so for every $\mathbf{v}\in\mathbb{R}^2$
$$df_{\mathbf{x}}(\mathbf{v})=\lim\limits_{t\to 0}\frac{f(\mathb... | When $f((r\cos\ t,r\sin\ t)) = r \sin\ (2t)$ where $r>0$ and $0\leq
t<2\pi$ and $f(0,0)=0$, then $$ df\ (\cos\ t,\sin\ t)=\lim_r\ \frac{
f((r\cos\ t,r\sin\ t)) -0 }{r} =\sin\ (2t) $$
Hence all directional derivatives exist. Here assume that $f$ is
differentiable So
$$ df\ -(\cos\ t,\sin\ t) =df\ (\cos\ (\pi +t),\sin\ (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Equivalent formulations of Hilbert's Nullstellensatz I'm trying to figure out this proof. Can anyone help out?
We know the following statement of the theorem:
For any idea $I \subset \mathbb{C} [X_1,\ldots,X_n],\mathbb{I}(\mathbb{V}(I)) = \sqrt{I}$
We want to show that the above statement implies this one:
Let $k$ be a... | Note that for an ideal $I$ in a commutative ring, $\sqrt{I}=(1) \Rightarrow I=(1)$ (because if some power of an element is a unit, it must be a unit itself), this implies that in your situation $\Bbb{I}(\Bbb{V}(F_1,\ldots,F_m)) \neq (1)$, so that $\Bbb{V}(F_1, \ldots, F_m) \neq \varnothing$, because $\Bbb{I}(\varnothin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Gradient of $X \mapsto \mbox{Tr}(AX)$ I know that the gradient of $X \mapsto \mbox{Tr}(XA)$ is $A^T$. However, how does this change if we had a scenario where $A$ and $X$ are swapped. Is the gradient $X \mapsto \mbox{Tr}(AX)$ the same?
Also, how does this extend if we have more matrices? We can just assume everything... | Theorem: ${\mathrm{d} f({X})= \text{trace}(M^T \mathrm{d} {X}) \iff \frac{\partial f}{\partial {X}} = M}$
In your case,
$$\mathrm d \ \text{trace}(AXB) = \text{trace}(\mathrm d (AX B)) = \text{trace}(A \ \mathrm d X\ B) = \text{trace}(B A \ \mathrm d X)$$
and thus we identify $(BA)^T = A^T B^T$ as the derivative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Showing Dirichlet Integral exists The problem I'm trying to do is the following, Spivak Chapter 19 Problem 43.
Problem:
a) Use integration by parts to show that
$$\int_a^b \frac{\sin x}{x}d{x}=\frac{\cos a}{a}-\frac{\cos b}{b} -\int_a^b\frac{\cos x}{x^2}d{x}$$
and conclude that $\int_0^\infty \frac{\sin x}{x}$ exists. ... | For (a), notice that $\lim_{x\to 0}\frac{\sin(x)}{x}=0$ implies that $\frac{\sin(x)}{x}$ is continuous and bounded in $(0,a]$ and therefore it is integrable in $[0,a]$.
In $[a,+\infty)$, with $a>0$,
$$\int_a^{+\infty} \frac{\sin x}{x}d{x}=\frac{\cos a}{a}-\lim_{b\to +\infty}\frac{\cos b}{b} -\int_a^{+\infty}\frac{\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Maximal length of a finite real sequence Suppose that $(a_n)_{1 \le n \le N}$ is a finite sequence of reals such that the sum of any 7 consecutive terms is (strictly) negative and the sum of any 11 consecutive terms is (strictly) positive.
What is the maximal length of this finite sequence of reals?
I tried creating fi... | $x_1+\dots+x_7<0$ and $x_8+\dots+x_{14}<0$, so $x_1+\dots+x_{14}<0$. But $x_4+\dots+x_{14}>0$, so $x_1+x_2+x_3<0$.
$x_5+\dots+x_{11}<0$ and $x_1+\dots+x_{11}>0$, so $x_4>0$. From there is it easy to get a contradiction if you have 17 terms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Interesting inequalities that can be derived from Cauchy-Schwarz The way I learned it, the scalar product is defined the following way:
If $V$ is a vector space, the scalar product is a function $B:V \times V \to \mathbb{R}$ satisfying for all $u,v,w \in V$ and for all $\lambda \in \mathbb{R}$
1) $B(u,v)=B(v,u)$
2) $B(... | A standard kind of example is to consider the linear space of random variables $X$ such that $\mathbf{E}(|X|^2) < \infty$, modulo a.e. zero functions. This has an inner product
$$
\langle X,Y\rangle = \mathbf{E}(XY)
$$
Cauchy-Schwarz in this case implies that
$$
\operatorname{Cov}(X,Y) \le \operatorname{Var}(X)\oper... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2656812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
First part of the restriction map $H_{j}(G,A)\to H_{j}(H,A)$ on group homology Let $G$ be a group, $H$ be a subgroup of finite index and $A$ be a $G$-module.
I'm trying to understand the restriction map on homology $H_{j}(G,A)\to H_{j}(H,A)$ explicitly.
The first step would be to introduce a map $H_{j}(G,A)\to H_{j}(G,... | The morphism $\tau(a) = \sum_{x\in G/H}x\otimes_H x^{-1}a$ is independent of the choice of representatives $\{x \in G/H\}$ of $G/H$. Clearly if we have chosen another system of representatives $\{x'\in G/H\}$ then $x = x'h$ for some $h \in H$ and we have $\tau(a) = \sum_{x'\in G/H}x'\otimes_H x'^{-1}a = \sum_{x\in G/H}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove a function is differentiable with chain rule Let $A$ be an open set of $\mathbb{R}^n$ and let $f : A \rightarrow \mathbb{R}^m$. Fix $u \in \mathbb{R}^m$ and define $g : A \rightarrow \mathbb{R}$ by $g(x) = f(x) \bullet u$ for all $x \in A$, where $\bullet$ is the inner product. Show that if $f$ is differentiable ... | Your formula $g'(a) = f'(a) \bullet u$ is a little bit off, since $f'(a)$ is a linear map, and $u$ is a vector.
Since $h(y):=u\cdot y$ is linear we have
$$\eqalign{g(a+X)-g(a)&=u\cdot\bigl(f(a+X)\bigr)-u\cdot\bigl(f(a)\bigr)=u\cdot\bigl(f(a+X)-f(a)\bigr)\cr &=u\cdot\bigl(df(a).X+o(|X|)\bigr)=u\cdot\bigl(df(a).X\bigr)+o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Where did I go wrong in my approach? Consider the two inequalities: $y \le-1$ or $ y\ge 2$
My approach:
$y \le-1$ or $ y\ge 2$
iff $y+1\le0$ or $y-2 \ge 0$
iff $(y+1)(y-2)\le0$
On solving: $(y+1)(y-2)\le0$
I got $-1\le y\le2$
Where did I do wrong?
| The mistake has been pointed out in the comment.
Now the fix.
Case $1$: If $y+1 \le 0$ , then $y-2\le 0$, hence $(y+1)(y-1) \ge 0$.
Case $2$: if $y-2 \ge 0$, then $y+1 \ge 0$, hence $(y+1)(y-1) \ge 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Biggest $(-t,t)\subset A-A$ if $λ(Α)>0$ I am trying to solve a problem which states that if $λ(E)>1$ ($λ$ being the Lebesgue measure in $\mathbb{R}$) then there exists $x,y \in E$ such that $x\neq y $ and $ x-y \in \mathbb{Z}$.
From the proof of the Steinhaus Lemma we can see that $λ(Ε)>0\Rightarrow (-\frac{λ(Ε)}{2},... | It seems clear that the result you state is sharp. Let $E=(0,1)$; then $\lambda(E)=1$ but there do not exist $x,y\in E$ with $x\ne y$ and $x-y\in\Bbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to calculate $ \lim_{\alpha \to 1} (I-\alpha A)^{-1}( I- A)$? Suppose that for all $\alpha \in (0,1)$, the matrix $\mathbf I-\alpha \mathbf A$ is invertible, but that $\det(\mathbf I-\mathbf A)=0$. How do I calculate
$$ \lim_{\alpha \to 1} (\mathbf I-\alpha \mathbf A)^{-1}(\mathbf I- \mathbf A)?$$
| In the special case where $A_{n \times n}$ is a real symmetric matrix the limit can be explicitly computed in terms of the spectral decomposition of $A$.
The condition $\det(I-A)=0$ ensures $1$ is an eigenvalue of $A$.
The condition $\det(I-\alpha A) \neq 0$ for $\alpha \in (0,1)$ implies $\det(\frac{1}{\alpha}I -A) \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Value of $k$ such that $f(x) = kx$ has solutions I have two functions $ f(x) = \exp(x^2)$ and $g(x) = kx$. I need to find values of $k > 0$ such that $f(x)=g(x)$ has solutions.
Here is what I did :
*
*If $k < \exp(1)$, these two functions do not touch so there is no solution;
*If $k = \exp(1)$, these functions touc... | Let's start with the 2 functions:$$f(x)=\exp(x^2)\\g(x)=kx$$ Notice that for $k>0,x\le 0$ $f(x)\ne g(x)$, so we left only with positive $x$.
Let's start with no solutions, if we have no solutions and $f(0)>g(0)$ it means that $f(x)>g(x)$ for all $x$(by the intermediate value theorem).
So we search for some $k$ such tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Let $f(x)$ be a differential real function defined on the real line. If $f(0)=0$ and $f'(x)=[f(x)]^2$, then $f(x)=0$ foi any $x$. Again, $f:\mathbb{R}\to\mathbb{R}$ is differentiable, $f(0)=0$, and $f'(x)=[f(x)]^2$ for every $x$. A friend suggested the following argument:
If exists $c$ such that $f(c)\neq0$, there exis... | Say $f_1$ is a solution of the differential equation. Let $J \supseteq \{0\}$ be an interval of maximal length with $f_1|_{J}=0$. By continuity, it is closed.
Assume $b=\sup J < \infty$. Then, by Picard-Lindelöf there exists an open interval containing $b$, say $\tilde J$, such that $f_1$ agrees with the 0-function on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
If $n$ is a multiple of $8$, then the number of sets of size divisible by $4$ is $2^{n-2} + 2^{(n-2)/2}$ Question in Cameron, Combinatorics:
If $n$ is a multiple of $8$, then the number of sets of size divisible by $4$ is $2^{n-2} + 2^{(n-2)/2}$
We are given a property derived from the Binomial theorem: For $n > 0$ ,... | Let $A_2$ be the number of subsets with size divisible by 2, and $A_4$ the number with size divisible by 4. Let $B$ be the number with size divisible by 2 but not 4. You have established that $A_2=A_4+B=2^{n-1}$.
Using the binomial theorem we have $$2^{n/2}=(1+i)^{n}=\sum_{k=0}^n{n\choose k}i^k=\sum_{k=0\bmod4}+\sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Can someone help me through the steps of how to do this problem? I find it quite difficult. This is a pretty hard inequality word problem for me. Help regarding what steps I need to take to solve this problem is greatly appreciated.
Roberto plans to start a new job. In preparation, he decides that he should spend no m... | Let $x$ and $y$ be the number of hours spent on job and homework respectively. Now we have, $x+y\le 30$ and $2x\le y$ adding both we get $3x+y\le 30+y\implies x\le 10$. Hence maximum hours he should spend at his job is $10$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2658103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Conditional Probability (The cookie Poroblem) Gentlemen,
I have small confusion in finding donditional probability in the "Cookies Problem" describe below:
Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at ran... | When we say something is “given,” we can say that we already observed it. In this case, we already observed you choosing bowl 1, so we don’t need to consider the probability of you choosing it. So if I asked you, what’s the probability that you pull the plain cookie GIVEN that you absolutely must pull from bowl 1, then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2658238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Solve the trigonometric equation: $\cos (3x)-\sin(x)=\sqrt 3(\cos (x)-\sin(3x))$
Solve the trigonometric equation:
$$\cos (3x)-\sin(x)=\sqrt 3(\cos (x)-\sin(3x))$$
My answer is contradictory to Wolfram Alpha.
Because, W.A. gives me:
$x = \pi n - \frac {11 \pi}{12}, n \in \mathbb{ Z}$
$x = \pi n - \frac {7 \pi}{12... | $$\dfrac\pi{12}+\pi k=\pi n-\dfrac{11\pi}{12}$$
$$\iff k=n-1$$
Now for odd $k,k=2m+1$(say)
$\dfrac\pi8+\dfrac{\pi k}2=\dfrac\pi8+\dfrac{\pi(2m+1)}2=m\pi+\dfrac{5\pi}8=(m+1)\pi-\dfrac{3\pi}8$
For even $k,k=2m$(say), $\dfrac\pi8+\dfrac{\pi k}2=m\pi+\dfrac\pi8=(m+1)\pi-\dfrac{7\pi}8$
So, there must be mistake in the W.A. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2658354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What we can say about these two elements $x= yxy^{-1} $? Let $x$ and $y$ are two elements of some group with relation $x = yxy^{-1}$. What we can say about $x$ and $y$?
I can say that $x$ and $y$ commute because $xy = yx$. What are other things we can say about $x$ and $y$? I think we can't say that $y$ is self-conjuga... | You can say anything that follows just from the fact that they commute, and nothing else without more information.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2658531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Solution of Differential equation as an integral equation I was looking for the solution of the following problem.
Prove that if $\phi$ is a solution of the integral equation
$$y(t) = e^{it} + \alpha \int\limits_{t}^\infty \sin(t-\xi)\frac{y(\xi)}{\xi^2}d\xi,$$
then $\phi$ satisfies the differential equation
$$y'' + (1... | Take the derivative of $y$ twice using the equation,
$$y(t) = e^{it} + \alpha\int\limits_{t}^\infty \sin(t-\xi)\frac{y(\xi)}{\xi^2}d\xi$$
Then plug what you get into
$$y''+(1+\frac{\alpha}{t^2})y$$
and simplify to get $0$ and thus show the two are equivalent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2658660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Inducing differentiable structure via continuous maps Consider a map $f: (X,\mathcal{O}_X) \to Y$, with the domain being a topological space with topology $\mathcal{O}_X$ and the codomain merely a set $Y$. We can induce many topologies on $Y$, however, the most natural one is the finest (or coarsest, I can never remem... | That's a neat idea, but I don't think it can work as stated. Here's why:
Think of $f$ as defining an equivalence relation on $M$ where $a\sim b$ whenever $f(a)=f(b)$. Then the coarsest topology on $X$ in which $f$ is continuous will be the quotient topology $M/\sim$, for which $f$ will be the quotient map (which is tru... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2658903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\sum_{p\leq n}\sum_{q\leq N}\sum_{n\leq N;p|n,q|n}1=^? \sum_{pq \leq N}\big( \frac N{pq}+O(1)\big)+\sum_{p \leq N}\big( \frac N{p}-\frac N{p^2}\big)$ I was studying Marius Overholt 'A course in Analytic Number Theory'. There in the section of "Normal order method". The proposition he is going to prove is
$Var[w]=O(l... | Split into two cases: $p \neq q$ and $p = q$.
If $p \neq q$, $p|N$, $q|N$ give you $\lfloor \frac N{pq}\rfloor = \frac N{pq} + O(1)$ numbers satisfying the condition $n \leq N$, $p | n$, $q | n$. As long as $pq \leq N$ you will get at least one number and thus should be included in your summand. Note that $pq$ and $qp$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Generalization of convexity This is not a homework, this is just something that came to my mind recently. Assume $f$ is a sufficiently nice function.
We know that
$$\frac{df}{dx} \geq 0 \iff f(x_2) \geq f(x_1) \text{ for } x_2 \geq x_1$$
$$\frac{d^2f}{dx^2} \geq 0 \iff \frac{f(x_1) + f(x_2)}{2} \geq f(\frac{x_1 + x_2}{... | I think we can play with the Schwarzian derivative (we work on $[0;\infty[$):
We have :
$$(Sf)(x)=\frac{f'''(x)}{f'(x)}-1.5(\frac{f''(x)}{f'(x)})^2$$
Here we assume that the Schwarzian is positive so :
$$\frac{f'''(x)}{f'(x)}\geq1.5(\frac{f''(x)}{f'(x)})^2$$
We can rewrite the inequaliy like this :
$$\frac{f'''(x)}{f''... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Basis for Matrix A
Find a basis for all $2\times2$ matrices $A$ for which
$A\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ = $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.
Maybe I'm dumb-- but isn't $A$ just the $0$ matrix? In which case, the base is simply the $0$ matrix as well?
| Your guess is very intuitive, but let's check this rigorously:
$$\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}1&1\\1&1\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$$
and we end up with a system of equations $$\begin{cases}a+b=0\\a+b=0\\c+d=0\\c+d=0\end{cases}$$
Do you think you can find a basis for the solut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How many positive integer solutions satisfy the condition $y_1+y_2+y_3+y_4 < 100$ In preparation for an upcoming test I have come across the following problem and I am looking for some help with it just in case a question of its kind comes up on a evaluation. Thanks!
How many positive integer solutions satisfy the con... | The problem is equivalent to put 4 separator between the 99 inner intervals between the 100 objects in such way that, for example starting from the left
*
*$y_1>1$ first group of objects
*$y_2>1$ second group of objects
*$y_3>1$ third group of objects
*$y_4>1$ fourth group of objects
*$y_5>1$ fifth group of obje... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
$f(x, \theta)= \frac{\theta}{x^2}$ with $x\geq\theta$ and $\theta>0$, find the MLE Let $X$ be a random variable with density $$f(x, \theta)= \frac{\theta}{x^2}$$ with $x\geq\theta$ and $\theta>0$.
a) Show if $S=\min\{x_1,\cdots, x_n\}$ is a sufficient statistics and if it is minimal.
b) Find the Maximum Likelihood Esti... | Comment. This is for intuition only. It seems your conversation with @V.Vancak (+1) has taken care of (a).
Hints for the rest: Using the 'quantile' (inverse CDF) method, an observation $X$ from your Pareto distribution can be simulated as
$X = \theta/U,$ where $U$ is standard uniform.
In the simulation let $\theta = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Do singular distributions have any real-world applications? Do singular probability distributions have any real-world applications, or are they just a pure-mathematical curiosity? I can't imagine a real quantity that they would describe. But on the other hand, singular functions do have surprising applications, e.g. re... | Here is one application that might interest you. Consider an integrable function f on $\mathbb R$ such that $2f(x)=3f(3x)+3f(3x-1)$ a.e.. It turns out that $f=0$ a.e.. There seems to be no simple way of doing this even if you assume that f is smooth function. There is an elegant proof by relating it to cantor set and a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Are d1 and lift metric equivalent distances? I need help proving that $d1\not\equiv d$, where d and d1 are defined as follows:
\begin{equation}
\label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande}
d(x,y) = \left\{
\begin{array}{ll}
|x_{2}-y_{2}| & \mathrm{if \ }x_{1}=y_{1} \\
|x_{2}|+|x... | $D=\{0\}\times \{y:1<y<3\}$ is the open $d$-ball of radius $1$ centered at the point $(0,2).$ It is not open in the $d_1$-metric.
For any $p=(x,y)\in \Bbb R^2$ and any $r>0$ the open $d_1$-ball $B_{d_1}(p,r)$ contains points $(x',y')$ with $x'\ne x. \;$ E.g. $(x+r/2,y)\in B_{d_1}(p,r). $ So no non-empty open $d_1$-b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find the range of $\angle A$ in a triangle. In a triangle $ABC$ line joining circumcenter and orthocenter is paralel to line $BC$. Find the range of $\angle A$
Solution i try:
From figure $HD =2R\cos B\cos C= OE$ and $\displaystyle \angle COE =A$
$ 2R\cos B\cos C=R\cos A$ here $R$ is circumradius of circle
$2\cos B \... | As $C=\pi-A-B$, you have: $\cos C=-\cos(A+B)=\cos A\cos B-\sin A\sin B$. Substituting that into your equation $-2\cos B\cos C+\cos A=0$ gives:
$$
\cos A(1+\cos^2B)-2\sin A\sin B\cos B=0,
$$
that is:
$$
\cos A(2+\cos 2B)-\sin A\sin 2B=0,
$$
or:
$$
\tan A ={2+\cos 2B\over \sin 2B}.
$$
Notice that we can choose $B$ such t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Basic confusion regarding definition of relative homology group $H_p(K;L)$ I am currently self-studying "Topology and Geometry for Physicists" by Nash and Sen, and have encountered a confusion regarding the definition of the relative homology group $H_p(K;L)$. The definition given is as follows :
The relative $p$-dimen... | As you say, this statement is incorrect. What I would guess is actually meant is that the elements of $Z_p(K;L)$ are of the form $z_p+C_p(L)$ (where $z_p\in C_p(K)$), since $Z_p(K;L)\subseteq C_p(K;L)=C_p(K)/C_p(L)$. An element of $H_p(K;L)$ is then represented by an element of $Z_p(K;L)$ which has this form, modulo ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Schwartz kernel theorem and order of distribution Let $\mathcal{S}(\mathbb{R}^N)$ be a space of Schwartz functions and let $T: \mathcal{S}(\mathbb{R}^N)\times \mathcal{S}(\mathbb{R}^M) \to \mathbb{C}$ be a bilinear functional such that there exist $c_1,c_2>0$ and for every $g\in\mathcal{S}(\mathbb{R}^M)$ it holds
$$\f... | A small complement to Jochen's answer. Separate continuity implies full continuity, essentially by the uniform boundedness principle. This then implies the existence of seminorms $||\cdot||_{E_1}$ and $||\cdot||_{E_2}$ so that
$$
|T(f,g)|\le cst\ ||f||_{E_1} ||g||_{E_2}
$$
What is impossible in general is to have the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If there is a point $p \in M$ such that $f(p) = g(p)$ and $df_p = dg_p$ then $f = g$. Let $M,N$ Riemannian manifold conected and $f,g:M \rightarrow N$ two isometries. If there is a point $p \in M$ such that $f(p) = g(p)$ and $df_p = dg_p$ then $f = g$.
Comments:
I'm considering the set $A = \{q \in M ; f(q) = g(q) \ ... | Consider $x\in A$, there exists a neigborhood $U$ of $A$ such that $y\in U$ implies that $y=\exp_x(v), v\in T_xM$, $f(\exp_x(v))=\exp_{f(x)}(df_x(v))=\exp_{g(x)}(dg_x(v))$ since $f$ and $g$ are isometries, where $\exp_x(v)$ is defined as follows: let $c(t)$ be the geodesic such that $c(0)=x$ and $c'(0)=v, \exp_x(v)=c(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Bit strings with at most two consecutive identical digits I am trying to develop a recurrence relation for $T(n)$, the number of bit strings of length $n$ with a maximum of two consecutive $0$s or $1$s
I find the recurrence relation
$$T(n) = 2T(n-1) + T(n-2)$$
Is this right?
| Consider the following graph:
For each binary string we start from the blue state (empty string) and move along the graph according to the next character (I didn't mark them to the arrows but it should be obvious which is which, anyway it doesn't matter for the calculation).
Consider the walks of length $n$ along the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Projectile: $v^*w^*=gk$ for minimum launch velocity A projectile launched from $O(0,0)$ at velocity $v$ and launch angle $\theta$, passes through $P(k,h)$. The velocity of the projectile at $P$ is $w$. The slope of $OP$ is $\alpha$, i.e. $\tan\alpha=\frac hk$, and the length of $OP$ is $R$.
$\hspace{4cm}$
Let $v^*$ b... | Because $w_x=v_x$ and $v_y^2=w_y^2-2gh$, the minimum speed in the origin implies you arrive with the minimum possible speed to point $(k,h)$.
I think the following picture shows a symmetry which could be useful. The minimum possible speed to reach $(k,h)$ is attained when you throw the projectile in the direction bisec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Show that the function $u(x)=\arctan(\frac{1}{x})$ is in $L^p(0,+\infty)$ $ \ \forall p>1$ I need to show that $||u(x)||_{L^p}$ is finite.
$$||u(x)||_{L^p}^p=\int_0^{+\infty}|u(x)|^pdx=\int_0^{+\infty}|\arctan\bigg(\frac{1}{x}\bigg)|^pdx=\int_0^{+\infty}\arctan^p\bigg(\frac{1}{x}\bigg)dx$$
At this point I thought a sub... | $$
\text{arctan}^p\left(\frac{1}{x}\right) \underset{(+\infty)}{\sim}\frac{1}{x^p}
$$
and the function $\displaystyle x \mapsto \frac{1}{x^p}$ is integrable on $\left[1,+\infty\right[$. Then
$$
\text{arctan}^p\left(\frac{1}{x}\right) \underset{(0^{+})}{\sim}\left(\frac{\pi}{2}\right)^p
$$
So $\displaystyle x \mapsto \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving very basic property of surfaces in $\mathbb{R^3}$
Let $X: U \subset \mathbb{R^2} \mapsto\mathbb{R^3}$ be a regular parameterized surface, i.e
a) $X$ is $C^{\infty}$;
b) The differential of $X$ at any $q \in U$, $dX_q:\mathbb{R^2} \mapsto \mathbb{R^3}$ is injective.
Prove that if $F$ is an invertible, $C^{\inft... | $\overline{X}$ is clearly differentiable because it's the composition of differentiable maps. Similarly, $d\overline{X}_q = dF_{X(q)}\cdot dX_q$, the composition of injective maps, hence injective, as desired.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $p,q $ prime numbers s.t. $p+p^2+...+p^{10}-q=2017$
Find $p,q $ prime numbers s.t. $$p+p^2+\cdots+p^{10}-q=2017.$$
It's easy to see that $p=2; q=29$ is solution.
There exists another solutions?
| Partial answer:
LHS is odd only when $q$ is odd. Consider the two cases:
*
*$p$ is even
*$p$ is odd
For the first case, $p=2$, forcing $q=29$.
For the second case, $p=2a+1$ and $q=2b+1$, with $a,b\in\mathbb{N}$, so we solve $$\left(\sum_{i=1}^{10}p^i\right)-q=2017$$ giving $$\frac{p^{11}-p}{p-1}=2016-2b\implies (2a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Find $\lim\limits_{x \to \infty} x\sin\frac{11}{x}$ Find $$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\right)$$
We know $-1\le \sin \frac{11}{x} \le 1 $
Therefore, $x\rightarrow \infty $
And so limit of this function does not exist.
Am I on the right track? Any help is much appreciated.
| $$\sin(11/x)\underset{(+\infty)}{\sim}11/x$$
What can you deduce ?
Note : What you have stated is good however with the product with $x$ you cannot conclude that it converges or diverges with what you wrote
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 0
} |
Infimum of infima of all subsets the following characterization seems trivial intuitively but it takes some time to prove it.
Informal version of the problem
Given a set $A$ and a family of subsets of $A$ whose union is $A$, is the infimum of $A$ equal to the infimum of all the infima of those subsets?
Formal version ... | Let's work in a partially ordered set $S$, with $A\subseteq S$. Also we know that $A=\bigcup_{i\in I} A_i$ and
*
*$a=\inf A$ exists;
*for each $i\in I$, $a_i=\inf A_i$ exists.
Then we claim that $a=\inf\{a_i:i\in I\}$.
Part 1 $a$ is a lower bound for $A_i$, hence $a\le a_i$ for every $i$, because $a_i$ is the gre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Are there any false variants of the Collatz conjecture for which the probability heuristic works? One of the supporting arguments for the Collatz conjecture is the probability heuristic, which states roughly that because the collatz operations tends to decrease numbers over time, it probably doesn't diverge.
Are there ... | Let $A$ be any infinite subset of $\mathbb{N}{\,\setminus}\{1\}$, with positive density less than $1/2$.
For $a \in A$, let $s(a)$ be the least element of $A$ which is greater than $a$.
Define $f:\mathbb{N}\to \mathbb{N}$ by
$$
f(n)=
\begin{cases}
s(n)&\text{if}\;n\in A\\[4pt]
1&\text{otherwise}\\
\end{cases}
$$
Then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Continuity of Probability Measure and monotonicity In every textbook or online paper I read, the proof of continuity of probability measure starts by assuming a monotone sequence of sets $(A_n)$. Or it assumes the $\liminf A_n = \limsup A_n$
But what about the following proof. It seems we don't need this property (mono... | I think the following is a complete proof of this theorem.
Divide the proof into 3 steps.
Step 1: If $\left\{ A_n \right\} _{n=1}^{\infty}$ is increasing, we have $\lim_{n\rightarrow \infty} \mathbb{P} \left( A_n \right) =\mathbb{P} \left( A \right) $;
Step 2 ：If $\left\{ A_n \right\} _{n=1}^{\infty}$ is decreasing, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Does converge in distribution imply limit being finite almost surely? If I have a sequence of random variable $x_n$ converge in distribution to a standard normal random variable, i.e.,
$$x_n\overset{d}{\to}N(0,1)$$
Does the following hold? Why or why not?
$$p(\lim_{n\to\infty}x_n<\infty)=1$$
Edited on 02/22/2018
More ... | The Law of Iterated Logarithm:
https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm
show that the opposite is true: $P\{\limsup x_n =\infty\} =1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Suppose $\bar{a}\in (\mathbb{Z}/n\mathbb{Z})$*. Prove that $\gcd(\bar{ab},n)=\gcd(\bar{b},n)\forall_{b\in (\mathbb{Z}/n\mathbb{Z})}$ Suppose $\bar{a}\in (\mathbb{Z}/n\mathbb{Z})$*. We want to prove that $\gcd(\bar{ab},n)=\gcd(\bar{b},n)$, so we show that for a divisor $d\in\mathbb{Z}: d|\bar{ab} \land d|n \iff d|\bar{b... | Here is a different take.
If $a$ is a unit mod $n$, then the ideal $(ab)$ is the same as the ideal $(b)$.
The size of the ideal $(b)$ is given by the additive order of $b$ mod $n$, which is $n/\gcd(b,n)$.
Therefore, $(ab)=(b)$ implies $n/\gcd(ab,n)=n/\gcd(b,n)$ and so $\gcd(ab,n)=\gcd(b,n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Differentiability at a point theorem for function of two variables I came across this theorem in calculus: If fx and fy exist near (a,b) and are continous at (a,b) then f(x, y) is differentiable at (a,b)
What confuses me is that when I look at solutions to questions that require you to use the above theorem, the soluti... | As for the example question:
Since we have:
$$f(x,y)=\sqrt{x+3y}$$
we take:
$$\begin{align*}
f_x(x,y)&=\frac{1}{2\sqrt{x+3y}}\\
f_y(x,y)&=\frac{3}{2\sqrt{x+3y}}
\end{align*}$$
which are both continuous - and, of course, well-defined - for every $(x,y)\in\mathbb{R}^2$ such that $x+3y>0$. So, since $(1,2)$ is such a poi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to get ray to segment distance For collision detection i used simple point QP to line segment S0-S1 closest distance test, with means of ortohonal projection like this:
s0s1 = s[1] - s[0];
s0qp = qp - s[0];
len2 = dot(s0s1, s0s1);
t = max(0, min(len2, dot(s0s1, s0qp))) / len2; // t is a number in [0,1] describing t... | Thanks to all of our efforts! Here is working solution for what i need:
// convenient mapping for neural network distance sensor;
scalar get_distance_inverse_squared(const point& qp, const point& d, const segment& s) {
auto s0s1 = s[1] - s[0];
auto s0qp = qp - s[0];
auto dd = d[0] * s0s1[1] - d[1] * s0s1[0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Spectral radius Volterra operator with an arbitrary kernel from $L^2$ How to prove that the spectral radius $r \left( V_K \right)$ of a Volterra operator with an arbitrary kernel is zero.
$$V_K : L^2 \left[a,b \right] \rightarrow L^2 \left[a,b \right]$$
$$ (V_Kf)(x) = \int_a^x K \left(x,y \right) f(y) dy, $$
where $K\l... | $$
|V_K^nf|^2= \\
= \left|\int_{a}^{x}K(x,x_{n-1})\cdots\int_{a}^{x_2}K(x_2,x_1)\int_{a}^{x_1}K(x_1,x_0)f(x_0)dx_0 dx_1\cdots dx_{n-1}\right|^2 \\
\le \left[\int_{a}^{x}\cdots\int_{a}^{x_2}\int_{a}^{x_1}|K(x,x_{n-1})\cdots K(x_2,x_1)K(x_1,x_0)||f(x_0)|dx_0dx_1\cdots dx_{n-1}\right]^{2}
\\ \le\left(\int_a^x\cdots\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Characteristic Polynomial of Restriction to Invariant Subspace Divides Characteristic Polynomial I am interested in finding a proof of the following property that does not make reference to bases, and ideally doesn't use facts about determinants that depend on the block structure of a matrix.
Let $T \in L(V,V)$ be a l... | The characteristic polynomial does not change if we extend the scalars. So we may assume that the basic field is algebraically closed.
Fact: the exponent of $(x-\lambda)$ in $P_A(x)$ equals the dimension of the subspace
$$V_{(\lambda)}\colon =\{v \in V \ | \ (A-\lambda I)^N v= 0 \text{ for some } N \}$$
(the general... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
} |
Integral of a shifted Gaussian distribution with an error function In the course of computing a convolution of two functions, I have simplified it to a single variable integral of the form
$$\int_0^\infty xe^{-ax^2+bx}\mathrm{erf}(cx+d) dx$$
where $\mathrm{erf}$ is the error function defined as $$\ \mathrm{erf}(x) = \f... | Not a full answer, more a long comment, but it might get you started. Define $$f_n:=\int_0^\infty x^n e^{-ax^2+bx}erf(cx+d)dx$$so we want $f_1=-\frac{1}{b}\partial_b f_0$. Defining $u:=\text{erf}(cx+d),\,v:=e^{-ax^2+bx}$, integration by parts gives $$(b+\frac{2a}{b}\partial_b)f_0=-2af_1+bf_0=-\text{erf}(d)-\frac{2c}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Probability that you have exactly one correct answer on a test I have two questions from a practice test which I have some concerns about:
Assume you write a multiple-choice exam that consists of 100 questions. For each question,
4 options are given, one of which is the correct one. If you answer each of the 100 que... | Why multiply by 100?
If you get the first one right, there are $3^{99}$ ways to get all the other problems wrong. But you could also only get the second problem right, with $3^{99}$ ways of getting all the other problems wrong, or only the third right. . . on to only getting the 100th question right with $3^{99}$ ways ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Double sided modulus equation: $|2x-1| = |3x+2|$ Considering this equation $$|2x-1| = |3x+2|$$
My question is what is the reasoning behind taking only the positive and negative values of one side of the equation to solve the problem. I have seen many people do that and it seems to work.
For e.g., $$2x-1 = 3x+2$$
... | In this example you actually can reduce this to two cases. Either the expressions inside the absolute values share the same sign, or they have opposite signs.
If you assume the left is negative and the right is positive.
$-(2x-1) = 3x + 2$
And the case the the right is positive and the left is negative.
$2x -1 = -(3x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
If $2k$ divides $P(n,k)$? Show that for $5\le k <n$, $2k$ divides $n(n-1)(n-2)..............(n-k+1)$
Answer:- My Attempt:- The product $n(n-1)(n-2)..............(n-k+1)= ^nP_k$
We know, $^nP_k=k\,! \times ^nC_k$
Since $5\le k $, $2$ is always a divisor of $k\,! $.
Again $k$ itself is a divisor of $k\,!$.
Thus $2k$ div... | You have some errors.
$2$ is a divisor of $k!$, not a multiple of $k!$.
Similarly, $k$ is a divisor of $k!$, not a multiple of $k!$.
Also, it's not automatic that the product of two divisors of $k!$ is a divisor of $k!$.
Of course, in this case, assuming $k > 2$, it's clear that $2$ and $k$ are distinct factors of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The fixed field of a conjugate of a subgroup $G'$ of the Galois Group of a normal extension is the image of the fixed field of $G'$ under automorphism Suppose that $K/F$ is a normal extension of fields of finite degree and consider $G' \leq \mbox{Gal}(K,F)$. If $\mathcal{F} = K_{G'}$ is the fixed field of $G'$, then sh... | Some of your notation is unusual: $G'$ usually denotes the derived
group of $G$ while the fixed field of $H$ is usually denoted $K^H$
rather than $K_H$.
I'll use standard notation: let $G=\textrm{Aut}(K/F)$, $H$
be a subgroup of $G$ and $K^H$ be its fixed field. Let $\tau\in G$.
Then $a\in K^{\tau H\tau^{-1}}$ iff $\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Induction and Integration by Parts
Let $T: C([0,2]) \to C([0,2])$ be given by
$(Tf)(x) = \int_0 ^x f(t)dt$ Prove $$(T^n f)(x) = \frac{1}{(n-1)!} \int_0 ^x (x-t)^{n-1}f(t)dt$$ for $n \in \mathbb{N}$.
I am attempting this with induction and integration by parts. However, I am struggling with the induction step to sh... | \begin{align*}
(T^{2}f)(x)&=\int_{0}^{x}(Tf)(t)dt\\
&=\int_{0}^{x}\int_{0}^{t}f(u)dudt\\
&=\int_{0}^{x}\int_{0}^{x}\chi_{[0,t]}(u)f(u)dudt\\
&=\int_{0}^{x}\int_{0}^{x}\chi_{[u,x]}(t)dtf(u)du\\
&=\int_{0}^{x}(x-u)f(u)du.
\end{align*}
Similarly, we have
\begin{align*}
(T^{n+1}f)(x)&=\int_{0}^{x}(T^{n}f)(t)dt\\
&=\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finite groups with only one conjugacy class of maximal subgroups
Let $G$ be finite. Suppose that all maximal subgroups of $G$ are conjugate. Then $G$ is cyclic.
I was stuck, then I find one solution. In Jack’s answer, it was mentioned that “one conjugacy class of maximal subgroups in fact implies that there is only o... | In the answer you mentioned $M$ refers to an element of the conjugacy class rather than the conjugacy class itself. In that answer since each Sylow $p$-subgroup of $G$ is contained in a maximal subgroup (which is conjugate to $M$) while all Sylow $p$-subgroups are conjugate, it follows that $M$ contains a Sylow $p$-sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Eigenvalues of a sum of power of a matrix $A$ is defined as a real $n×n$ matrix. $B$ is defined as:
$$B=A+A^2+A^3+A^4+ \dots +A^n$$
What's the relation between eigenvalues of $A$ and eigenvalues of $B$?
Can anyone give me some materials?
| HINT
If $A$ is diagonalizable, say $A = VDV^{-1}$ then
$$
B = \sum_{k=1}^n \left(VDV^{-1}\right)^k
= \sum_{k=1}^n VD^kV^{-1}
= V \left(\sum_{k=1}^n D^k \right)V^{-1}
$$
and $D$ is a diagonal matrix. Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Find $\lim_{x\to \infty} \left[f\!\left(\sqrt{\frac{2}{x}}\,\right)\right]^x$. Let $f:\mathbb{R} \to \mathbb{R}$ be such that $f''$ is continuous on $\mathbb{R}$ and $f(0)=1$ ,$f'(0)=0$ and $f''(0)=-1$ .
Then what is $\displaystyle\lim_{x\to \infty} \left[f\!\left(\sqrt{\frac{2}{x}}\,\right)\right]^x?$
When I was solv... | This is a partial answer to your question and I'm not expecting any upvote, just to make it easier:
It seems for any polynomial with the degree equal or higher than $2$ this works fine: $f(x)=a_nx^n+a_{n-1}x^{n-1}+...-\dfrac{x^2}{2}+1$
It satisfies all conditions required by the question:
$f(0)=1, f'(0)=0, f''(0)=-1$
W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
A question about $(\mathbb{Z}/p\mathbb{Z})[X]$ Let $p$ be prime and $f, g \in (\mathbb{Z}/p\mathbb{Z})[X]$. Prove:
$$(\forall x \in \mathbb{Z}/p\mathbb{Z}: f(x) = g(x) ) \iff f - g \in (X^p - X)$$
My approach:
First $\implies$. As $p$ is prime, $\mathbb{Z}/p\mathbb{Z}$ is a field, so it has no zero divisors, and $f - ... | Nice work! It seems like you only need to show
$$
\prod_{i=1}^{p-1}(X-i)=X^p-X.
$$
Instead of expanding the product, we will simply factor the right hand side. From your second implication, you know that $X^p-X$ has $p$ roots in $\mathbb{Z}/p\mathbb{Z}$ by Fermat's little theorem. But you are working over a field, so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve recurrence relation. I've got an recurrent function to solve.
$T_1 = 1$
$T_n = 2T_{\frac n2} + \frac{1}{2}n\log n$
I've got a tip to this excercise to determine additional variable as $k = 2^n$, where $ k = 1, 2, 3, ...$
But after some calulations I'm wondering if $k=2^n$ can i say that $2^{k-1} = n - 1$ ?
I reci... | We can prove by induction that
$$T_n = 2^kT_{\frac{n}{2^k}} + \frac{k}{2}n\log(n)-\sum_{i=0}^{k-1}\frac{in}{2}$$
This clearly holds for the base case $k = 1$. So assuming it holds for some $k$, we find:
\begin{align}
T_n &= 2^kT_{\frac{n}{2^k}}+\frac{k}{2}n\log(n) - \sum_{i=0}^{k-1}\frac{in}{2}\\
&= 2^k\left(2T_{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that every $n$-element set has the same number of subsets with even and odd cardinality without creating a bijection I need to prove that the number of even and odd subsets of a finite set if always equal. This is not a problem once we assume that $n$ is odd. Then, every "even" subset of the set defines in a uniq... | A bijection is a combinatorial way IMO. But there's a simple inductive argument. It's true for $n=1;$ suppose true for $n$ and take a set $S$ of size $n+1.$ Let $a$ be any element of $S$. Now $S- a$ is a set of size $n$, so has the same number of odd and even subsets. Therefore $S$ has the same number of odd subsets no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
What are the necessary conditions on integers $a, b$ for the sum of the fractions below to be reduced?
*
*$a/3+b/6$
*$a/20+b/12$
*$a/10+b/12$
For 1., $2a+b$ would have to be a multiple of 2 or 3. $2a$ must be even, and for $2a+b$ to be divisible 3, then $b$ would have to be odd. I can show other obvious relatio... | Notate the first fraction's denominator as $c$ and the second fraction's denominator as $d$. Then set $g = \gcd(c, d)$. If $g > 1$ and, then if $\gcd(a + b, g) > 1$, you'll be able to reduce the sum of the two fractions.
To work through your second example: $g = \gcd(20, 12) = 4$. Then suppose $a = 9$ and $b = 7$. Then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Proof of Limit Using Unusual Definition Let $\lbrace a_n \rbrace$ be a real sequence. We say $\lim_{n\to\infty} a_n=\infty$ provided that:
$$\forall K>0, \exists N\in \mathbb{N} \forall n \ge N:a_n>K$$
Use this definition, as well as the Archimedean property, to prove that:
$$\lim_{n\to\infty}n^3-4n^2-99n=\infty$$
In ... | Write : $a_n = n^3 - 4n^2 - 99n = n(n^2 - 4n - 99)$.
Now, use a trivial bound : $n \geq 1$ for all $n \in \mathbb N$.
This gives $a_n \geq n^2 - 4n - 99 = (n-2)^2 - 103$.
Therefore, $a_n \geq (n-2)^2 - 103$ for all $n$. We will call this $(*)$
A small observation : if $a, b > 2$ are two numbers and $a > b$, then $a-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How Are the Solutions for Finite Sums of Natural Numbers Derived? So, I've been learning set theory on my own (Lin, Shwu-Yeng T., and You-Feng Lin. Set Theory: An Intuitive Approach. Houghton Mifflin Co., 1974.) and have come across infinite sums of natural numbers. Since I took Algebra II many years ago, I've known o... | Bernoulli numbers are used to find the sum of $k$th powers of first $n$ natural numbers. A very accessible pdf is available on https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjSvPWxwYD8AhXymeYKHXR4DDwQFnoECBMQAw&url=https%3A%2F%2Fwww.whitman.edu%2Fdocuments%2FAcademics%2FMathema... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2664218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 4
} |
If $a_n \in \Bbb R$ s.t. $|a_n|$ diverges while converging conditionally.Then prove that the series ${a_n}^+ $ diverges. I was trying to solve the following problem :
Let $a_n \in \Bbb R$ s.t. $\sum_{n=1}^\infty {|a_n|} = \infty$ and $\sum_{n=1}^m {a_n} \to a \in \Bbb R$ as $m \to \infty$. Now let, ${a_n}^+ = max\{a_n ... | Hint:
We are given that $\sum_{k=1}^n a_k = \sum_{k=1}^n a_k^+ - \sum_{k=1}^n a_k^- $ converges.
If $\sum_{k=1}^n a_k^+$ converges then so does $\sum_{k=1}^n a_k^- = \sum_{k=1}^n a_k^+ - \sum_{k=1}^n a_k $.
Note that $|a_k| = a_k^+ + a_k^-$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2664335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Computing $\iint_{\mathbb R^2}\exp\left(u\frac{xy}{\sqrt{x^2+y^2}}+v\frac{x^2-y^2}{2\sqrt{x^2+y^2}}-\frac12(x^2+y^2)\right)dxdy$ I'm trying to work on another solution to this question by computing the moment generating function:
$$M(u,v)=\iint_{\mathbb{R}^2}\frac1{2\pi}\exp\left(u\frac{xy}{\sqrt{x^2+y^2}}+v\frac{x^2-y... | After changing to polar coordinates, make the change of variables $\theta=\phi/2$, to obtain that $$\begin{align*}M(u,v)&=\int_0^{\infty}\int_{0}^{2\pi}\frac1{2\pi}\exp\left(\frac{r}2(u \sin2\theta + v\cos 2\theta)\right) r\cdot\mathrm{e}^{-\frac{1}{2}r^2}\,dr\,d\theta\\&=\int_0^{\infty}\int_{0}^{4\pi}\frac1{4\pi}\exp\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2664469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $I$ is an ideal of the ring $R$ then show that if $R$ has no non-zero divisors then $R/(l(I)\cap r(I))$ has the same property. If $I$ is an ideal of the ring $R$ then show that if $R$ has no non-zero divisors then $R/(l(I)\cap r(I))$ has the same property.
Here $l(X):=\{r\in R:rx=0,\forall x\in X\}$ and $r(X):=\{r\i... | If $R$ has no non-zero divisors then $l(I)=(0)=r(I)$, so $$R/r(I)\cap l(I)=R/(0)\cong R$$ which by assumption has no non-zero divisors.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2664705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A rhombus is inscribed in triangle ABC
A rhombus is inscribed in triangle $ABC$ with $A$ as one of its vertices. Two sides of the rhombus lie along $AB$ and $AC$, with $AC=6, AB = 12$, and $BC = 8$. Find the length of a side of the rhombus.
So far by using heron's formula I got that the area of $ABC$ is $\sqrt {455}... | Let $R$ be a vertex of the rhombus on the side $BC$.
Then, since $PR$ is parallel to $AC$, we see that $\triangle{BPR}$ and $\triangle{BAC}$ are similar.
So, we have
$$BP:BA=PR:AC\implies 12-s:12=s:6\implies s=4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2664809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Question on nomenclature of ... mappings? I would like to understand "in english", what this sentence is saying here:
I understand what $R^3$ means, but I am not sure I understand the rest...
Thanks!
EDIT:
The image is from this paper.
| The pre-print is really badly written, not stating what $\Omega$ is. Anyway it's easy to get the idea. $\Omega$ represent a surface which is supposedly the camera receiver. The projection is a map that goes from the 3D space to the camera receiver. The back-projection is aimed to reconstruct the projected 3D-space from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2664913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $P_n(1)=1$ calculate $P'_n(1)$ in Legendre polynomials $P_n(x)$ is in $[-1,1]$ and $P_n(1)=1$ .The problem is getting $P'_n(1)$.
On Wikipedia it says that it is $\frac{n(n+1)}2$.
I derive the problem showed here
How could I prove that $P_n (1)=1=-1$ for the Legendre polynomials?
in order to get P'(n) but it didn't h... | Alternatively, it follows directly from Legendre's differential equation of
$$(1 - x^2) y'' - 2xy' + n(n + 1) y = 0.\tag1$$
Since $y = P_n(x)$ is a solution to (1) we have
$$(1 - x^2) P''_n(x) - 2x P'_n (x) + n(n + 1) P_n (x) = 0.$$
Setting $x = 1$ in the above equation leads to
$$P'_n (1) = \frac{n(n + 1)}{2} P_n (1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Finite difference: problem on edge of Dirichlet and Neumann boundary I'm trying to solve the non-homogeneious heat equation using a finite difference scheme. The grid on which to solve this looks like this:
N N N N N N N N N
N N
N N
D D
D D
D D
N ... | I managed to solve this by slightly altering the problem. In a small region around the edge of the two types of boundaries, I let the boundary condition vary linearly between the two types. This made the solution have a continuous derivative and solved the accuracy problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
integral $ \int_0^{\frac{\pi}{3}}\mathrm{ln}\left(\frac{\mathrm{sin}(x)}{\mathrm{sin}(x+\frac{\pi}{3})}\right)\ \mathrm{d}x$ We want to evaluate
$ \displaystyle \int_0^{\frac{\pi}{3}}\mathrm{ln}\left(\frac{\mathrm{sin}(x)}{\mathrm{sin}(x+\frac{\pi}{3})}\right)\ \mathrm{d}x$.
We tried contour integration which was not h... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Solving ODE $yy'=2x^3$ with $y(1)=3$ I'm stuck solving $yy'=2x^3$ with $y(1)=3$
I know that I'm looking to get the equation into the form of: $y' + P(x)y = Q(x)$ and then find the integrating factor $e^{\int(P(x)dx}$, but then how do I find my $P(x)$ in this case? And what do I do after with the initial condition?
| The differential equation
$$yy'=2x^3$$ is separable.
Integrate both sides and you get $$ \frac {y^2}{2} = \frac {x^4}{2}+c $$
Solve for $y,$ and with initial value $ y(1)=3,$ we get $$ y=\sqrt {x^4+8}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How do you solve the following nonlinear equations? How to solve the following system of equations
$$
\begin{cases}
a+c=12\\
b+ac+d=86\\
bc+ad=300\\
bd=625\\
\end{cases}
$$
| Well, $b=\cfrac {625}{d}$ from your last equation, and $a=-c+12$ from your first equation.
Substitute $b$ and $a$ into your second equation to get $\cfrac{625}{d}+(-c+12)(c)+d=86$
Do the same for your third equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Accumulation point in the definition of a limit of a function It is stated that the following is the definition of a limit of a function is:
Let f:D→ℝ be a function and let c be a accumulation point of D. Then we say L∈ℝ is the limit of f at c written limx→cf(x)=L if ∀ϵ>0 there exists a δ>0 such that ∀x∈D with 0<∣x−c∣<... | If $c$ is not an accumulation point, then for some $\delta_0 > 0$, $B_{\delta_0}(c) = \{x: |x-c| < \delta_0\} = \{x\}$, in which case the notion of a limit existing at that point does not capture the essence of what one wants to define. In particular, the definition breaks down and becomes vacuously true and hence we s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to show that the set is at most countable? $f(x)$ is a real-valued function on $\mathbb{R}$. How to show that the set $E=\{x\in\mathbb{R}: \lim_{y\to x}f(y)=+\infty\}$ is at most countable?
| $a\in E$ means that for every $M\in\Bbb N$, we can pick $u,v\in\Bbb Q$ with $u<a<v$ and $f(x)>M$ for all $x\in(u,v)\setminus\{a\}$.
Write $(u,v)=\phi(a,M)$.
We may assume wlog. that $\phi(a,M+1)\Subset \phi(a,M)$.
Start with $E_0:=E$ and assume $E_n$ is uncountable.
As $\phi(a,n)$ can take only countably many values, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Can we always establish whether an infinite series converges or diverges? I'm currently working with infinite series for my calculus class, and I'm wondering whether we always (in theory) can establish whether a series is divergent or convergent? Of course, it might be computationally hard, but is there a class of seri... | No, there are some criteria (for example $|a_n|/|a_{n+1}| \rightarrow l<1$) you can sometimes use, but even when you have one of those, on the limit cases (for example if $|a_n|/|a_{n+1}| \rightarrow 1$, for this last criterium), you'll have to prove it "by hand" (meaning there is no general way to do so)...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
} |
Subspace of $\mathcal{L}(V)$ For which vector space the set of non-invertible operators $T\colon V\longrightarrow V$ is a subspace of $\mathcal{L}(V)$?
I know sum of two non-invertible matrices is not a non-invertible. That means set of all non-invertible matrices will not form a subspace of any vector space.
Hence, ... | No, your assumption is not correct. Observe that$$\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}=\begin{pmatrix}1&0\\0&1\end{pmatrix}.$$Therefore, the sum of two singular matrices can be invertible.
Using this example as a model, it is not hard to prove that the answer to your original questi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given that $f(x^3)=[f(x)]^3$ for $f:K\to K'$ with $f(x+y)=f(x)+f(y)$, $f(1)=1$ and $K$ and $K'$ two fields with characteristic not equal$2$ and $3$. Given that $f(x^3)=[f(x)]^3$ for $f:K\to K'$ with $f(x+y)=f(x)+f(y)$, $f(1)=1$ and $K$ and $K'$ two fields with characteristic different from $2$ and $3$ show that $f$ is... | Having $f((1+x)^3)=(1+f(x))^3\implies 1+3f(x^2)+3f(x)+f(x^3)=1+3f(x)^2+3f(x)+f(x)^3$, we have $f(x^2)=f(x)^2$, since $char~K\neq 3$. Then applying $f((x+y)^2)=(f(x)+f(y))^2$, we have $f(xy)=f(x)f(y)$, since $char~K\neq 2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to prove an increasing sequence that converges is bounded above by its limit I am trying to prove that an increasing sequence that converges to $ L$ is bounded above by its limit.
By using $a_n \le a_{n+1}$ and the definition of limit of a sequence, I can prove that for $\epsilon > 0$ , $ a_n \lt {L + \epsilon} $ ... | HINT
You can easily show that if for some n $a_n>L$ then by definition of limit $a_n$ must decrease which is impossible.
You only need to formalize this idea by setting “assume exists n such that ...then by definition of limit...contradiction”.
Notably
*
*suppose $\exists n_1$ such that $a_{n_1}>L$ with $d=a_{n_1}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find the formula for $n*S_n$ Let $1*S_n$ denote $1+2+3+4+5...+n.$
Let $2*S_n$ denote $1*S_1 + 1*S_2 + 1*S_3 +... + 1*S_n.$
Similarly, let $3*S_n$ denote $2*S_1 + 2*S_2 + 2*S_3 +... + 2*S_n.$
and so on.
Find the formula for $n*S_n.$
Note: $n*S$ is not multiplication. It is a new notation.
| An example computation to help you find a general formula for $k\star S_n$ which may be proven by induction. Observe that
$$
3\star S_n=\sum_{m=1}^n2\star S_{m}=\sum_{m=1}^n\sum_{j=1}^m1\star S_j=
\sum_{m=1}^n\sum_{j=1}^m\sum_{i=1}^j\binom{i}{1}
$$
At this point use the identity
$$
\sum_{i=0}^n\binom{i}{k}=\binom{n+1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is it true that if $A^4 = I$, then $A^2 = \pm I$? So for linear algebra, I either need to prove that, for all square matrices, if $A^4 = I$, then $A^2 = \pm I$, or find a counterexample of this statement. Can anyone help please?
Thanks!
| This answer expands and expounds upon Botund's answer above; if $S= \{ \pm 1, \pm i \}$, then $x^4 = 1$ for every $x \in S$.
Consequently, if $D$ is any $n$-by-$n$ diagonal matrix with diagonal entries from $S$, then $D^4 = I_n$; however, the diagonal entries can be chosen such that $D^2 \ne \pm I_n$. For example, sele... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
How to solve $\sum_{k=1}^{\infty} ke^{-k}$? I am interested in
$$\sum_{k=1}^{\infty} ke^{-k}$$
And I can get the closed form on Wolfram Alpha, $\frac{e}{(e-1)^2}$, but I am curious how to derive it. It doesn't appear to be a typical geometric series.
| There are many ways to evaluate
$S
= \sum_{k=1}^{\infty} kx^k
$.
This is possibly the simplest.
$xS
= x\sum_{k=1}^{\infty} kx^k
= \sum_{k=1}^{\infty} kx^{k+1}
= \sum_{k=2}^{\infty} (k-1)x^{k}
$
so
$\begin{array}\\
S-xS
&= \sum_{k=1}^{\infty} kx^k-\sum_{k=2}^{\infty} (k-1)x^{k}\\
&= x+\sum_{k=2}^{\infty} kx^k-\sum_{k=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
A given line has an origin of 5 and forms a 22º angle with the X axis, and is tangent with a point P on a given circle. What are the coordinates of P? A given line has an origin of 5 and forms a 22º angle with the X axis. It is also tangent with a point P on a given circle. What are the coordinates of P?
| HINT
We need to find points on the circle with slope of the tangent equal to the slope of the line.
From the figure we have that the circle is centered in the origin thus to find P it suffice to find the equation of the line through the origin and ortogonal to the given line.
Remember that the condition for orthogonali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Algebra 2 Equation of growth and decay, Equation of arithmetic sequence
*
*Why do I need a $1$ in this equation of growth and decay? $A=P(1\pm r)^t$
*Why must I divide by $2$ to find the sum of an arithmetic sequence?
$S(n) = \frac{n}{2}(a_1 + a_n)$
| 1) because for $t=0$ we have $P=A$
2)because in the arithmetic sequence the addition of two terms equidistant from the extreme elements $a_1$ and $a_n$ is the same, and we have $n/2$ sums of this kind.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Quotient of Gaussian Integers via Third Ring Isomorphism It's known that $\mathbb{Z}[i] / (1+i) \cong \mathbb{Z}_2$.
I'm having difficulties with my attempt.
So we have $2 \in (1+i)$ since $2 = (1+i)\cdot(1-i)$. Thus $(2) \subset (1+i) \subset \mathbb{Z}[i]$. This looks like a third isomorphism problem so, $\mathbb{Z... | There is another isomorphism theorem (maybe the second one in your numeration) which you can use:
If $A\subseteq B$ are rings and $I\subset B$ is an ideal, then we have
$$ (A+I)/I \cong A/(I\cap A) $$
You can apply it for $A=\mathbb{Z}$, $B=\mathbb{Z}[i]$ and $I=(1+i)$. Since $(1+i)\cap \mathbb{Z}=(2)$, $\mathbb{Z}+(1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
von Neumann algebra Let $M$ be a subset of $B(\mathcal{H})$ (the space of bounded linear operators) such that $M'$ is a von Neumann algebra.
As we know if $M$ is invariant under involution, then $M'$ is a von Neumann algebra. My question is about the converse of it. Is $M$ invariant under $*$-operation, if $M'$ is a v... | Fix some $T\in B(\mathcal H)$ that is normal but not self-adjoint, and put $M=\{T\}$. Then $M'$ is a von Neumann algebra, but $M$ is not self-adjoint.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The Central Limit Theorem and the Scaled Sample Mean My introductory probability book gives the following two theorems:
Theorem 1.
For large $n$, the distribution of $\bar{X}_n$ is approximately $N(\mu, \sigma^2/n)$.
Theorem 2.
The CLT says that the sample mean $\bar{X}_n$ is approximately Normal, but since the sum ... | $$\Bbb{E}(W_n) = \Bbb{E}(n \bar{X}_n) = n \Bbb{E}(\bar{X}_n) \stackrel{\text{Thm 1}}{\approx} n \mu \\
\mathrm{var}(W_n) = \mathrm{var}(n \bar{X}_n) = n^2 \mathrm{var}(\bar{X}_n) \stackrel{\text{Thm 1}}{\approx} n^2 \cdot \frac{\sigma^2}{n} = n \sigma^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Functions: Induced Set functions Let $f:X\longrightarrow Y$ be a function, $A,A_1,A_2$ be subsets of $X$ and $B,B_1,B_2$ subsets of $Y$.
Prove that if $f$ is one-to-one then $f\displaystyle\left(\bigcap^\infty_{n=1}{A_n}\right)= \bigcap^\infty_{n=1}{f(A_n)}$
This is what I have so far, I'm pretty sure I'm right up unt... | You you've done so far is correct. It's only while proving the reverse inclusion that you'll need to use the one-to-one hypothesis. In fact, take the null function from $\mathbb Z$ into itself, take $A_1=\{0\}$ and take $A_2=\{1\}$. Then $A_1\cap A_2=\emptyset$ and $f(A_1)\cap f(A_2)=\{0\}$. So, in this case$$f(A_1\cap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Showing $\sum\limits_{n = 1}^∞\log(1+(-1)^{n-1}\frac{1}{n})$ is convergent Show that $$\sum_{n = 1}^\infty \log\left(1+(-1)^{n-1}\frac{1}{n}\right)$$ converges.
I want to say that the convergence/divergence of this series is equivalent to the convergence/divergence of $$\sum(-1)^{n-1}\frac{1}{n}.$$ Without the sign ter... | Just to add a note: Since $\ln(1+x)\leq x$ for all $x$, the convergence of
$$\sum_{n=1}^{\infty} \ln(1+x_n)$$
is, in general, implied by that of
$$\sum_{n=1}^{\infty} x_n.$$
(In fact, if $x_n>0$, it is possible to say more; since the convergence of the first sum is euivalent to that of $\prod_{n=1}^{\infty} (1+x_n)$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Embedding of a cubic field into $\mathbb{R}$ I'm attempting problem IV.3B in Samuel Pierre's Algebraic Theory of Numbers. The problem is as follows:
Let $K$ be a cubic field such that $r_1 = r_2 = 1$. Suppose $K$ is
imbedded in $\mathbb{R}$.
(a) Show that the positive units of $K$ form a group isomorphic to
$\math... | (1) every positive real has a positive square root. Also if $u_1$
and $u_2$ are the other conjugates of $u_2$ then $u_2=\overline{u_1}$
and $uu_1u_2=1$. Therefore $|u_1|=x^{-1}$ etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Laplace transform of Bessel function I am stuck in a question, and don't know where to start. I have to obtain the Laplace transform of $J_0(t)$, I have to let:
$$a_n=\int_{0}^{\pi}(\sin \theta)^{2n}d\theta$$
And now wish to show that:
$$a_n= \frac{(2n)!}{2^{2n}(n!)^2}\pi$$
My idea was:
I know that:
$$J_0(t)=\sum_{n=0}... | I do not quite follow your train of thoughts, so I will start from scratch. Given the following definition of $J_0$
$$ J_0(t)=\sum_{n\geq 0}\frac{(-1)^n t^{2n}}{n!^2 2^{2n}}\tag{1} $$
it is trivial that $J_0$ is an entire function. Since $\mathcal{L}(t^{2n})(s)=\frac{(2n)!}{s^{2n+1}}$ we formally have
$$ \mathcal{L}(J_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Folland Real Analysis Chapter 4 Exercise 15 I'm studying for a test (and prelims) and have been working through Folland. I've been a bit stuck on the following problem. $\overline{A}$ denotes the closure of $A$, $A^o$ the interior, and $g\in C(A)$ means $g$ is continuous on $A$.
If $X$ is a topological space, $A\subset... | For $x\in A^\circ$, pick $V$ to be an open neighbor of $g(x)$. Then by continuity of $g$, we can find $U\subset A$ open in $A$ such that $g(U)\subset V$. Then $x\in U\cap A^\circ$ with $g(U\cap A^\circ)\subset V$ where $U\cap A^\circ$ is open in $X$.
For $x\in A-A^\circ$, we have $g(x)=0$. Let $V$ be an open neighbor o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Formula for ellipse with two tangents intersecting with two points Assume that I have four points $P_1, P_2, P_3, P_4$. These points lie on the 2d plane and take the form $P_i = (x_i, y_i)$
Assume that I define line $L_{ij}$ as the line passing through $P_i$ and $P_j$.
How do I find the coefficients $a$ and $b$ in the... | If you're after an ellipse that passes through 2 points with given tangents at those points then this may be of use to you[1].
[1] Roundest ellipse with specified tangents
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Why does this innovative method of subtraction from a third grader always work? My daughter is in year $3$ and she is now working on subtraction up to $1000.$ She came up with a way of solving her simple sums that we (her parents) and her teachers can't understand.
Here is an example: $61-17$
Instead of borrowing, maki... | I think this method is awesome, it might even be easier than the classical method in some cases. Consider the following 'easy' subtraction, where the digit of the first number is bigger than the corresponding digit of the second number.
\begin{align}462-231&=(400+60+2)-(200+30+1)\\
&=(400-200)+(60-30)+(2-1)\\
&=200+30+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "297",
"answer_count": 18,
"answer_id": 6
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.