Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is the difference between line search and gradient descent? I understand the gradient descent algorithm, but having trouble how it relates to line search. Is gradient descent a type of line search?
| The following figure shows the hierarchy of some of the line search optimization methods for quadratic forms. Indeed, the methods are categorized based on choosing the descent direction $p_k$ and the length step $\alpha_k$. Just recall that
\begin{align}
\phi(x) &= \frac{1}{2}x^{\top}A \, x + b^\top x + c,\\
x_k &= x_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What do the eigenvalues of a matrix tell us about the original matrix? I have a problem of...
Let $A$ be a 2x2 matrix such that it is not invertible and 2 is an
eigenvalue of $A$.
a) Find all eigenvalues of $A+I$.
b) Prove or disprove A+I is invertible.
Since it's not invertible, it has an eigenvalue of 0. So I... | You don't have to find the original matrix to answer the question. As you said the eigenvalues must be $\lambda =0,2$. Claim, $\mu =1,3$ are the eigenvalues of $A+I$. Lets check, Let $x$ be an eigenvector of $A$ corresponding to the eigenvalue 2, then $$(A+I)x=Ax+x=2x+x=3x.$$Thus, $\mu=3$ is an eigenvalue for $A+I$. Si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\cos\alpha+\cos2\alpha+\cdots+\cos n\alpha=\frac{1}{2}\left(\frac{\sin\left(n+\frac{1}{2}\right)\alpha}{\sin\frac{1}{2}\alpha}-1\right)$ I have to prove using mathematical induction that:
$$\cos\alpha+\cos2\alpha+\cdots+\cos n\alpha=\frac{1}{2}\left(\frac{\sin\left(n+\frac{1}{2}\right)\alpha}{\sin\frac{1}{2... | Hint: Write
\begin{align}
\cos\alpha = \operatorname{Re} e^{i\alpha}
\end{align}
then the sum becomes
\begin{align}
\operatorname{Re}\left(e^{i\alpha}+e^{i2\alpha}+\ldots+e^{in\alpha} \right)
\end{align}
which is a geometric series.
Edit: It's not hard to see
\begin{align}
e^{i\alpha}+e^{i2\alpha} + \ldots + e^{in\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof of a Epigraph that is not convex
Let the $g: [-2, 2] \rightarrow \mathbb{R}$ be defined by
$g(x) = \begin{cases} 2x^2 \quad \text{ if } -2 \leq x \leq 0 \\
x+2 \quad \text{if } 0 < x \leq 2 \end{cases}$
Prove by definition: "Epi $g$ is not convex"
I'm not sure how you prove the above. I tried by stating "E... | $$ {\rm epi}\ g=\{ (x,y)|g(x)\leq y \} $$
Note that $(0,0),\ (2,4)\in {\rm epi}\ g$ Hence if ${\rm epi}\ g$ is
convex then $$ \frac{1}{2} \{ (0,0)+(2,4)\}=(1,2)\in {\rm epi}\ g$$
But $$ g(1)=3\Rightarrow (1,t)\in {\rm epi}\ g,\ t\geq 3$$
It is a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $\sin 2x = 2\sin x \cos x$ using Taylor Series and Cauchy products I have that the Taylor series of $\sin x$ and $\cos x$ are
\begin{equation*}
\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\
\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}
\end{equation*}
which I understand yields the pro... | Note that
$$
\begin{align}
c_{2n+1}&=\sum_{k=0}^{n} \frac{(-1)^{k}(-1)^{n-k}}{(2k+1)!(2n-2k)!}\\
&=\sum_{k=0}^{n} \frac{(-1)^{n}(2n+1)!}{(2n+1)!(2k+1)!(2n-2k)!}\\
&=\sum_{k=0}^{n} \frac{(-1)^{n}}{(2n+1)!}{2n+1 \choose 2k+1}
\end{align}
$$
and that
$$
\begin{align}
\sum_{k=0}^n{2n+1 \choose 2k+1}&=\sum_{k=0}^n({2n \choo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove by induction that $\sum\limits_{i=1}^n \frac{1}{n+i} \leq \frac{3}{4}$
Prove by induction that $\sum\limits_{i=1}^n \frac{1}{n+i} \leq \frac{3}{4}$.
I have to prove this inequality using induction, I proved it for $n=1$ and now I have to prove it for $n+1$ assuming $n$ as hypothesis, but this seems impossible t... | Note that you do not use induction in the above.
Another way:
Define $$S_n := \sum_{i=1}^n \frac{1}{n+i}$$
Then $$S_{n+1} = \sum_{i=1}^{n+1} \frac{1}{n+1+i} = \sum_{i=1}^{n+1} \frac{1}{n+(i+1)} = \sum_{i=2}^{n+2} \frac{1}{n+i}.$$
Hence $$S_{n+1} = S_n - \frac{1}{n+1} + \frac{1}{n+n+1} +\frac{1}{n+n+2}.$$
Now proceed...... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
How would you go about this algebra problem? (order of operations?) Okay, so my teacher gave us this to solve:
$$a + b/c = 2$$
In this equation he explained that there were many ways to do it. However, we were told to do it with at least two negative numbers. I was a bit stuck at this one, however, more than a couple o... | Your teacher is wrong, the order of operation applies always.
$$-1-1/(-1)=0$$
as that means
$$-1-\frac{1}{-1}=-1-(-1)=-1+1 = 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Prove the following equality: $\int_{0}^{\pi} e^{4\cos(t)}\cos(4\sin(t))\;\mathrm{d}t = \pi$ Prove the following equality:
$$
\int_{0}^{\pi} e^{4\cos(t)}\cos(4\sin(t))\;\mathrm{d}t = \pi
$$
| Hint :
Consider the other integral
$S=\int_0^\pi e^{4cos(t)}sin(4sin(t))dt$
and the sum
$C+iS=\int_0^\pi e^{4(cos(t)+isin(t))}dt$
$=\int_0^\pi e^{4e^{it}}dt$
put $4e^{it}=u$ or $t=-iln(u)+i2ln(2)$
the integral becomes
$i\int_{-1}^{1}\frac{e^udu}{u}$
I'll surely be downvoted.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
If for every $z,$ either $|f(z)| \le 1$ or $|f'(z)| \le 1,$ then $f$ is a linear polynomial I am working on the following exercise:
Let $f$ be entire and assume that for every $z,$ either $|f(z)| \le 1$ or $|f'(z)| \le 1$ (or both). Then $f$ is a linear polynomial.
I have a few questions about this. First, I believe I... | I think there’s a mistake in the hint.
Inspired by the hint,I think we can prove as follow:
$\forall x\in \mathbb{C}\backslash\{0\}\ $,suppose $\ \vert f(z)\vert>1\ $,then$\ \vert f’(z)\vert\leq 1.\ $
$Let\ t_0=inf\lbrace t_1\in [0,1]\vert\ \vert f’(tz)\vert\leq 1, \forall t\in[t_1,1]\rbrace.$
Since$\ f(z)\ $is entire,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $E$ and $F$ are connected subsets of $M$ with $E\cap F\ne\emptyset$, show that $E\cup F$ is connected. If $E$ and $F$ are connected subsets of $M$ with $E\cap F\ne\emptyset$, show that $E\cup F$ is connected.
My attempt:
Suppose $E\cup F$ is disconnected. $\boldsymbol{\Rightarrow \exists} \ open \ sets\ A, B\ne \emp... | This hint should help you solve the problem: Since $E\cap F\neq \varnothing$, we have a point $x\in E\cap F$. In particular, $x\in E$ as well as $x\in F$. Now, consider the connected component containing $x$. Does it contain $E$? Does it contain $F$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there an entire function with $f(\mathbb{Q}) \subset \mathbb{Q}$ and a non-finite power series representation having only rational Coeffitients I'm trying to answer the following question:
Is there an entire function $f(z) := \sum \limits_{n=0}^\infty c_nz^n$ such that
*
*$f(\mathbb{Q}) \subset \mathbb{Q}$
*$\fo... | If you allow meromorphic functions (and as a consequence, finite radius of convergence), you have
$$
\frac{1}{1-x}=1+x+x^2+x^3+\cdots
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\frac{1}{\pi} \int_a^b \frac{1}{\sqrt{x(1-x)}}dx$ I can't calculate the following integral:
$$\frac{1}{\pi} \int_a^b \frac{1}{\sqrt{x(1-x)}}dx,$$
where $[a, b] \subset [0,1]$.
Can someone, please, give me a hint?
Thank you!
| Anoteher simple substitution exploiting the symmetry of the integrand: $x=1/2 + u \implies dx=du$.
$$\int\frac{1}{\sqrt{x(1-x)}}dx=\int\frac{1}{\sqrt{(1/2+u)(1/2-u)}}du$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Degree of field extension $\mathbb{Q}\left(\sqrt{1 + \sqrt{3}}\right):\mathbb{Q}$ I think the degree
$$ \left[\mathbb{Q}\left(\sqrt{1 + \sqrt{3}}\right):\mathbb{Q}\right] $$
is equal to four, but how do I find the minimum polynomial of such an extension? If I square the term I take care of one square root, but squarin... | $x = \sqrt{1 + \sqrt 3} \implies x^2-1 =\sqrt{3} \implies x^4-2x^2-2 = 0$.
This polynomial is irreducible by Eisenstein's criterion, since $2$ divides all the coefficients except the first, and $2^2=4$ does not divide the constant. Hence, the degree of the extension, is the degree of the minimal polynomial, which is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving that an arithmetic function is multiplicative, but not completely multiplicative?
Define an arithmetic function $\rho$ by $\rho(1)=1$ and $\rho(n)=2^m$ where m is the number of distinct prime numbers in the prime factorization of n. Prove that $\rho$ is multiplicative, but not completely multiplicative.
This... | Just show that $\rho(2)^2\ne \rho(2^2)$. Clearly $\rho(2)=\rho(4)=2$, so you're done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1974955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving area of triangle formed at parallelogram midpoint is 1/4 of the parallelogram? ABCD is a parallelogram . X is the midpoint of AD & Y is the midpoint of BC. Show that the area of $\triangle {ABX}$ is $\frac{1}{4}$ the area of ABCD
Can you help me with this proof? Where should I start? I think It should be by pr... | The length of perpendicular for the triangle and parallelogram is the same.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
positive elements in $\mathbb{M}_n(A)$ are unltraweakly dense in the positive part of $\mathbb{M}_n(A^{**})$ I try to read the book of C*-algebra and finite- dimensional approximations. In the proof of Theorem 2.3.8, i can't understand the question that positive elements in $\mathbb{M}_n(A)$ are unltraweakly dense in t... | It is true in general that $M_n (A)^+ $ is ultraweakly dense in $M_n(A'')^+$. The first thing is to notice that $M_n (A'')=M_n (A)''$. So we can simply work with an inclusion $A\subset A''$ with $A $ (wot, sot, ultra wot) dense.
Note first that, by using Kaplanski, the unit ball of $A $ is sot (and so, wot and ultra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Second derivative of bump function? My question is?
Does there exist a function $f\in C^{2}\left(\mathbb{R}\right)$ such that
$$\begin{cases}
f\left(x\right) & =1\quad\textrm{when }\left|x\right|<1,\\
f\left(x\right) & =0\quad\textrm{when }\left|x\right|\geq2,\\
f\left(x\right) & \in\left[0,1\right],\forall x\in\mathb... | Observation: Since $f$ is continuous, $\lim_{|x|\rightarrow 1^-}f(x)=1$, so the first inequality is not strict. (Alternately, $f^{-1}(1)$ is closed).
Therefore, $f(1)=1$ and $f(2)=0$. So, by the mean value theorem, there is some $c\in(1,2)$ such that $f'(c)=\frac{f(2)-f(1)}{2-1}=-1$.
Since $f$ is constant for $|x|<1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why won't a series converge if the limit of the sequence is 0? Just thinking about it in terms of logic, shouldn't the series of a sequence whose limit as $n$ approaches infinity is 0 converge?
I know that the $n$th term test for divergence says that if a series is convergent, then the limit of its sequence is 0 and I ... | Do you think the series
$$1+\frac12+\frac12 + \frac14+\frac14+\frac14+\frac14 + \frac18 + \cdots$$
converges? Note, there are $2$ terms equal to $\frac12$, $4$ terms equal to $\frac14$, $8$ terms equal to $\frac18$ and so on, with $2^i$ terms equal to $\frac{1}{2^i}$ for each $i\in\mathbb N$.
You will probably agree th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 6,
"answer_id": 1
} |
Is there a statement independent from PA and does not increase the consistency strength? There are many known independence results of $\mathsf{PA}$, for example, Goodstein theorem, Paris-Harrington theorem, and the reflection principle. But these examples imply the consistency of $\mathsf{PA}$.
I guess not all independ... | You can attack this kind of question by thinking about the Lindenbaum-Tarski algebra for PA. This can be viewed as a partial order $\preceq$ on the set of all sentences in the language of arithmetic, where we have $\phi \preceq \psi$ if and only if $PA + \psi \vdash \phi$. (It is possible to use the opposite order, but... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Is it possible for an integral between $a$ and $a$ to have a value other than 0 Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ with a constant $a \in \mathbb{R}$ such that
$$\int_{a}^{a} f(x) \, dx \neq 0 \quad$$
holds?
| Assuming that the integral is defined, the answer in NO.
$$\int_{a}^{a} f(x) dx =\int_{\{a\}} f(x) dx$$ and since the integral on the right is taken over the set of measure $0$, the integral is $0$.
However, notice that if instead of Lebesgue measure we put a counting measure on $\mathbb{R}$ (call it $\mu$), then
$$\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 1
} |
Interesting mathematical artifact: Equality sign wrong for exponential function. I found an interesting case where it seems like an equality sign works wrong.
Let's consider the following construction:
$\frac{1+\Lambda}{2} e^{i\Lambda \phi}$ where $\Lambda = \pm1$, so $\Lambda^2=1$.
Then I apply Euler formula:
$\frac{1... | The problem is the division by zero:
The identity $\tfrac{1+\Lambda}{2}e^{i\Lambda\phi}=\tfrac{1+\Lambda}{2}e^{i\phi}$ holds as you have shown.
For $\Lambda=1$, this implies $e^{i\Lambda\phi}=e^{i\phi}$, which is obviously correct. However, for $\Lambda=-1$, this means that $\tfrac{1+\Lambda}{2}=0$ and therefore, the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1975969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Can an unbounded sequence have a convergent cesaro mean? I was wondering if an unbounded sequence may have a convergent cesaro mean ($\frac{1}{n}\sum_{k=1}^n a_n$). I was maybe thinking of $$a_n = (-n)^n$$
as a sequence having a convergent mean, but I might be wrong. Anyways, how would you proceed to prove such an intu... | Take the following example
$u_{2n}=-\sqrt{n}$
and
$u_{2n+1}=\sqrt{n}$
then
$v_{2n}=0$
and
$v_{2n+1}=\frac{\sqrt{n}}{2n+1}$.
$(u_n)$ is unbounded.
$(v_n)$ goes to $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1976222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
combinatorics: sum of product of integer compositions I am trying to solve a problem from Stanley's book, it says:
Fix $k,n \in \mathbb{P}$. Show that:
\begin{align}
\sum a_1 a_2 \cdots a_k = \binom{n+k-1}{2k-1}
\end{align}
where the sum ranges over all compositions $(a_1 , a_2 , \ldots , a_k)$ of $n$ into $k$ parts.
I... | Compositions of $n$ into $k$ parts can be represented as a string of $n$ stars and $k-1$ bars. The $k-1$ bars break the stars into $k$ groups, and the number of stars in the $i^{th}$ group represents $a_i$.
For each of these compositions, we are adding $a_1a_2\dots a_k$. This represents the number of ways to choose one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1976347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Show that $L(HK) = L(H) + L(K) - L(H\cap K) $ where L is the length of the composition series of the group and H and K are normal subgroups of G. I already proved that $L(G) = L(H) + L(\dfrac{G}{H})$ and then Im asked to prove that $L(HK) = L(H) + L(K) - L(H\cap K) $ but I fail to see a connection with what I already... | Hint: Remember that $$\frac{HK}{K}\simeq\frac{H}{H\cap K}$$ and shows that $L(M/N)=L(M)-L(N)$ for any subgroup $N$ of the group $M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1976502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find integer that satisfies two congruences Find the integer between $0$ and $29\times 23$ $= 667$ that satisfies the two following congruences:
$x ≡ 15$ (mod $23$)
$x ≡ 1$ (mod $29$).
| You can just look for a pattern. Each time you add $29$ to a number, the remainder when divided by $23$ increases by $6$, so you can see that:
$$1≡1\pmod {23}, \;30≡7\pmod {23}, \;59≡13\pmod {23},\;\;...\;\;291≡15\pmod {23}$$ and there's your answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1976590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Difficult limit problem involving sine and tangent I encountered the following problem:
$$\lim_{x\to 0} \left(\frac 1{\sin^2 x} + \frac 1{\tan^2x} -\frac 2{x^2} \right)$$
I have tried to separate it into two limits (one with sine and the other with tangent) and applied L'Hôpital's rule, but even third derivative doesn'... | $$\frac{1}{\sin^2x}-\frac{1}{x^2}=\frac{x-\sin x}{x^3}\frac{x+\sin x}{x}\frac{x^2}{\sin^2x}\to\frac{1}{6}\cdot 2\cdot 1$$
Now try the remaining terms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1976711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Can we show that all $2 \times 2$ matrices are sums of matrices with determinant 1? I came across a paper on the Sums of 2-by-2 Matrices with Determinant One. In the paper, which I have conveniently indicated here for reference, the author claims, but without proof, that a $2 \times 2$ is a sum of elements of the speci... | The matrix decomposition into a matrices is not unique. For example, $$
\begin{bmatrix}
a&b\\c&d
\end{bmatrix}= \begin{bmatrix}
a&-1\\1&0
\end{bmatrix}
+
\begin{bmatrix}-1&b\\0&-1
\end{bmatrix}
+
\begin{bmatrix}
1&0\\c&1
\end{bmatrix}
+\begin{bmatrix}0&1\\-1&d\end{bmatrix}.
$$
is another such decomposition into matri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1976815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Particular solution of $y''+4y=x\cos x$
Find both the general solution $y_h$ and a particular solution $y_p$ for
$$y''+4y=x\cos x$$
So far I've got $y_h$ from factoring the characteristic polynomial:
$$y_h=C_1\sin2x+C_2\cos2x$$
But the $y_p$ part troubles me, any help?
| For $y_p$ use method of variation of parameter
$$
\begin{aligned}
& W=\left|\begin{array}{cc}
y_{1} & y_{2} \\
y_{1}^{\prime} & y_{2}^{\prime}
\end{array}\right| \\
& W=\left|\begin{array}{cc}
\sin 2 x & \cos 2 x \\
2 \cos 2 x & -2 \sin 2 x
\end{array}\right|=-2 \\
\therefore y_{p} &=-y_{1} \int \frac{y_{2} x}{w} d x+y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
When to stop rolling a die in a game where 6 loses everything You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwi... | The question is missing the concept of utility, a function that specifies how much you value each possible outcome. In the game, the utility of ending the game with a certain score would be the value you place on that outcome. Although you could certainly argue it is implied in the question that the utility of a score ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 11,
"answer_id": 8
} |
Sum of reciprocals of the perfect powers
*
*What is the sum of all of the numbers which can be written as the reciprocal of an integer that is also a power greater than one, excluding the powers of one ?.
*I'm asking about the sum of numbers which can be written as powers. $$
\mbox{This sum would start off as}\quad
... | Your sum contains the sum of all reciprocals of non-unit natural number powers of primes, that is, every number of the form $\frac{1}{p^k}$ for prime $p$, and $k\in \{2,3,...\}=\mathbb{N}\setminus\{1\}$. Since each number of this form is uniquely determined by the choice of $p$ and $k$, we obtain
$\sum_{p}\sum_{i=2}^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 1
} |
Radical series and simple factors in the composition series The context is a finitely generated modules of a finite dimensional $\mathbb{K}$-algebra $A$, where $\mathbb{K}$ is an arbitrary field.
Now, if we consider $A$ as a left $A$-module as usual, this module $A$ has many composition series which, according to the J... | All isotypes of simple left $A$-modules already appear in $A/rad(A)$. This is because the two rings have the "same set" of simple modules, and since $A$ is Artinian, each one is a direct summand of $A/rad(A)$.
Since you can take any of those semisimple radicals and express it as a composition series, you can chain them... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given $f''(x) = -4f(x)$ for all $x \in$ $\mathbb{R}$ and $f(0) = f'(0) = 1$, how do I find $f(x)$ Let $f$ be a function so that $f''(x) = -4f(x)$ for all $x \in \mathbb{R}$ and $f(0) = f'(0) = 1$.
i) Prove that $2f(x)\sin(2x)+f'(x)\cos(2x)=1$ for all $x \in \mathbb{R}$
ii) Prove that $2f(x)\cos(2x)-f'(x)\sin(2x)=2$ for... | If we put $t=2x$ and if prime denotes derivative with respect to $t$ then it is easy to see that we have $f''+f=0$ and this has a unique solution $f= f(0)\cos t + f'(0)\sin t$ and note that given conditions imply that $f(0)=1,f'(0)=1/2$ so that $f(x) = \cos 2x +(1/2)\sin 2x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Show that $f(g(x)) = x$ and $g(f(x))= x$, with $f(x) = x^e \bmod n$ and $g(x) = x^d \bmod n$ I want to solve the following problem:
Let $d$ and $e$, both natural numbers, be each others inverses modulo $\varphi(n)$, where $n = p\cdot q$ is a product of two different prime numbers $p$ and $q$. Let $M = \{0,1,2,\dots,(n... | I think I figured it out. First I have to prove that $x^{k\varphi(n) + 1} \equiv x \pmod{n}$, even when I don't know if $\gcd(x,n) = 1$.
We look at the system
\begin{align}
\begin{cases}
y \equiv x \pmod p \\
y \equiv x \pmod q
\end{cases}
\end{align}
Since $q$ and $p$ are two different prime numbers, they are rel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If f is injective can you prove this relation? Given that f maps E to F and A is a subset of E, prove that if f is injective then f^-1(f(A)) is a subset of A.
Actually, im good at performing similar proofs, but i didn't understand why should f be injective? I proved it without using this given and didn't know how to b... | Choose $x\in f^{-1}(f(A))$. Then $f(x)\in f(A)$. Hence there exists $a\in A$ such that $f(x)=f(a)$. Injectivity of $f$ then implies that $x=a$. In particular, $x\in A$. Hence, $f^{-1}(f(A))\subset A$.
To show that injectivity is really needed, consider for instance the non-injective function
$$f:\{a,b,c\}\rightarrow\{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Are these two measures same? Let $f_k :\{0,1\}^{\infty} \to \{0,1\}^k$ denote the projection map onto first $k$ component. Now given a probability measure $P$ on the measure space $( \Omega=\{0,1\}^{\infty},P(\Omega))$ we can push forward this probability measure using $f_k$ and we obtain a probability measure on $(\{0... | The condition on $\mathbb{P}_1$ and $\mathbb{P}_2$ means that they agree on all cylinder sets, i.e. sets of the form
$$ E_n=\{\omega\in \Omega:\omega_1=a_1,\dots,\omega_n=a_n\} $$
where $n$ is a natural number and $a_1,\dots,a_n\in\{0,1\}$. The cylinder sets (together with the empty set) form a $\pi$-system which gene... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove or disprove: if $f_n$ converges uniformly on $A$, then $f_n$ converges uniformly on $\overline{A}$. I am interested in the statement
Let $A$ be a set and $f_n$ be a sequence of functions defined on $\overline{A}$. Assume that $f_n$ pointwise converges to a function $f$ on $\overline{A}$. If $f_n$ converges unif... | False.
Let $A=\mathbb Q$ and define $f_n(x)=f(x)=0$ for each $x\in\mathbb Q$ and $n\in\mathbb N$. If $x\in\mathbb R\setminus \mathbb Q$, then let $f_n(x)=x/n$ for each $n\in\mathbb N$ and $f(x)=0$. Clearly, $(f_n)_{n\in\mathbb N}$ converges pointwise to $f$ on $\overline A=\mathbb R$. However, convergence is not unifor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Basis of a finite free $R$-algebra starting with $1$ , $R$ is local Let be $A$ be a finite free $R$-algebra where $R$ is a local ring. Does $A$ always have a basis $a_1, \ldots, a_n$, where $a_1=1$?
| Yes, that's true: More generally, if $k:=R/{\mathfrak m}$ is the residue field of $R$ and if $M$ is a finite free $R$-module such that $\{\overline{m_1},\ldots,\overline{m_n}\}$ is a basis of $M\otimes_R k=M/{\mathfrak m}M$, then ${\mathscr B}:=\{m_1,\ldots,m_n\}$ is a basis of $M$. For the proof, note that the base ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving two lines are perpendicular Given $z_1,z_2,z_3 $ and $z_4$ are complex numbers, prove that the line joining $z_1,z_2$ and the line joining from $z_3,z_4$ are perpendicular iff $Re\{(z_1-z_2)(\bar z_3-\bar z_4)\}=0$. Try not to use polar form.
I try to start with writing $Re\{(z_1-z_2)(\bar z_3-\bar z_4)\}=Re\{... | Let $z_1-z_2=ae^{i\theta}$ and $z_3-z_4=be^{i\phi}$, then
$$(z_1-z_2)\overline{(z_3-z_4)} = abe^{i(\theta-\phi)}$$
$$\operatorname{Re}[(z_1-z_2)\overline{(z_3-z_4)}] = ab\cos (\theta-\phi)$$
\begin{align*}
\operatorname{Re}[(z_1-z_2)\overline{(z_3-z_4)}]=0
& \iff \cos (\theta-\phi) =0 \\
& \iff \theta-\phi=\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Separate equation $\sin3x\mathrm{d}x+2y(\cos3x)^3\mathrm{d}y$ Find solution by using separable equation $$\sin3x\mathrm{d}x +2y\cos^33x\mathrm{d}y=0$$
So far I got this
$$\frac{\sin3x}{\cos^33x}\mathrm{d}x + 2y\mathrm{d}y = 0$$
$$ 2y\mathrm{d}y = -\frac{\sin3x}{\cos^33x}\mathrm{d}x$$
Integrate both sides
$$\int2y\mathr... | Hint:$$y^2=-\int\frac{\sin3x}{\cos^33x}dx$$
$$\cos 3x=u\Rightarrow-3\sin3xdx=du\Rightarrow-\sin3xdx=\frac{du}{3}$$
$$y^2=\int\frac{du}{3u^3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find partial derivatives in very abstract case when F is just F(x1,x2) and how to express it correctly? I've just started learning differential calculus and there's one task that I don't completely understand. It sounds like:
"Given $Y=F(x_1,x_2)+f(x_1)+g(x_2)$, find $\frac {\partial Y}{\partial X_1}$, $\frac {... | $$Y=F(x_1,x_2)+f(x_1)+g(x_2) \\ \implies \frac{\partial}{\partial x_1} Y = \frac{\partial}{\partial x_1} \left[F(x_1,x_2)+f(x_1)+g(x_2)\right] \\ \begin{align}\implies \require{enclose}\enclose{box}{\frac{\partial Y}{\partial x_1}} &= \frac{\partial F(x_1,x_2)}{\partial x_1}+ \frac{\partial f(x_1)}{\partial x_1}+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Need some hints for proving a logarithmic inequality. $$\frac{\log_ax}{\log_{ab}x} +
\frac{\log_b{x}}{\log_{bc}x} +
\frac{\log_cx}{\log_{ac}x} \ge 6$$
Did as you suggested and got this, im stuck again:
$$\log_ab + \log_bc + \log_ca \ge 3$$
| If you write $\frac{\log_a x}{\log_{ab} x}=\frac{\frac{\ln x}{\ln a}}{\frac{\ln x}{\ln ab}} = \frac{\ln ab}{\ln a} = 1+\frac{\ln b}{\ln a}$, and something similar with the other fractions, you can the use the familiar inequality :
$$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\ge3$$
which can be derived easily from the inequali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Show that if two maximal values are equal on continuous functions, then there exists $\psi \in [a,b]$ with $f(\psi) = g(\psi)$ Let $f,g : [a,b] \rightarrow \mathbb{R}$ be continuous. We know that $f$ and $g$ have maximal values, as they are continuous on a closed interval. Let $M_f$ be the maximal value of $f$, and $M_... | Suppose that $f(x_1) = M_f$ and $g(x_2) = M_g$ and assume without loss of generality that $x_1 < x_2$. Now consider $h:=f-g$ restricted to the interval $[x_1,x_2]$. Note that
$$
h(x_1) = f(x_1) - g(x_1) = M_g - g(x_1) \geq 0
$$
and
$$
h(x_2) = f(x_2) - M_f \leq 0
$$
So if $h(x_1) = 0$ or $h(x_2) = 0$ we are done. Else,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1978935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Are Cartesian and spherical coordinates smoothly compatible? And is the transition map a global diffeomorphism? Consider the transition from spherical coordinates $(r, \theta, \varphi)$ to Cartesian coordinates $(x, y, z)$, given by the map
$$F:(0,\infty) \times [0, \pi] \times [0, 2 \pi) \to \mathbb R^3,\qquad (r,\th... | What you can use here is
1) A smooth map whose differential at $p$ is an isomorphism is a diffeomorphism in a neighbourhood of $p$.
2) An bijective local diffeomorphism is a diffeomorphism.
So in particular from the Jacobian determinant you see that $F$ is a local diffeomorphism at any point for which $r\neq 0$ and $\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Group a range of integers such as no pair of numbers is shared by two or more groups This is a duplicate of another question from StackOverflow. I've been advised to post it on Mathematics by another user who clearly has more experience in combinatorics than myself, and, although I have my doubts, I hope he is right.
... | Talking about pairs of numbers makes me naturally think of edges in graphs. Each pairing can be represented by an edge of the graph. Since we want all possible pairings (without repeats), we're looking at the desired state of a complete graph on $N$ vertices, $K_N$. Each group is a complete subgraph of $G$ vertices. th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
trouble with double/iterated integration of $\int^1_0[\int^1_0v(u+v^2)^4du]dv$ I have:
For $\int^1_0v(u+v^2)^4du$:
u substitution (using x instead since there's a u in there already) with $x=(u+v^2), dx/du=1$
$v\int^1_0x^4=v\frac{1}{5}x^5=v\frac{(u+v^2)^5}{5}-\frac{v(0+v^2)^5}{5}|^1_0=\frac{(v+v^3)^5}{5}-\frac{v^{15}}{... | An alternate approach would be to change the order of integration. This will greatly reduce the difficulty of the integral. As the limits of the inner integral do not depend upon $v$ this trivially becomes:
$$\int^1_0[\int^1_0v(u+v^2)^4\ dv]\ du$$
$$=\int^1_0\left[\frac{1}{10}\bigg((u+v^2)^5\bigg)_0^1\right]\ du$$
$$=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solving a system of equations involving functions Solve this system of equations:
$$
\left\{\begin{array}{cccccr}
\displaystyle{\mathrm{f}\left(x\right)} & \displaystyle{+} &
\displaystyle{3\mathrm{f}\left(x - 1 \over x\right)} & \displaystyle{=} &
\displaystyle{7x} & \displaystyle{\qquad\qquad\qquad\qquad\left(\mathr... | When given such a system of equations, you are expected to find a function (here $f$) that satisfies all those equations for all values of $x$.
Let $\frac{x-1}x=y$ and $\frac1{1-x}=z$. Then
$$f(x)+3f(y)=7x\tag1$$
$$f(y)+3f(z)=7y\tag2$$
$$f(z)+3f(x)=7z\tag3$$
Eliminate $f(z)$ by subtracting $(3)$ from $(2)$ thrice:
$$f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Second order linear differential equations : Particular Integral
How do they find the particular Integral. These two lines are not giving me any clue. Please someone explain me.
| The method of variation of parameters, however, provides one extra term.
$y_p=-\frac{1}{4}\mathrm{e}^{-2x}+\frac{x}{2}\mathrm{e}^{-2x}$
Solution Procedure:
Two fundamental solutions are $y_1(x)=\mathrm{e}^{-2x}$ and $y_2(x)=\mathrm{e}^{-4x}$.
Thus, $y_p=y_2(x)\int\frac{y_1(x)f(x)}{W(y_1,y_2)}\mathrm{d}x-y_1(x)\int\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$V = V_1\oplus V_2\oplus....\oplus V_k$, show that $f V =f V_1\oplus f V_2\oplus...\oplus f V_k$. Let $T$ be a linear operator on the vector space $V$ over the field $F$. If $f$ is a polynomial over $F$ and $a$ is in $V$, let $fa = f(T)a$. If $V_1, . . . , V_k$ are $T$-invariant sub-spaces and $V = V_1\oplus V_2\oplus.... | You almost completed one part of the question. Since the $V_i$ are $T$-invariant, they are also $f(T)$ invariant and so $fv_i = f(T)v_i \in f(T)(V_i) = fV_i$ which shows that any vector $fa \in fV$ can be written as a sum of vectors from $fV_i$. What is left is to show that the sum is in fact a direct sum, or, in other... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Functional equation (N to N) Find all $f : \mathbb{N} \to \mathbb{N}$ which satisfy the equation:
$f(d_1)f(d_2)...f(d_n)=N$
Where $N$ is a natural number and $d_i, 1 \leq i \leq n$ are all of the divisors of $N$.
| Here is my attempt.
Any $N$ will have a prime power factorisation, so its divisors are all powers of primes, or a product of prime powers.
Consider $N$ = $p^2$ with $p$ prime. Taking $f(p) = p$ and $f(1) = 1$ as given, we have $f(1)f(p)f(p^2)=p^2$, and so $f(p^2) = p$. Generalising, $f(p^k)=p$, for any positive int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If the fourier integral of a function exists, is it always equal to the fourier cosine integral of that function? I read that every function which is piecewise continuous and absolutely integrable has a Fourier integral representation which reduces to a Fourier sine or cosine integral accordingly as the function being ... | A function $f(x)=f_{e}(x)+f_{o}(x)$ where $f_e$ is even, $f_o$ is odd:
$$
f_e(x)=\frac{f(x)+f(-x)}{2},\;\; f_o(x)=\frac{f(x)-f(-x)}{2}.
$$
If $f$ is even, then $f_e=f$ and $f_o=0$. If $f$ is odd, then $f_e=0$ and $f_o=f$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why the $\lambda$-generalized eigenspace is invariant? The following comes from a text book, I am very confused about the last sentence,
For a matrix $A$, a subspace $V$ is invariant w.r.t $A$ if $AV\subseteq V$. From my understanding we need to show $\forall x\in V_{\lambda_i},Ax\in V_{\lambda_i}$, i.e. $(A-\lambda... | Indeed, you need to show $(A-\lambda_iI)^n(Ax)=0$. But this is easy since if $f(A)$ is a (analytic) function of operator $A$, then $[f(A),A]=0$, i.e. you have an operator identity $f(A) \cdot A = A \cdot f(A)$ and thus
$$(A-\lambda_iI)^n(Ax)=\left[(A-\lambda_iI)^n\cdot A\right]x=\left[A\cdot (A-\lambda_iI)^n\right]x=A\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the following limit problem I'm trying to find the following limit
$$ \lim_{x\to 0} \left({1-\sin x } {\cos x}\right)^{{ \csc 2x} }$$
How to prove the above limit equals $e^{-{1\over2}}$?
| $$ \lim_{x\to 0} \left(1-\sin x \cos x\right)^{\csc 2x}=\lim_{x\to 0} \left(1-\sin x \cos x\right)^{\frac{1}{\sin 2x}}=$$
$$ \lim_{x\to 0} \left(1-\sin x \cos x\right)^{\frac{1}{\sin x\cos x}\frac{1}{2}}=e^{-1\cdot\frac{1}{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Bellman Ford Algorithm Clarifications I'm a little hung up on the Bellman-Ford algorithm. Here is my current understanding and some questions:
1) The root is defined as a source node that has only outgoing paths from it and the goal of the algorithm is to find a path from this source node to every other node in the gra... | The selection of source node depends on you. You can choose any node to be your source irrespective of its non zero in-degree. The algorithm tries to generate the shortest path distance from your selected source node to all other nodes in the graph.
It is safe to assume that the graph is connected . For the same you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Pairing function for ordered pairs Is there a pairing function like Cantor's (https://en.wikipedia.org/wiki/Pairing_function) that would map ordered pairs (of integers) to different integers? ie:
(M, N) -> L1
(N, M) -> L2
Where L1 != L2
All input integers could be positive, but the output does not have to be positive..... | Cantor's pairing function should already work, as does a prime number encoding, e.g.
$$
\langle M, N \rangle = 2^M 3^N
$$
Example:
$$
\langle 1, 2 \rangle = 2^1 3^2 = 18 \\
\langle 2, 1 \rangle = 2^2 3^1 = 12
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integrating on open vs. closed intervals What is one difference in the values of
$$\int\limits_{\left[0,1\right]}y\, dx$$$$\int\limits_{\left(0,1\right)}y\, dx$$
and how would you calculate the values? For the sake of simplicity, let $y=x$. Conceptualizing integration as the area bounded by the function, the $x$-axis a... | It should be intuitive that $\displaystyle \int_{(0, 1)} f(x) \ dx = \int_{[0, 1]} g(x) \ dx$ where $g(x) = \begin{cases} f(x) & \ \text{ if }\ x \in (0, 1) \\ 0 & \ \text{ if } \ x \in \{0, 1\}\end{cases}$. We claim that $\displaystyle \int_{[0, 1]} f(x) \ dx = \int_{[0, 1]} g(x) \ dx$, or more generally, changing... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 0
} |
Show that a given harmonic function has a specific form Suppose $u$ is a harmonic function in $\mathbb{R}^n$ and satisfies
$$|u(x)| \leq C |x|~\forall x \in \mathbb{R}^n.$$
Show that $u(x)=q \cdot x$ where $q$ is a constant vector.
I'm struggling with this problem. I tried defining a new function, $v(x)=u(x)/{x}$ for ... | Hint: Because $u$ is harmonic, it is real analytic, and so it can be uniformly approximated in some large open ball by a high-degree polynomial. Deduce that most of the polynomial's coefficients are zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $aI have the following GRE question that I have some trouble seeing.
If $g$ is a function defined o the open interval $(a,b)$ such that $a < g(x) < x$ for all $x \in (a,b)$, then $g$ is
A) an unbounded function
B) a nonconstant function
C) a nonnegative function
D) a strictly increasing function
E) a polynomial fu... | Differentiation is not monotonic, unlike integration.
GRE Subject Test in Mathematics - Where can I find related past papers, solutions to those, sample tests, advice, books, apps or other resources?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find the sum of $\sum\limits_{n=1}^{\infty} \frac{x^{2n}}{(2n)!}$ Find the sum of $\sum_{n=1}^{\infty} \dfrac{x^{2n}}{(2n)!}$ on its interval of convergence.
We can see that the domain of convergence is $D=R$. Then let:
$$f(x)=\sum_{n=1}^{\infty} \dfrac{x^{2n}}{(2n)!}$$
$$f'(x)=\sum_{n=1}^{\infty} \dfrac{x^{2n-1}}{(2n... | Note that your equation should be $f''(x)=f(x)+1$.
Using the series for $e^x$, we get
$$
\begin{align}
e^x&=\sum_{k=0}^\infty\frac{x^k}{k!}\tag{1}\\
e^{-x}&=\sum_{k=0}^\infty(-1)^k\frac{x^k}{k!}\tag{2}
\end{align}
$$
Average $(1)$ and $(2)$
$$
\frac{e^x+e^{-x}}2=\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}\tag{3}
$$
Subtract... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = \frac{1+\sqrt{5}}{2} = \phi$, is this a coincidence? I was playing around with square roots today when I "discovered" this.
$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = x$
$\sqrt{1 + x} = x$
$1 + x = x^2$
Which, via the quadratic formula, leads me to the golden ratio.
Is there an... | If you play around a little more, you will also notice that:
$$
\frac{1+\sqrt{5}}{2} = 1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+ \ldots} } } }
$$
Which simplifies to $x = 1+ \frac 1x \implies x^2=x+1$.
It's no coincidence. I mean to say, it comes directly from the equation itself.
Just to give you another ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Image and preimage of a function
Given the function $f(x)=x^{2}-4x-5$, $A=[0,3)$ and $B=[0,1]$, find $f(A)$ and $f^{-1}(B)$.
I found $f(A)$ by looking at the graph $f([0,3))=[-9,-5]$ but how would I calculate this without the graph? If I plug in $0$ into $f$ I get $-5$, and if I plug in $3$ I get $-8$, when I should... | More general if $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = ax^2 + bx + c, a \gt 0$ then $f$ is decreasing on $(-\infty, - \frac b {2a}]$ and increasing on $[- \frac b {2a}, + \infty)$. You can use this together with the fact that $f$ is continuous to get the images you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What's the difference between $∀x\,∃y\,L(x, y)$ and $∃y\,∀x\,L(x, y)$?
*
*Everybody loves somebody. $∀x\,∃y\,L(x, y)$
*There is somebody whom everybody loves. $∃y\,∀x\,L(x, y)$
What's the difference between these two sentences? If they are same, can I switch $\exists y$ and $\forall x$?
| If $L$ satisfies 2., then it necessarily satisfies 1. Therefore you can switch $\exists y$ and $\forall x$ to go from 2. to 1., but not the other way around. Counterexample: let $L$ be a relation over set $S=\{a,b,c\}$, and suppose $L(a,b)$, $L(b,c)$, $L(c,a)$. You can easily verify that 1. holds here, but 2. does not.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 3
} |
are the integers modulo 4 a field? Basically are the integers mod 4 a field? I want to know because I am reading a text and it has a problem assuming the integers modulo any number are a field
| If $q=p^n$, then $\mathbf F_q$ denotes the field with $q$ elements.
You have to know that for any integer $n\ge 1$, there exists a finite field with $p^n$ elements, and this field is unique up to an isomorphism.
It is even unique in the still more restrictive sense: a field with $p^n$ elements is unique within a given... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Polynomial n variables differentiable on $\mathbb{R^n}$ How would I define a polynomial of $n$ variables? And how would I go on to prove that any polynomial in $n$ variables is differentiable on $\mathbb{R^n}$? (assuming this function is continuous on $\mathbb{R^n}$)
I'm struggling to find a formal definition and I ass... | Aloizio's answer should already be enough (and you should accept it), but explicitly, we'd have $$p(x_1,\cdots,x_n) = \sum a_{i_1\cdots i_n} x_1^{i_1}\cdots x_n^{i_n}.$$Since $$\Bbb R^n \ni (x_1,\cdots,x_n) \mapsto x_i \in \Bbb R$$is differentiable for each $i$, it follows that $p$ is differentiable as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculus: Finding limit $$\lim_{x\to0}\frac{\displaystyle\\x^3}{\sin^3x}.$$
Do I use L'Hopital's rule for this? But I can't seem to find the answer.
| Hint: Since $x \mapsto x^3$ is continuous and $x \mapsto 1/x$ is too, we have $$\lim_{x \to 0 }\frac{x^3}{\sin^3x} = \left( \frac{1}{\color{red}{\lim_{x \to 0}\frac{\sin x}{x}}}\right)^3.$$That limit in red you absolutely must know.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Fourier transform properties (integration) proof From Signals and Systems _ Alan V. Oppenheim
There's a property of fourier transform states as below.
Fourier transform of $\int_{-\infty}^\tau x(\tau) d\tau $ equals to $\frac{ X(j\omega)}{j\omega} + \pi \delta(\omega)X(0)$
Can someone prove this?
| I know it is kind of late for answering the question, but it might help somebody else.
I would approach it using the convolution property and the Heaviside Step Distribution u(t).
First of all, notice that:
$$f(t)*u(t) = \int_{-\infty}^{+\infty}f(s)u(t-s)ds$$
Since, for $t-s < 0 \Longrightarrow s > t$, the integrand is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
When is a function who is defined on two disjoint sets are continuous? Okay so let $f$ be a function, $A$ and $B$ two sets that are disjoint.
$f$ is continuous on $A$
$f$ is continuous on $B$.
My teacher told me that $f$ is continuous on these two disjoint sets if and only if they are mutually separated, meaning no one... | Definitely false: an easy general method for making counterexamples is take $A$ and $B$ as disjoint subsets of some larger topological space $X$. Then take a function $f^*:X\rightarrow Y$ that is continuous, and restrict its domain to $A\cup B$ to get a new function $f:A\cup B\rightarrow Y$. This function is continuous... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to prove that $a\le b$ , if $ a0$? How to prove that $a\le b$ , if $ a<b+c$ for each $c>0$?
I tried to prove it with the reductio ad absurdum method and with the trichotomy property of two real numbers $a,b$ : $a=b$ ,$a<b$ or $a>b$. But I didn't make it.
Any advice would be helpful !
| Suppose $a>b$, then $a-b>0$. Let $c=b-a$ and we have
$$a<b+(a-b)=a$$
We found a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1981885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Why is this corollary of Liouville true I know that if $f(z)$ is an entire function and $f(z)$ is bounded, then $f(z)$ is constant. I also know how this can be used to prove the fundamental theorem of algebra.
However, in my books after Liouville it states as a corollary
Suppose that f is entire and n a natural number ... | By Cauchy's Integral formula, we see that
\begin{align}
|f^{(n)}(z)| \leq C \int_{C_R} \frac{|f(\zeta)|}{|z-\zeta|^{n+1}}\ |d\zeta| \leq \int_{C_R} \frac{(1+|\zeta|^n)}{\left||\zeta|-|z| \right|^{n+1}}|d\zeta| \leq C\frac{(1+R^n)R}{|R-|z||^{n+1}} \leq C
\end{align}
for all $z$. Hence $f^{(n)}(z)$ is constant.
Moreover... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a commonly used notation for flipped composition? We have $ (f \circ g) = x \mapsto f(g(x)) $ however since I read left to write it always seems backwards to me. Is there a symbol like $ ( g \ggg f) = x \mapsto f(g(x)) $.
It is especially grating in situations like.
$$
\require{AMScd} \begin{CD} X @>{f \circ... | Some math books (especially abstract algebra) may write composition in the reverse order: $\sigma\tau$ means: first $\sigma$ then $\tau$. For notation they write
$$
x^{\sigma \tau} = \big(x^\sigma\big)^\tau
$$
For example, this may be seen with field automorphisms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
erdos.sdslabs problem Let $n$ be the largest positive integer, such that $n!$ can be expressed as the product of $(n−2014^{2015})$ consecutive integers.
Let $x$ be equal to $n$ mod $38980715857$. Find x mod $8037517$ ?
Question source is this.
.
| as we know that value of n is equal to(by solving it with number theory)
n=(2014^2015!)+1
now,
given that x=n mod38980715857.
by solving it with help of wolfram alpha we get,
x=38980715856,
further xmod8037517is equal to:
6795923
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to translate geometric intuitions about vector fields into algebraic equations By considering stereographic projections, I was asked to find a smooth vector field on $S^2$ which vanishes at 1 point, and one that vanishes at 2 points. The intuition, I think, for the vanishing at 1 point is to have all vectors emanat... | First, to address the question in your title: I think the only honest answer is that there is no "standard algorithm" for translating intuitions into equations. It takes lots of practice and lots of trial and error. Try to stretch your geometric intuition as far as you can, and then try to write down formulas to prove ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$n_1,...,n_k$ pair coprime $\!\iff\! {\rm lcm}(n_1,...,n_k)=n_1...n_k$ [lcm = product for coprimes] $n_1,...,n_k$ pairwise coprime $\iff LCM(n_1,...,n_k)=n_1...n_k$
Recently, I was told this as part of a larger proof concerning direct products of groups. I am wondering why this is true.
| Can you prove the following two statements?
*
*If $a \mid c$ and $b \mid c$ with $\gcd(a,b) = 1$ then $ab \mid c$.
*If $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $gcd(a,bc) = 1$.
Then, use induction to show that $n_1 \dots n_k$ divides $\text{lcm}(n_1,\dots,n_k)$. Since $n_1\dots n_k$ is a common multiple that divi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Roots of the polynomial $x^{p-1}+x^{p-2}+\ldots+x+1$ when $p$ is a prime. I'm studying Galois Theory and I have some doubts about the roots of the polynomial $x^{p-1}+x^{p-2}+\ldots+x+1$ when $p$ is a prime.
Let $\zeta_n$ be an $n$th root of unity, i.e, $\zeta^n-1$, and $\zeta_n \neq 1$. Then, as $(x^n-1)=(x-1)(x^{n-1}... | It's easy to show that $\zeta_p^k$ is a root of $f(x) = x^p - 1 = (x-1)(x^{p-1} + x^{p-2} + \cdots + x + 1)$ for all $1 \leq k < p$. Since it is obviously not a root of $(x-1)$, it must be a root of the other factor of $f$.
P.S. it's an important / not-too-difficult exercise to show that $\operatorname{Gal}(\mathbb{Q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to prove $Hom_A(P_0,X),Hom_A(P_1,X)$ are injective right $R$-modules? I asked a question here:https://mathoverflow.net/questions/252724/how-to-prove-hom-ap-0-x-hom-ap-1-x-are-injective-right-r-modules. But there are no responses and I really want to know how to solve it. So I repost it here. Hope for any help.
Let ... | Let $P$ be a projective $A$-module. Then we have $P\oplus Q=A^{(I)}$, for some set $I$ (free module), so $\def\H{\operatorname{Hom}_A}\H(P,X)$ is a direct summand of $\H(A^{(I)},X)\cong(\H(A,X))^I\cong X^I$ (direct product).
A direct product of injective modules is injective, as well as direct summands thereof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
In how many different ways can boys and girls sit a desks such that at each desk only one girl and one boy sits? There are $n$ boys, $n$ girls and $n$ desks. In how many different ways can boys and girls sit a desks such that at each desk only one girl and one boy sits?
I have a solution, but I have a little doubt tha... | To remove from unanswered queue:
Yes, you are correct. Approaching via multiplication principle, first choose whether the boy is on the left, or the right for each desk in sequence. Then in sequence, choose which boy is at the desk and which girl is at the desk for a final total of:
$$2^n(n!)(n!)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove by Induction with Summation Struggling with this , especially with the double summation; if anyone can help it be much appreciated!
$$\forall n \in \Bbb N : \quad \sum_{i=1}^n \sum_{c=1}^i c = \sum_{i=1}^n i(n − i + 1) .$$
It needs to be answered in the following format:
1- Prove for n=1
2- assume n=k
3- prove fo... | Rewrite the identity for $n+1$:
*
*Left hand side
$$\sum_{i=1}^{n+1} \sum_{c=1}^i c =\sum_{i=1}^{n} \sum_{c=1}^i c+\sum_{c=1}^{n+1} c.$$
*Right hand side
$$\sum_{i=1}^{n+1} i(n+1 − i + 1)=\sum_{i=1}^{n+1} i(n− i + 1)+\sum_{i=1}^{n+1}i=\sum_{i=1}^{n} i(n− i + 1)+\sum_{i=1}^{n+1}i.$$
As the additional terms are the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1982891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Using $3$ points create new coordinate system and create array of points on $XY$ plane in original coordinate system This is my first post on stack exchange :) Basically what I am trying to do is something called palletizing in the robotics world. Given $3$ points I would like to create a new coordinate system and the... | Here is a possible approach to your problem. Let $A$, $B$, and $C$, be the 3 points of interest.
1. Define your coordinate system by three orthogonal unit vectors with $A$ as the origin. Perhaps defined as
$V_1 = \frac{B-A}{\|B-A\|}$
$V_2 = \frac{V_1 \times (C-A)}{\|V_1 \times (C-A)\|}$
$V_3 = V_1 \times V_2$
2. Defin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Given a compact $S$ in $\mathbb R^m$ with the following property: For every pair of points $a, b \in S$ and for every $\varepsilon > 0$ there exists a finite set of points $\{x_0, x_1,..., x_n\}$ in $S$ with $x_0 = a$ and $x_n = b$ such that $\|x_k - x_{k-1}\|<\varepsilon$ for $k = 1, 2, . . , n$. Prove or disprove: ... | We have that $S=A\cup B$ where $A,B$ are non-empty open sets such that $A\cap B=\emptyset$. Then $A=B^c$ and $B=A^c$, and therefore, by definition, $A$ and $B$ are closed sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many ways are there to arrange two digits into an n digit number, when both digits must be used? I know the answer to this question is $2^n-2$, but I am unsure of how this answer is gotten. Could someone please explain to me how the answer is gotten step by step? Also, how many ways would there be to arrange 3,4, o... | If the number must contain both digits then the number of ways is
$$
\sum_{k=1}^{n-1} \left(
\begin{array}{c}
n\\
k
\end{array}\right) = 2^n - 2
$$
Suppose the two digits are $a$ and $b$, $a\neq b$. Then the number of ways to choose $k$ of the digits to be equal to $a$ is
$$
\left(
\begin{array}{c}
n\\
k
\end{array}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Dirac delta distribution and fourier transform Dirac delta distribution is defined as
$f(t_{0})=\int_{-\infty }^{\infty } \! f(t)\delta(t-t_{0}) \, dt $ where $f(t)$ is smooth function. Then my question is:
:Calculate Fourier transform $\hat \delta(\omega)$ from $\delta (t-t_{0})$
Solution:
$$\hat \delta(\omega)=\frac... | We have \begin{align}f(t_{0})=:\int_{-\infty }^{\infty } \! f(t)\delta(t-t_{0}) \, \text{d}t &&(1) \\ \hat g(\omega):=\widehat{g(\cdot)}(\omega):=\int_{-\infty }^{\infty }e^{-j \omega t}g(t) \ \text{d}t &&(2) \end{align}
and therefore we get:
$$\widehat{\delta(\cdot-t_0)}(\omega) \stackrel{(2)}{=} \frac{1}{\sqrt{2 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How can I prove $d_1(x,y) \leq n d_\infty (x,y)$ $$d_\infty (x,y) = \max{|x_i - y_i| | i=1,2,...,n}$$
$$d_1 (x,y)= \sum_{i=1}^n |x_i - y_i|$$
$$d_\infty : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$$
$$d_1: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$$
$$x,y\in \mathbb{R}^n$$
I would like to pr... | As C. Falcon says, you should change $d_{\infty}$ by $d_{2}$.
By Cauchy-Schwarz we have $$(\sum_{i=1}^n (|x_i - y_i|\cdot 1))^2\le (\sum_{i=1}^n |x_i - y_i|^2)(1^2+\cdots 1^2).$$
Then $$\sum_{i=1}^n |x_i - y_i|\le \sqrt{n}(\sqrt{\sum_{i=1}^n |x_i - y_i|^2}).$$
Hence $d_{1}(x,y)\le \sqrt{n}d_{2}(x,y)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Examples of quasi-projective varieties that are not (topologically) quasi-affine I'm trying to think of a quasi-projective variety that is not isomorphic to a quasi-affine one. I image that it must be $Y \subseteq \mathbb{P}^n$ of at least $n \geq 3$, and maybe $\operatorname{dim} Y \geq 2$ as well. I am also intereste... | I think $\mathbb{P}^2 - pt$ is an example, but I would have to think longer about why this can't be quasi-affine.
(I think you can argue that if it were quasi-affine, it would have global functions, but if that were the case, you would be able to find global functions on $\mathbb{P}^2$, which can't be. Hartog's extens... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
I throw $4$ dice, what is the probability of having at least one $6$? Say that I have an event that happens with probability $p$ -- lets say the probability of having a $6$ when I throw a die. Is there a formula that tells me the probability of have at least one of these events if I have $n$ simultaneous trials? Let's ... | We can use something called complementary counting to find the probability that an event doesn't happen. Then we subtract this probability from $1$ to find the probability that the event does happen. In your example, if you have $n$ trials and the probability of getting a 6 is $p$ on each trial, then the probability th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to convert this Cartesian double integral to polar coordinates? So most of the converting cartesian to polar integrals I've seen (both on this website and in class) involve some sort of circular integral region.
I was doing practice problems for my exam and I ran across one that does not and I'm stuck on how to so... | Don't try to do this sort of thing by "pure algebra" - always draw the region of integration. If you do this you will see easily that $\theta$ varies from $\pi/4$ to $\pi/2$. So we have
$$I=\int_{\pi/4}^{\pi/2}\int_?^? x^2\,J\,dr\,d\theta$$
where $J$ is the Jacobian. To find the limits for $r$, draw a line on your d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving distance between bounded and compact set in $\mathbb{R}$ Let $E,F \in \mathbb{R}$ be two non-empty closed sets, with $E$ bounded. Show that there are points $x \in E, y \in F$ such that $\text{dist}(E,F) = \lvert x - y\rvert\cdot\text{dist}(E,F)$ is defined as $\inf\{\lvert x - y \rvert: x \in E, y \in F\}$
I k... | Since
\begin{align}
\operatorname{dist}(E, F) = \inf\{|x-y| : x \in E, y \in F\},
\end{align}
then there exists a sequence of pairs $(x_n , y_n) \in E\times F$ such that $|x_n-y_n| \rightarrow \operatorname{dist}(E, F)$.
Now, since $\{x_n\}\subset E$ is bounded then it contains a convergent subsequence say $\{x_{n_k}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1983960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are these derivatives correct (the respective functions involve a square root, fraction and expansion)? Question: Use rules of differentiation to answer the following. There is no need to simplify your answer.
*
*If $y = 3x{\sqrt x}$, find $\frac{dy}{dx}$
My working:
$y = 3x^{1+\frac{1}{2}}$
$y = 3x^{\frac{3}{2}}... | Your work for Question 1 is fine. For Question 2, use the Quotient Rule (you seem to have assumed that the derivative of a fraction is the derivative of the numerator over the derivative of the denominator, which isn't the case):
\begin{align}
f' \left ( x \right ) & = \frac{5x \cdot 2 - \left ( 2x - 1 \right ) \cdot 5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Compute operator norm of $l^1$ bounded operator I have a problem with the following exercise:
We have the operator $T: l^1 \to l^1$ given by
$$T(x_1,x_2,x_3,\dots)=\left(\left(1-\frac11\right)x_1, \left(1-\frac12\right)x_2, \dots\right)$$ for $(x_1,x_2,x_3,\dots)$ in $l^1$. Showing that this operator is bounded is eas... | $$\left|\left| T((x_n)_{n\in\mathbb{N}})\right|\right|_{\ell_1}=\left|\left| \left(\left(1-\frac{1}{n}\right)x_n\right)_{n\in\mathbb{N}}\right|\right|_{\ell_1}=\sum_j \left|\left(1-\frac{1}{j}\right)x_j\right|\leqslant \sum_j |x_j |=||(x_n )_{n\in\mathbb{N}} ||_{\ell_1}$$
hence
$$||T||\leqslant 1$$
but $$||T||\geqslan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Square class of algebraic extension of finite fields This is Q17 in Chapter 2 of the book "Introduction of quadratic forms over fields".
Let $F$ be an algebraic extension of a finite field $\mathbb F$. Show that $|F^*/(F^*)^2| \le 2$.
If the extension is finite, then $F$ is itself a finite field and indeed $|F^*/(F^... |
Corollary 3.6 Let $F=\mathbb{F}_q,$ ($q$=odd).
A) If $q\equiv 1(\mod 4)$, then $W(F)\cong \mathbb{Z}_2[\dot{F}/\dot{F}^2].$
B) If $q\equiv 3(\mod 4)$, then $W(F)\cong \mathbb{Z}_4.$
Since all finite algebraic extensions of a finite field are finite, then the theorem applies, and you can use $I/I^2\cong\dot{F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Trigonometric inequality $\cos x+ \sin x>0$ Solve the inequality: $\cos x+ \sin x >0$
Why can't I square this to get $\sin 2x>0$? And what is the first step here then?
| $$\cos x+ \sin x >0\Rightarrow \frac{\cos}{\sqrt2}+\frac{\sin x}{\sqrt2}=\sin(\frac{\pi}{4}+x)\gt0$$
It follows $$x\in\bigcup\space\left]-\frac{\pi}{4}+2k\pi,\frac{3\pi}{4}+2k\pi\space\right[$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
What is the number that, when divided by $3$, $5$, $7$, leaves remainders of $2$, $3$, $2$, respectively? What is the number?
The LCM of the divisors is 105. I think this has something to do with the Chinese remainer theorem, but I am not sure how to apply this knowledge.
| We can solve in a simple way without the Chinese Remainder Theorem.
Let $n = 3x + 2 = 5y + 3 = 7z + 2$.
Then, $5y + 3 \equiv 3x + 2 \equiv (\mod 3) \implies y \equiv 1(\mod 3)y$
Hence let $y = 3k + 1$ which gives $n = 5y + 3 = 15k + 8$.
Again $n = 15k + 8 \equiv 7z + 2 (\mod 7) \implies k \equiv 1 (\mod 7)$
Hence let ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Find basis vectors of the vector space of all $4 \times 4$ magic squares I'm taking a course in linear algebra and I need to solve this problem:
Let's define a magic square as a matrix whose sums of all the
numbers on a line, a column and on both the main diagonal and the main
anti-diagonal are the same.
*
*Prove th... | Here's another basis, with an easy proof of linear independence: the entry marked with a star is the only nonzero entry in that location in any of the eight matrices.
$$\pmatrix{1*&0&0&0\cr0&0&0&1\cr0&1&0&0\cr0&0&1&0\cr}\quad\pmatrix{0&1*&0&0\cr0&0&0&1\cr0&0&1&0\cr1&0&0&0\cr}\quad\pmatrix{0&0&1*&0\cr0&0&0&1\cr0&0&1&0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Definition of Homeomorphism Background:
Let $(X, \tau_X)$ and $(Y,\tau_Y)$ be topological spaces. A bijection $\gamma:X\to Y$ is called a homeomorphism if:
*
*$\gamma$ is continuous, and
*$\gamma$ has some continuous inverse $\gamma^{-1}:Y\to X$.
Also recall that $\gamma$ is (topologically) continuous if the prei... | Let $\langle X,\tau\rangle$ be any space. Let $D=\{0,1\}$ with the indiscrete topology $\{\varnothing,D\}$, and let $Y=X\times D$ with the product topology $\tau_Y$. The projection map $\pi_X:Y\to X:\langle x,d\rangle\mapsto x$ induces a bijection from $\tau_X$ to $\tau_Y$, but it’s not a homeomorphism. Specifically, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convergence of metrics Let $U\subset\mathbb{R}^2$ be an open neighborhood of the origin and let $f_n:U\rightarrow \mathbb{R}_{>0}$, $n\in \mathbb{N}$ be a sequence of differentiable functions which uniformly converges on $U$ to an integrable function $f:U\rightarrow \mathbb{R}_{>0}$.
Fix two point $p_1,p_2\in U$ and c... | This equality is true. For each $\epsilon>0$ we can choose a path $\gamma$ between $p_1$ and $p_2$ such that $l_g(\gamma)< d_g(p_1,p_2)+\epsilon$. Since $f_n$ converges to $f$ uniformly, we can pick an $N>0$ such that for each $n>N$ we have also $l_{g_n}(\gamma)<l_g(\gamma)+\epsilon$. Therefore $d_{g_n}(p_1,p_2)\leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Find the biggest potential P (Particle) can have along this the curve C? The curve $C$ is given by $x=t×cos(t)$, and $y=sin(t)$ $C$ $R^2$,
where $t$ $∈$ $R≥0$.
*
*Find the parametrization of the curve.
*Find the biggest potential P can have along this curve.
This may be a part of the exercise as well:
*
*V... | *
*The answer is in the question:
\begin{cases}
x=t\cos t\\
y=\sin t
\end{cases}
*$P$ is under the influence of $\vec{F}=(x+2xy,y+x^2-y^2)$. A potential of $\vec{F}$ is a mapping $f(x,y):\mathbb{R}\rightarrow \mathbb{R}$ such that
$$
\vec{F}=\nabla f
$$
Solving for $f$ yields
$$
f(x,y)=\frac{x^2}{2}+x^2y+\frac{y^2}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the coefficient of $x^{80}$ in the power series expansion $\dfrac{x^2(1+x^2+x^5)}{(1-x)^2(1+x)}$ Find the coefficient of $x^{80}$ in the power series expansion $$\dfrac{x^2(1+x^2+x^5)}{(1-x)^2(1+x)}.$$
I don't know how to find coefficients in power series, solutions are greatly appreciated!
| Write it as $$\frac{x^2(1+x^2+x^5)}{(1-x)(1-x^2)}=x^2(1+x^2+x^5)\left(\sum_{h=0}^\infty x^h\right)\left(\sum_{s=0}^\infty x^{2s}\right)=\\=(x^2+x^4+x^7)\sum_{k=0}^\infty x^k\#\left\{(h,s)\,:\,0\le h,s\wedge h+2s=k\right\}=$$
if you call $a_k:=\#\left\{(h,s)\,:\,0\le h,s\wedge h+2s=k\right\}$, it continues as
$$=\sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Algebraic closures and monomorphisms Let $F$ be a field and $f\in F[X]$, an irreducible polynomial over $F$, $C$ an algebraic closure of $F$ and $a,b \in C$ two roots of $f$.
In an exercise I proved the existence of an $F-$monomorphism $\tau :C \longrightarrow C$ such that $\tau (a)=b$.
But I also have to prove that $\... | Hint: Any element $c\in C$ is algebraic over $F$ and hence is a root of some nonzero polynomial $g\in F[X]$. Observe that $\tau$ must map roots of $g$ to roots of $g$, and there are a finite number of roots of $g$ in $C$.
(More generally, this line of argument shows that if $C$ is any algebraic extension of a field $F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Number of ways a 6-letter word contains at least 1 "a" and at least 1 "b, c, or d". Repeats are allowed. I tried to approach this problem by using complementary counting. The number of ways to get a 6-letter word with no restriction is $26^6$. Then we can subtract the number of ways where the word does not contain "a, ... | Casework usually works when you want complementary counting, but the problem has a few details to take care of.
Case (1): No $a,b,c,$ or $d$.
Case (2): Yes $a$, no $b,c,$ or $d$.
Case (3): No $a$, yes $b,c,$ or $d$.
These cases are mutually exclusive, and so overcounting is not possible. The sum of these cases is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solve the differential equation $ \dfrac{ d^2x }{ dt^2 } + 6 \dfrac{ dx }{ dt } + 9x = 4t^2 + 5 $ using variation of parameters Would solving $ \dfrac{ d^2x }{ dt^2 } + 6 \dfrac{ dx }{ dt } + 9x = 4t^2 + 5 $ using variation of parameters require integration by parts or can I solve it without knowing integration by part... | Considering the general case of $$I_k=\int t^k e^{r t}\,dt$$ Using integration by parts $$u=t^k \implies du=k t^{k-1}\,dt$$ $$dv=e^{r t}\,dt\implies v=\frac{e^{r t}}{r}$$ makes $$I_k=\frac{t^k e^{r t}}{r}-\frac k r\int t^{k-1} e^{r t}\,dt=\frac{t^k e^{r t}}{r}-\frac k r I_{k-1}$$ with $I_0=\frac{e^{r t}}{r}$.
If you a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Taylor Series of trigonometric function I searched quite a bit online but could only find the MACLAURIN SERIES of $\sin x$ and $\cos x$:
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots$$
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+\cdots$$
Can anyone explain how we can expre... | The Taylor expansion of any function about any point can be found by replacing $x$ with $x-c$, assuming you want a $c$-centered expansion. You also would need to evaluate your derivatives at this point $c$ as opposed to zero. Check the general formula for a Taylor expansion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Alternative solution and generalization to a puzzle "gasoline crisis". Suppose that on a circular route, the gas stations located along the route contain just enough gas for one full trip. Prove that if one starts at the right gas station with an empty tank, one can complete the route.
The solution that is offered is:
... |
I suppose station $k$ is not unique, there can be multiple stations with the right amount of fuel?
That's right. If there are multiple "minimal" stations, then the driver can start at any of these stations. By the time the driver arrives at the next minimal station, she will just have run out of gas, so she can fill ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.