Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$\frac{f(x)-f(0)}{g(x)-g(0)}=\frac{f'(\nu(x))}{g'(\nu(x))} $ ,the value of the limit: $\lim_{x \to 0^+} \frac{\nu(x)}{x} $ Good evening,
I thought a lot about this issue.
I think I have to apply Lagrange, Taylor.
Can someone help me to calculate this limit?
$$f,g \in C^2 [0,1]: \\ f'(0)g''(0) \ne f''(0) g'(0) \\ g'(... | $\nu(0)=0$ because $0\leq \nu(x)\leq x$. Then cross-multiply, and take $O(x)$ terms.
From your last line, everything is evaluated at zero:
$$(f'+f''x/2)(g'+g''\nu'x)\approx(f'+f''\nu'x)(g'+g''x/2)\\
f'g'+(f''g'+2f'g''\nu')x/2+Ax^2\approx f'g'+(2g'f''\nu'+f'g'')x/2+Bx^2\\
f''g'+2f'g''\nu'=2g'f''\nu'+f'g''\\
\nu'=1/2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Linear transformation invariant wrt. maximum. All matrices are real. Define the operator $\max$ on matrices as a function that returns the largest value in each row.
Consider a matrix $F$ of size $n \times l$. The matrix has the property that any vector $v$ of the form $v(i) = F_{i,q(i)}$ for any mapping $q$ is in the ... | Independent of the property of the matrix $F$, it is always possible to find such a matrix $C$. Pick any "rectangular permutation matrix" (Not sure if that is an well-established term) $C$, which has in every row exactly one entry with $1$ and else zero entries and in every column at most one entry with $1$ and otherw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the zeros of $f(x)=x^3+64$ $$f(x)=x^3+64$$
Again, I am really not sure how to do this I tried to factor but it clearly was not the right answer
| At the precalculus level, any cubic polynomial (i.e. a function of the form $f(x) = ax^3 + b x^2 + cx + d$) that you are asked to find the roots of will almost certainly have one "obvious root".
In this example, can you find a value of $x$ such that $f(x) = 0$ - i.e. $$x^3 = -64$$
Once you have this one root, let's cal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to differentiate the function $f(\mathbf x) = \|\mathbf x\|^2 \mathbf x$?
Let $f:\mathbb R^n\to\mathbb R^n$ be given by the equation $f(\mathbf x)=\|\mathbf x\|^2 \mathbf x$. Show that $f$ is of class $C^\infty$ and that $f$ carries the unit ball $B(\mathbf 0;1)$ onto itself in a one-to-one fashion. Show, however,... | Hint: To make things easier for you, let's work on $n=2$ as always...
$f(x) = f(x_1,x_2) = \begin{pmatrix} (x_1^2+x_2^2)x_1 \\ (x_1^2+x_2^2)x_2 \end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
A non-vanishing one form on a manifold of arbitrary dimension So the problem I have is:
Let $\theta$ be a closed 1-form on a compact Manifold M without boundary. Further suppose that $\theta \neq 0$ at each point of M. Prove that $H^{1}_{dR}(M)\neq 0$.
The only approach I see to doing this is finding a closed loop in M... | Suppose there exists $f: M\rightarrow \mathbb{R}$ such that $df = \theta$. What happens at a maximum of $f$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to adopt the Woodbury matrix identity to this matrix formula The Woodbury matrix identity is defined as follows:
$$
{(A+UCV)}^{-1}=A^{-1}-A^{-1}U{(C^{-1}+VA^{-1}U)}^{-1}VA^{-1}
$$
I want to use the Woodbury matrix identity theorem to change the following matrix formula
$$
W={(XX^T+\lambda G)}^{-1}XY
$$
into the fol... | I'm not sure the different forms of W, as stated, are equivalent. For one thing, they do not appear to be equivalent when the matrices involved are replaced by scalars. To illustrate, let $X=a$, $G=b$ and $Y=c$. Then,
$$
\begin{eqnarray*}
W{}={}{(XX^T+\lambda G)}^{-1}XY &{}\implies{}&W{}={}\frac{ac}{a^2{}+{}\lambda b}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How many time the digit 6 appear when we count from 6(base 8) to 400 (base 8)? How many time the digit 6 appear when we count from 6(base 8) to 400 (base 8)?
I am not sure if I am going in the right path. I want to find the most accurate approach of solving this problem.
6(base 8)=6(base 10). Also, 400(base 8)= 256 (ba... | Your answer is correct. Your idea of starting from $1$ instead of $6$ makes it easy: One-eighth of the third-place (units) digits are $6$, and one-eighth of the second-place (eights) digits are $6$. So the answer is $\dfrac{400_8}{8} + \dfrac{400_8}{8} = 64_{10}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Getting wrong answer trying to evaluate $\int \frac {\sin(2x)dx}{(1+\cos(2x))^2}$ I'm trying to evaluate $$\int \frac {\sin(2x)dx}{(1+\cos(2x))^2} = I$$
Here's what I've got:
$$t=1+\cos(2x)$$
$$dt=-2\sin(2x)dx \implies dx = \frac {dt}{-2}$$
$$I = \int \frac 1{t^2} \cdot \frac {-1}2 dt = -\frac 12 \int t^{-2}dt = -\frac... | Perhaps simpler:
$$\begin{cases}1+\cos2x=1+2\cos^2x-1=2\cos^2x\\{}\\\sin2x=2\sin x\cos x\end{cases}\;\;\;\;\;\;\;\;\;\;\;\implies$$
$$\int\frac{\sin2x}{(1+\cos2x)^2}=\int\frac{2\sin x\cos x}{4\cos^4x}dx=-\frac12\int\frac{(\cos x)'dx}{\cos^3x}=-\frac12\frac{\cos^{-2}x}{-2}+C=$$
$$=\frac14\sec^2x+C\ldots\ldots\text{and y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
} |
Evaluating the limit $\lim_{x\to\infty}\frac{(1+1/x)^{x^2}}{e^x}$ Could anybody show me step by step why the following equality holds?
$$\lim_{x\to\infty}\frac{(1+1/x)^{x^2}}{e^x} = e^{-1/2}$$
The most obvious method gives you 1 as an answer, but I understand that only the limit of $(1+1/x)^x$ is $e$, but expression is... | Compute the limit of the logarithm of your function:
$$
\lim_{x\to\infty}\log\frac{(1+1/x)^{x^2}}{e^x}=
\lim_{x\to\infty}(x^2\log(1+1/x)-x)
$$
Now set $x=1/t$:
$$
\lim_{x\to\infty}(x^2\log(1+1/x)-x)=
\lim_{t\to0^+}\frac{\log(1+t)-t}{t^2}
$$
The limit becomes $\displaystyle\lim_{t\to0^+}\frac{t-t^2/2+o(t^2)-t}{t^2}=-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
(Kelley's General Topology) Exercise G chapter 1. I am finding difficultes in solving the following exercise written on the Kelley's book as in the title. Could anyone help me?
Thanks in advance.
If $A$ is dense in a topological space and $U$ is open, then $U \subseteq \overline{(A \cap U)}$.
| Suppose that $x\in U$. Since $A$ is dense, $x$ is contained in the closure of $A$. It follows that there exists a net $(x_{\alpha})_{\alpha\in D}$ in $A$ (where $D$ is some index set directed by a relation $\geq$) such that $x_{\alpha}\to x$. By the definition of convergence of nets, since $x\in U$ and $U$ is open, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Does the following type of "SVD" exist? SVD of $A$ gives $U$, $\Sigma$ and $V$ such that $A = U \Sigma V$.
I am interested in a different problem. Given an $A$ and $\Sigma_1,\ldots,\Sigma_{n-1}$ diagonal matrices, such that we know that
$$A = U_1 \Sigma_1 U_2 \Sigma_2 U_3 \Sigma_3 \ldots U_n$$ for some unitary matrices... | In general this can't be done. For example, in the $2 \times 2$ case with $V = \pmatrix{0 & 1\cr 1 & 0\cr}$ and $\Sigma_3 = \Sigma_1$,
$$ I \Sigma_1 I \Sigma_2 V \Sigma_1 V = V \Sigma_1 V \Sigma_2 I \Sigma_1 I $$
In addition, you can always multiply the $U$'s by scalars of absolute value $1$ whose product is $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove every derived set is closed if this is the case for singleton sets Suppose that for each $x \in X$, the set of accumulation points of $\{ x \}$ is closed. Then for each $S \subseteq X$, the set of its accumulation points is closed.
This is the last part of exercise $D$ of chapter 1. I managed to solve all the pre... | Suppose $S \subseteq X$ and $x \notin S'$. We need to find some open $U$ containing $x$ and such that $U$ misses $S'$.
$x \notin S'$ means that there exists some open $O$ containing $x$ such that $O \cap S \subseteq \{x\}$.
If $O \cap S = \emptyset$, we can pick $U = O$.
If $O \cap S = \{x\}$, then use that $X \setmin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is continuous $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$? Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous. Is $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$?
What metric spaces we can use instead of $\mathbb{R}$?
I guess we have same result for $f:\mathbb{R}^n \... | This holds for $f:X\to\mathbb R$ if $X$ is a connected space. For each $x\in X$, $f^{-1}([f(x),\infty))$ is closed by continuity, and open by the condition on local minima. This set is nonempty because it contains $x$, hence it equals $X$ by connectedness. Thus for all $y\in X$, $f(y)\geq f(x)$. Because $x$ and $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Proving ratio of subsequent terms in sequence <1, then limit of sequence tends to 0 If $\lim_{n \rightarrow \infty} \left|\frac{u_{n+1}}{u_n}\right|=|a|<1$, prove that $\lim_{n \rightarrow \infty} u_n=0$.
We have to prove that for $\forall \epsilon>0: |u_n|<\epsilon, \forall n>N\in\mathbb{N}$
I started by:
$$\lim_{n \r... | I don't think you can say $|u_N - 0| \leq \frac{\epsilon}{2}$ without implicitly assuming $u^* = 0$. Here's what I would do.
Start from the point where you have defined $u^*$. Let $a < 1$ be the limit of the ratio of terms of the sequence. Then we have that $$\lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Characterization of Matrices Diagonalizable by Matrices P such that P times P^Transpose is Diagonal Let $M$ be a square matrix with complex entries.
What is a characterization of $M$ such that $M = P^{T} D P$, where both $D$ and $P^{T} P$ are diagonal matrices?
For example, such a characterization includes all real sym... | It is a theorem due to Takagi that any complex (entrywise) symmetric matrix may be written as $M=PDP^T$, and $P$ may be chosen to be unitary.
As others have noted, it is clear that only symmetric matrices can be factored in this way, so that symmetry is both sufficient and necessary.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How find the maximum possible length of OC, where ABCD is a square, and AD is the chord of the circle? Given a circle $o(O(0,0), r=1)$. How to find the maximum possible length of $OC$, where $ABCD$ is a square, and $AD$ is the chord of the circle?
I have no idea how to do this, can this be proved with simple geometry?
|
With the diagram as labeled, we see that
$$\begin{align}
|\overline{OC}|^2 &= \cos^2\theta + ( \sin\theta+2\cos\theta )^2 \\
&= \cos^2\theta + \sin^2\theta + 4 \cos\theta\sin\theta + 4 \cos^2\theta \\
&= 3+2 \sin 2\theta + 2 \cos 2\theta \\
&= 3+2\sqrt{2}\left( \sin 2\theta \cos45^\circ + \cos 2\theta \sin45^\circ \r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
strongly continuous mapping implies bounded mapping Hi does anyone know how to show the result that if we have a relexive Banach space $X$ and a mapping $A: X \rightarrow X^{*}$ (not necessarily linear), which is strongly continuous, which means $$u_{n} \rightharpoonup u~~~\text{in }X\implies A(u_{n}) \rightarrow A(u)... | Thanks for responses in the comments. Is this then okay?
Proof:
Assume there is some bounded set $B \subset X$, where $A(B)$ is unbounded in $X^{*}$. Then choose a sequence $\{ A(u_{n}) \}_{n \in \mathbb{N}} \subset A(B)$ such that $\| A(u_{n}) \| > n$. Note then that $\{ u_{n}\}_{n} \subset B$ is bounded in $X$, so b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is it that an ideal is homogeneous if and only if it is generated by homogeneous elements? Hartshorne says the following on pg. 9
An ideal is homogeneous if and only if it is generated by homogeneous elements.
Take $\langle x+y,x^3+y^3\rangle$. It is generated by homogeneous elements. But how is it homogeneous? ... | If an ideal I is homogeneous, and if a possibly non-homogeneous P(x,y) element belongs to I, each homogeneous component of P(x,y) belongs to I. Therefore, given any system of generators of $I$, each of their homogeneous components belo,gs to $I$ – in other words, $I$ is generated by the homogeneous components of a syst... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $C^\infty(\mathbb{R}^n)$ is dense in $L^2(\mathbb{R}^n, (1 + |\xi|^2)^s d\xi$
I would like to show that $C^\infty(\mathbb{R}^n)$ is dense in the space $L^2(\mathbb{R}^n, (1 + |\xi|^2)^s d \xi)$ (here, $s$ is an arbitrary element of $\mathbb{R}$).
I am familiar with the standard argument that if $\phi \in C... | Hint.
A possible approach is to use the fact that the Schwartz space $\mathcal S(\mathbb{R})$, the space of functions all of whose derivatives are rapidly decreasing, is dense in $L^2(\mathbb{R}^n, (1 + |\xi|^2)^s d \xi)$ (recall that a rapidly decreasing function is essentially a function $f(\cdot)$ such that $f'(\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
An analogue to Cantor's theorem Cantor's theorem states that for all sets $$|A| < |2^A|$$
I was interested in a similar proposition. If $A$ is a set, denote by $A! := \{f : A \rightarrow A \mid f \text{ is a bijection}\}$. Is it true in general that $|A| < |A!|$?
It is not to difficult to show that $|\mathbb{N}| < |\ma... | Taking a hint from Unit’s comment down below (ultimately coming from Factorials of Infinite Cardinals by Dawson and Howard, it seems), here’s a case differentiation proving the result in ZF (as far as I can tell).
In case $|A| = |2 × A|$: Mapping $2^A → (2×A)!,~ B ↦ σ_B$, where $σ_B$ swaps the two copies of $B$ in $2×... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Continuous map to a subspace I'm studying the book Topology, Geometry, and Gauge Fields by Gregory L Naber. This is for self study.
I'm trying to prove the first part of Lemma 1.1.2
Let $Y$ be a subspace of $Y'$. If $f:X \to Y'$ is a continuous map with $f(X) \subseteq Y$, then, regarded as a map into $Y$, $f:X \to Y$... | Because $f(X) \subseteq Y$, $f^{-1}(W) = f^{-1}(Y \cap W)$ for any set $W \subseteq Y'$.
(Proof: If $x \in X$ is such that $f(x) \in W$, then $f(x) \in Y \cap W$. Conversely, if $f(x) \in Y \cap W$, then $f(x) \in W$.)
So, in your proof, we have
$f^{-1}(U) = f^{-1}(Y \cap U') = f^{-1}(U')$, which is open.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does Green's $\mathcal{J}$-relation define a total order on the equivalence classes? In a semigroup $S$ we define $a\le_\mathcal{J} b$ iff $a=xby$ for some $x,y\in S^1$. Defining $a\equiv_\mathcal{J} b$ by $a\le_\mathcal{J} b$ and $b\le_\mathcal{J} a$ gives a partial order on $S/\equiv_\mathcal{J}$. Is this order alway... | No it is not always a total order.
For example, in the semigroup $S$ with elements $a$, $b$, and $c$ and multiplication in the table below:
\begin{equation*}
\begin{array}{c|ccc}
&a&b&c\\\hline
a&a&a&a\\
b&a&b&a\\
c&a&a&c
\end{array}
\end{equation*}
The $\mathscr{J}$-order on $S$ has $b$ and $c$ incomparable, $a\leq_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why do complex eigenvalues correspond to a rotation of the vector? We have a linear transformation $T: \Bbb R^m \to \Bbb R^n$ defined by $T(x)=Ax$ for $x \in \Bbb R^m$ and $A \in M_{n \times m}(\Bbb R)$. I understand why real-valued eigenvalues of $A$ correspond to scaling the length of the associated eigenvectors, bu... | For a real matrix, if $a + bi$ is an eigenvalue, so is $a - bi$. And in fact, the classic matrix with this pair of eigenvalues is
$$
\begin{bmatrix}
a & -b \\
b & a
\end{bmatrix}.
$$
You probably know some theorems that say you can sometimes diagonalize a matrix by changing basis. In the cases where diagonalization is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 2
} |
Polynomial with a root modulo every prime but not in $\mathbb{Q}$. I recently came across the following fact from
this list of counterexamples:
There are no polynomials of degree $< 5$ that have a root modulo every prime but no root in $\mathbb{Q}$.
Furthermore, one such example is given: $(x^2+31)(x^3+x+1)$ but I ... | If you just want an easy example of polynomial that has root modulo every prime but not in $\mathbb Q$ — just take e.g.
$$
(x^2-2)(x^2-3)(x^2-6)
$$
(it has this property since the product of two non-squares mod p is a square mod p).
One more interesting example is $x^8-16$ (standard proof uses quadratic reciprocity).
A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Counter example for Poincare inequality does not hold on unbounded domain The Poincare inequality states that if domain $\Omega$ is bounded in one direction by length $d>0$ then for any $u\in W_0^{1,p}(\Omega)$ we have
$$ \int_\Omega|u|^p\,dx\leq \frac{d^p}{p}\int_\Omega |\nabla u|^p\,dx $$
Now I assume the domain $U\... | Actually you can build an example out of almost any function: Just notice that if $u: B(0,1)\to \mathbb{R}$, then $u_r(x)=u(x/r)$ is defined in the ball of radius $r$ and you get
$$
\| u_r\|_{p,B(0,r)} = r^{n/p}\| u\|_{p,B(0,1)}, \quad \| \nabla u_r\|_{p,B(0,r)} =r^{-1} r^{n/p} \| \nabla u\|_{p, B(0,1)}.
$$
Translati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$\int \left[\left(\frac{x}{e}\right)^x + \left(\frac{e}{x}\right)^x\right]\ln x \,dx$ Integrate:
$$\int \left[\left(\frac{x}{e}\right)^x + \left(\frac{e}{x}\right)^x\right]\ln x \,dx$$
This question looks like of the form of $$\int\ e^x(f(x)+f'(x))\,dx,$$ but don't know how to get the proper substitution?
| Hint
Try with
\begin{align*}
y & = \left(\frac{e}{x}\right)^{x}\\
\ln y & = x[1-\ln x]\\
\frac{y^{'}}{y} & = -\ln x.
\end{align*}
Now for the first component think of a similar idea.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
definition of product of modules I have been given this definition of a product between modules:
If $I$ is an indexing set with $M_i$ as an $R$-Module then the product
$\prod \limits_{i \in I} M_i$ is defined as the set consisting of I indexed tuples $(x_i)_{i\in I}$ for each $i \in I$ which is made into an R module by... | Consider $\prod \limits_{i \in I} M_i$.
The sum is $$( a_i)_{i \in I } + ( b_i)_{i \in I } = ( a_i + b_i)_{i \in I }$$ The action of R : $$ r \cdot ( a_i)_{i \in I } = ( r \cdot a_i)_{i \in I }$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $Y^2 + X^2(X+1)^2$ is irreducible over $\mathbf R$
Show that $Y^2 + X^2(X+1)^2$ is irreducible over $\mathbf R$.
Are there some general tricks for avoiding barbaric computations in general case?
| If it were reducible it would factor in two non-unit factors. Either both of them are of degree 1 in $Y$ or one of them is of degree zero in $Y$ and the other is of degree 2 in $Y$.
In the second case we see that the factor that is of degree zero in $Y$ must divide $Y^2$. Therefore it must be a unit, which is a contra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Could one be a friend of all? The social network "ILM" has a lot of members. It is well known: If you choose any 4 members of the network, then one of these 4 members is a friend of the other 3.
Proof: Is then among any 4 members always one who is a friend of all
members of the social network "ILM"?
Note: If membe... | Suppose there isn't a person who knows everyone.
Then take vertex $v$, he doesn't know everyone, in particular there is a $w$ he does not know.
If $w$ doesn't know anybody we are done, if he does select a friend of $w$, Preferably somebody who does not know somebody besides $v$ (if everybody that knows $w$ knows everyb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Closed form for integral of inverse hyperbolic function in terms of ${_4F_3}$ While attempting to evaluate the integral $\int_{0}^{\frac{\pi}{2}}\sinh^{-1}{\left(\sqrt{\sin{x}}\right)}\,\mathrm{d}x$, I stumbled upon the following representation for a related integral in terms of hypergeometric functions:
$$\small{\int... | $$\sinh^{-1}(\sqrt{\sin x}) = \sum\limits_{n=0}^{\infty} \frac{(-1)^n (2n)!}{2^{2n}(n!)^2}\frac{{(\sin x)}^{(2n+1)/2}}{2n+1}$$
so the integral is equivalent to
$$\sum\limits_{n=0}^{\infty} \frac{(-1)^n (2n)!}{2^{2n}(n!)^2(2n+1)}\int_0^{\pi/2}(\sin x)^{(2n+1)/2}\,dx$$
You have
$$\int_0^{\pi/2}(\sin x)^{(2n+1)/2}\,dx=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Continuous bijection at boundary of open set Suppose $f:U \to V$ is a continuous bijection, where $U \subset \mathbb{R}^n$,$V \subset\mathbb{R}^m$ and $U$ is open. Suppose further that $U \ni x_n \to x \notin U$. Then $y_n:=f(x_n)$ may not necessarily converge. I have two questions:
*
*Define $d_n:= \text{distanc... | Both of them are incorrect. For example, for $n = m=1$, $U= (0,3\pi/2)$ and $f(x) = \cos x$. Then if $x_n \in U$ and $x_n \to 3\pi/2$, we have $y_n \to 0$ and $0$ is an interior point of $V = [-1, 1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to evaluate a sum which contains limit variables?
How to evaluate a sum which contains limit variables?
For example: $$\lim_{n\to\infty}\sum_{i=1}^n\frac{n-1}n\frac{1+i(n-1)}n $$
And would the result necessarily be rational, because each term appears to be the multiplication of two rational fractions?
| Notice that in the sum, each piece has a common factor of $\frac{n-1}{n^2}$ which can be pulled out of the sum as per the distributive property since it does not depend on the index, $i$.
$$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n \frac{n-1}{n}\frac{1+i(n-1)}{n} = \lim\limits_{n\to\infty}\left(\frac{n-1}{n^2}\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Simplifying $2^{30} \mod 3$ I simply cannot seem to get my head around this subject of mathematics. It seems so counter-intuitive to me.
I have a question in my book: Simplify $2^{30}\mod 3$
This is my attempt: $2^{30}\mod 3 = (2\mod 3)^{30}$ (using the laws of congruences).
$2\mod 3$ must be equal to $2$, right? $0\... | You are being asked to find the least non-negative residue modulo $3$ (or perhaps the residue with least absolute value. For the first the options are $0,1,2$ and for the second you can choose $-1,0,1$. The residue is essentially the same as the remainder on division by $3$.
Now $2^{30}=(2^{10})^3=(1024)^3$ is of the o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Confused by how to derive the derivative of $f(\boldsymbol{x})=g(\boldsymbol{y})$ I was watching an online tutorial and saw this derivation.
It seems the the author took the derivative with respect to y on left side and to x on right side. I thought dx should always be in the denominator and should on both side of the... | It's curious that you should call this a "derivation" because this is more accurate than saying that the derivative in the usual sense was taken. The construction used was the exterior derivative, or differential. This satisfies
$$df(x,y)=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$
for a function ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
What is the value of this continued fraction? I am curious about the value of the continued fraction $$1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\dots}}}}}.$$
*
*Can we evaluate it ?
*Is it a nice value ?
Clearly it should be a transcendental number. But I have no idea about calculate it.
| Here's the info on this continued fraction and others.
http://mathworld.wolfram.com/ContinuedFractionConstants.html
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
formula to calculate number of arch with certain angel could be fixed in a circle I'm looking looking for a formula to calculate how many arches with certain angle could be fixed around a circle or in circular formation. I want to use that formula to write a procedure for MSWlogo for designing purpose. applying trial a... | The argument of the normal to an arch changes by $a$ as we traverse that arch. The normal then gets turned back by $\pi-b$ by the angle. Thus, each arch is rotated $a+b-\pi$ from the previous one.
Thus, there should be
$$
n=\frac{2\pi}{a+b-\pi}
$$
arches before they repeat the argument of the normal.
In degrees rather... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is $\rightarrowtail$ used for? I have come across this symbol many times, but I am unsure as to how to correctly use it.
So I can read up on it, what is the name of this mapping function?
When would it be correct to use and when wouldn't you use it?
I think it may be used when you haven't specified a function for ... | It is usually used to denote an injection.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Non-additive-subtractive prime sequence Call the following a NON additive-subtractive prime sequence or lets name it Gary's sequence. It goes like this: let a(0)=2. The next term is defined as smallest prime number which cannot be expressed as the sum and/or difference of any previous terms (note that it means using an... | This seems identical to:
OEIS A138000 "Least prime such that the subsets of { a(1),...,a(n) } sum up to 2^n different values."
http://oeis.org/A138000
Discovered by S. J. Benkoski and P. Erdos, On weird and pseudoperfect numbers, Math. Comp. 28 (1974) 617-623
You're following in the footsteps of giants ;-)
Your next t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Can a function be continuous but not Hölder on a compact set? Is it possible to construct a function $f: K \to \mathbb{R}$, where $K \subset \mathbb{R}$ is compact, such that $f$ is continuous but not Hölder continuous of any order? It seems like there should be such a function--it would probably oscillate wildly, li... | Take a look at the functon
$$f(x) = \sum_{n=0}^\infty {\sin(2^n x)\over n^2}.$$
This function is continuous on the line. Much ugliness ensues when you try to show it's Hölder continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Does $\sum_{n\ge0} \sin (\pi \sqrt{n^2+n+1}) $ converge/diverge? How would you prove convergence/divergence of the following series?
$$\sum_{n\ge0} \sin (\pi \sqrt{n^2+n+1}) $$
I'm interested in more ways of proving convergence/divergence for this series.
My thoughts
Let
$$u_{n}= \sin (\pi \sqrt{n^2+n+1})$$
trying to ... | $$\lim_{n\rightarrow\infty}{\sin\left(\pi\cdot\sqrt{n^2+n+1}\right)}=\sin{\lim_{n\rightarrow\infty}\left(\pi\cdot\sqrt{n^2+n+1}\right)}=$$
$$=\sin\left(\pi\cdot\lim_{n\rightarrow\infty}\left(\sqrt{n^2+n+1}-(n+1)+(n+1)\right)\right)=$$
$$=\sin\left(\pi\cdot\left(\lim_{n\rightarrow\infty}\left(\sqrt{n^2+n+1}-(n+1)\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Let $G$ to be abelian group and $|G|=mn$ when $(m,n)=1$. $G_m=\{g\mid g^m=e\}$,$G_n=\{g\mid g^n=e\}$, prove isomorphism I want to prove $ f:G_n\times G_m\rightarrow G$ when $f(g,h)=gh $ is an isomorphism.
First of all I showed that $G_m,G_n$ are subgroups of $G$ (easy).
Now I want to show that for every $ a,b, \in ... | Suppose
$$f(a,b):=ab=e\implies a=b^{-1}$$
Buth te last equality is impossible as $\;a\in G_n\;,\;\;b\in G_m\;$ and thus the only possible element in both of them is the unity, i.e. $\;G_n\cap G_m=\{e\}\;$ .
For a counter example with $\;G\;$ non abelian take $\;G=S_3\;$ , though in this case $\;G_2\;$ is not a subgrou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Compute the contraction of a 1-form with a vector field Question: Let $\alpha$ be the $1$-form on $\mathbb{R}^3$ given by $\alpha=zdy-ydz$ and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,y,-z)$. Compute $i_\mathbb{X}\alpha$
Answer: $i_{(0,y,-z)}(zdy-ydz)=i_{(0,y,-z)}(zdy)-i_{(0,y,-z)}... | Your computation of $i_{\mathbb{X}}\alpha$ is correct. Let me explain some general facts that work on any manifold (at least locally), but keep it restricted to $\mathbb{R}^3$.
First note that a one-form $\alpha$ is a function on vector fields. Any one-form can be written as a linear combination of $dx$, $dy$, and $dz$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Convergence of series of $\sin(x/n^2)$ For the convergence of series of $\sin\left(\frac{x}{n^2}\right)$, is it enough to say that since, for large $n$, $$a_n:= \sin\left(\frac{x}{n^2}\right) \approx b_n:= \frac{x}{n^2},$$ so that $\displaystyle\lim_{n\to\infty} \dfrac{a_n}{b_n} \ \text{ exists}$, and by the limit comp... | Note it suffices we check this for $x\geqslant 0$. We can use that $\sin{x}n^{-2}\leqslant {x}{n^{-2}}$ and that if $x>0$ is fixed and $n>N$ large enough, $\sin({x}{n^{-2}})\geqslant 0$ (since $\sin$ is positive on $(0,\pi/2)$). Thus, by comparison, $$0\leqslant \sum_{n>N}\sin(xn^{-2})\leqslant x\sum_{n>N} n^{-2}<\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Convergence of $ \int_{-\infty}^\infty \cos(x\log(\lvert x\lvert ))\,dx $ Show that the improper integral
$$ \int_{-\infty}^\infty \cos(x\log(\lvert x\lvert ))\,dx $$
is convergent.
I rewrote it, using even function symmetry of cosine, as twice the integral from zero to +infinity. Now the argument of log is simply $x$... | Notice that it suffices to consider the convergence of
$$ \int_{1}^{\infty} \cos (x \log x) \, dx. $$
Now let $f : [1, \infty) \to [0, \infty)$ by $f(x) = x \log x$. This function has an inverse $g = f^{-1}$ which is differentiable. So we have
$$ \int_{1}^{R} \cos (x \log x) \, dx
= \int_{0}^{f(R)} g'(y)\cos y \, dy
= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the field of fractions and the integral closure of a subring of $\mathbb Z[x]$. Let $R$ be a subring of $\mathbb{Z}[x]$ consisting of polynomials such that the coefficients of $x$ and $x^2$ are zero.
*
*Find the field of fractions of $R$.
*Find the integral closure of $R$ in it's field of fractions.
*Show th... | I'm guessing your argument for $2$ is that you can use factorization in $\mathbf{Z}[x]$ to factor in $R$.
The problem is that when you use factorization in $\mathbf{Z}[x]$, the factors will may not be in $R$! In particular, consider the problem of factoring of $x^8$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Surface Integral without Parametrization
Let $S$ be the sphere $x^2 + y^2 + z^2 = a^2$.
*
*Use symmetry considerations to evaluate $\iint_Sx\,dS$ without resorting to parametrizing the sphere.
*Let F $= (1, 1, 1)$. Use symmetry to determine the vector surface integral $\iint_S$F$\cdot \, dS$ without parame... | Integrals of the form $$\iint x \, dS$$ refer to center of mass calculations. This double integral will give the $x$ coordinate of the center of mass of your surface, likewise for $y$ and $z$. However, the center of mass of a sphere is at its center, and this sphere is centered at the origin. Therefore, this surface in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Besides proving new theorems, how can a person contribute to mathematics? There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:
*
*Organizing known results into a coherent narrative in the form of lecture notes or a textbo... | Another way to contribute that hasn't been mentioned yet is advocacy. It is important to raise awareness on the importance of mathematics and scientific literacy in general. For instance, here are a few concepts that it would be useful to disseminate to the general public:
*
*that some knowledge of mathematics, stat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "80",
"answer_count": 9,
"answer_id": 7
} |
How does $\sum_{k=0}^n (pe^t)^k{n\choose k}(1-p)^{n-k} = (pe^t+1-p)^n$? How does $\sum_{k=0}^n (pe^t)^k{n\choose k}(1-p)^{n-k} = (pe^t+1-p)^n$?
Where $e$ is Euler's number and $p,n$ are constants. ${n\choose k}$ is the binomial coefficient.
If context helps, I'm currently trying to show that the moment generating funct... | The binomial theorem, or binomial identity or binomial formula, states that:
$$(x+y)^n=\sum\limits_{k=0}^n{n\choose k}x^{n-k}y^k$$ where $n\in\mathbb{N}$ and $(x,y)\in\mathbb{R}$ (or $\mathbb{C}$).
Then for $x=(1-p)$ and $y=pe^t$ you get your formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Non exact sequence of quotients by torsion subgroups $$0\rightarrow G_{1}\rightarrow G_{2}\rightarrow G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\bar{G_{i}}$ the quotient of $G_{i}$ by its torsion subgroup.
I want to show that $$0\rightarrow \bar{G_{1}}\rightarrow \bar... | Suppose your sequence is always exact. Then take $G_2$ any torsionfree group and $G_1$ any subgroup of $G_1$. Then also $G_1$ is torsionfree and exactness would imply that $G_2/G_1$ is torsionfree.
But for every abelian group $G$ there exists a surjection $F\to G$, with $F$ a free group (in particular torsionfree). So ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find solutions of $\alpha x^n = \ln x$ How can I find the solutions of $$\alpha x^n = \ln x$$ when $\alpha \in \mathbb{R}$ and $n\in \mathbb{Q}$? Or, if it is not possible to have closed form solutions, how can I prove that there exist one (or there is no solution) and that it is unique? (I'm particularly interested in... | In the case that $\alpha n\not =0$, we have
$$\alpha x^n = \ln x$$
$$ \alpha nx^{n}= n\ln x$$
$$ \alpha nx^{n}= \ln x^{n}$$
$$ e^{\alpha nx^{n}} = x^{n}$$
$$ 1= \frac{x^{n}}{e^{\alpha nx^{n}}}$$
$$ 1= x^{n}e^{-\alpha nx^{n}} $$
$$ -\alpha n= -\alpha nx^{n}e^{-\alpha nx^{n}} $$
$$ W(-\alpha n)= -\alpha nx^{n} $$
$$ -\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Is there an easy way to calculate $\int_0^{2\pi} \sin^2x\ \cos^4x\ dx $? I want to calculate the integral
$$\int_0^{2\pi} \sin^2x\ \cos^4x\ dx $$
by hand. The standard substitution $t=\tan(\frac{x}{2})$ is too difficult.
Multiple integration by parts together with clever manipulations can give the result.
But it woul... | Hint: $$e^{ix} = \cos x + i \sin x$$ and the binomial theorem.
Even further to simplify the algebra a bit initially you can use $$\sin^2x + \cos^2x = 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Is $1234567891011121314151617181920212223......$ an integer? This question came from that one and from that talk where it's noted that "integers have a finite count of digits", so that the "number" in the title is not at all a number (not integer nor rational or real) . This statement seems to me justified because I su... | Your number is not an integer. Suppose that you can have an integer with an infinite number of digits. Then by Cantor's diagonal argument you can show that the set of all integers in uncountable, which is a contradiction if you accept the Peano axioms as suggested by Ahaan S. Rungta. Thus your number cannot be an integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 3
} |
Find other iterated integrals equal to the given triple integral I want to find all the other five iterated integrals that are equal to the integral
$$\int_{1}^{0}\int_{y}^{1}\int_{0}^{y} f(x,y,z)dzdxdy$$
I have so far found these three
$$\int_{0}^{1}\int_{0}^{x}\int_{0}^{y} f(x,y,z)dzdydx$$
$$\int_{0}^{1}\int_{0}^{y}\... | The bounds of integration determine equations that bound a solid $S$ in three-dimensional space. After you integrate with respect to the first variable, you should orthogonally project $S$ along the axis specified by that first variable onto the plane spanned by the other two variables. That projection then determines... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Units in a ring of fractions
Let $R$ be a UFD and $D \subseteq R$ multiplicative set. What are the units in $D^{-1}R$?
I assume the answer should be $D^{-1}R^{\times}$, but I get stuck:
If $a/b$ is a unit, then there exists $c/d$ so that
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{1}{1} \Longleftrightarrow ac = bd,$$
bu... | We can assume $0\notin D$, or the ring $D^{-1}R$ would be trivial. Also it's not restrictive to assume $1\in D$.
Let $a/b$ be a unit in $D^{-1}R$. Then there exists $c/d\in D^{-1}R$ such that
$$
\frac{a}{b}\cdot\frac{c}{d}=\frac{1}{1}
$$
so
$$
ac=bd
$$
In particular, $ac\in D$, because $b,d\in D$. Thus $a$ divides an e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
reduction order method $xy''-(1+x)y'+y=x^2e^{2x}$, $y_1=1+x$ reduction order method $xy''-(1+x)y'+y=x^2e^{2x}$, being $y_1=1+x$ a solution of the homogeneous equation.
I made y=u(1+x) and got $u''(x^2+x)-u'(x^2+1)=x^2e^{2x}$
Then i did $u'=w$ and obtained $w=\frac{e^{2x}x^2+c_1e^xx}{(x+1)^2}$, however $u=\int{w }dx$ wi... | Your work so far looks correct, so to find $u=\displaystyle\int\frac{x^{2}e^{2x}+Cxe^x}{(x+1)^2}dx=\int\frac{(xe^x)(xe^x+C)}{(x+1)^2}dx$,
use integration by parts with $w=xe^x+C, dw=(x+1)e^x dx$ and $ dv=\frac{xe^x}{(x+1)^2}dx, \;v=\frac{e^x}{x+1}$
to get $\displaystyle u=(xe^{x}+C)\left(\frac {e^x}{x+1}\right)-\int e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $ \sum_{n=1}^\infty \ln\big(n\sin \frac{1}{n}\big)$ converges.
Prove that $\displaystyle \sum_{n=1}^\infty \;\ln\left(n\sin\frac{1}{n}\right)$ converges.
My Work:
$$\left|\ln \left(n \sin \frac{1}{n}\right)\right| \leq\left|\ln \left(n \sin \frac{1}{n^{2}}\right)\right| \leq\left|\ln \left(\sin \frac{1}{n^... | Let's amply use Taylor series to give the leading orders.
Note that $\sin\left( \frac{1}{n} \right) \approx \frac{1}{n} - \frac{1}{6n^3}$. So $n \sin \left( \frac{1}{n} \right) \approx 1 - \frac{1}{6n^2}$.
Note also that $\ln(1 - x) \approx x + x^2 + \dots$, so that
$$\ln\left( n \sin \frac{1}{n} \right) \approx \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$ Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$
I seem to struggle with this simple proof. All I managed to find is that ac+bd=-4 (which might not even be correct).
| 1st Method
$\begin{align}\left(a^2+b^2\right)\left(c^2+d^2\right)=1& \implies (ac+bd)^2+(ad-bc)^2=1\\&\implies (ac+bd)^2\le1\end{align}$
2nd Method
$\begin{align}\left(a^2+b^2\right)+\left(c^2+d^2\right)=2& \implies \left(a^2+c^2\right)+\left(b^2+d^2\right)=2\\&\implies 2(ac+bd)\le 2\qquad \text{(by A.M.- G.M. Inequal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 5
} |
How do I integrate: $\int\sqrt{\frac{x-3}{2-x}} dx$? I need to solve:
$$\int\sqrt{\frac{x-3}{2-x}}~{\rm d}x$$
What I did is:
Substitute: $x=2\cos^2 \theta + 3\sin^2 \theta$. Now:
$$\begin{align}
x &= 2 - 2\sin^2 \theta + 3 \sin^2 \theta \\
x &= 2+ \sin^2 \theta \\
\sin \theta &= \sqrt{x-2} \\
\theta &=\sin^{-1}\sqrt{... |
$$I=\int\sqrt{\frac{x-3}{2-x}}~{\rm d}x$$
Integrating
Let $x=2\cos^2t+3\sin^2t$, $dx=\sin2tdt$
$$I=\int\sqrt{\frac{-\cos^2t}{-\sin^2t}}\sin2tdt=\int2\cos^2tdt=\int(1+\cos2t)dt=t+\frac12\sin2t+c\\I=\underbrace{\cos^{-1}\sqrt{3-x}}_{\pi/2-\sin^{-1}\sqrt{3-x}}+\sqrt{x-2}\sqrt{3-x}+c\\I=\underbrace{\sqrt{x-2}\sqrt{3-x}}_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Convergence of $\int_{1}^{\infty} \frac{\sin x}{x^{\alpha}}dx$ For which values of $\alpha > 0$ is the following improper integral convergent?
$$\int_{1}^{\infty} \frac{\sin x}{x^{\alpha}}dx$$
I tried to solve this problem by parts method but I am nowhere near to the answer. :(
| As David Mitra mentioned in the comments you can split the integral up. First we do
$$\int_{1}^{\infty} \frac{\sin x}{x^{\alpha}}dx = \int_{1}^{\pi} \frac{\sin x}{x^{\alpha}}dx + \int_{\pi}^{\infty} \frac{\sin x}{x^{\alpha}}dx$$
Now the first part converges for sure (you can approxiamate it with $\frac{1}{x^a}$). For ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What are other methods to Evaluate $\int_0^{\infty} \frac{y^{m-1}}{1+y} dy$? I am looking for an alternative method to what I have used below. The method that I know makes a substitution to the Beta function to make it equivalent to the Integral I am evaluating.
*
*Usually we start off with the integral itself that ... | I think something might be wrong with your answer, suppose $m\geq 0$, let $v = 1+y$. Your integral becomes
$$I=\int\limits_1^{\infty} \frac{(v-1)^m-1}{v}dv . $$
We can then apply the binomial expansion and transform the integral into
$$I= \int\limits_1^{\infty} \frac{ \sum\limits_{i=0}^{m}\left[ {m \choose i} (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Find the angle if the area of the two triangles are equal?
Let $I$ be the incenter of $\triangle ABC$, and $D$, $E$ be the midpoints of $AB$, $AC$ respectively. If $DI$ meets $AC$ at $H$ and $EI$ meets $AB$ at $G$, then find $\measuredangle A$ if the areas of $\triangle ABC$ and $\triangle AGH$ are equal.
I played a... | This is an easy problem to solve with trilinear coordinates. We have $I=[1,1,1]$, $D=\left[\frac{1}{a},\frac{1}{b},0\right]$ and $E=\left[\frac{1}{a},0,\frac{1}{c}\right]$. Moreover, we have $H=[1,0,\mu]$ and $G=[1,\eta,0]$, with $\det(D,I,H)=\det(E,I,G)=0$, so:
$$ H=\left[\frac{1}{b}-\frac{1}{a},0,\frac{1}{b}\right],\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Must algebraic extensions of the same degree have subfields of the same degree? Let $\mathbb F$ be a field and let $\mathbb K_1$ and $\mathbb K_2$ be finite extensions of $\mathbb F$ with the same degree, that is, $[\mathbb K_1:\mathbb F]=[\mathbb K_2:\mathbb F]$. Now, assume that $\mathbb K_1$ contains a subfield of d... | Let $F=\mathbb{Q}$. Let $K_1$ be the splitting field of $f_1=x^4+8x+12$ over $\mathbb{Q}$, and let $K_2$ be the splitting field of $f_2=x^{12}+x^{11}+\cdots+x+1$ over $\mathbb{Q}$ (this is just the $13$th cyclotomic polynomial). Then
$$[K_1:F]=[K_2:F]=12$$
However, $\mathrm{Gal}(K_1/F)\cong A_4$ and $\mathrm{Gal}(K_2/F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to show that $ \int^{\infty}_{0} \frac{\ln (1+x)}{x(x^2+1)} \ dx = \frac{5{\pi}^2}{48} $ without complex analysis? The Problem
I am trying to show that $ \displaystyle \int^{\infty}_{0} \frac{\ln (1+x)}{x(x^2+1)} \ dx = \frac{5{\pi}^2}{48}$
My attempt
I've tried substituting $x=\tan\theta$, and then using the subs... | We have:
$$I=\int_{0}^{+\infty}\frac{\log(u+1)}{u^3+u}\,du =\int_{0}^{+\infty}\int_{0}^{1}\frac{1}{(u^2+1)(1+uv)}\,dv\,du$$
and by exchanging the order of integration, then setting $v=\sqrt{w}$:
$$ I = \int_{0}^{1}\frac{\pi +2v\log v}{2+2v^2}\,dv =\frac{\pi^2}{8}+\int_{0}^{1}\frac{v\log v}{1+v^2}\,dv=\frac{\pi^2}{8}+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 1
} |
Manipulating Partial Derivatives of Inverse Function In lectures we're told:
$$\dfrac {\partial y} {\partial x} = \dfrac 1 {\dfrac {\partial x} {\partial y}}$$
as long as the same variables are being held constant in each partial derivative.
The course is 'applied maths', i.e non-rigorous, so don't confuse me.
But anyw... |
The rule presumably fails because one of the partial derivatives is $0$
No, not because of that. Take
$$\xi = x - y \qquad \eta = x+y$$
then
$$x = \frac12(\xi+\eta) \qquad y = \frac12(\eta-\xi)$$
so $\dfrac {\partial x} {\partial \xi} = \dfrac12$ and $\dfrac {\partial \xi} {\partial x} = 1$.
The correct way to fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Limit involving square roots, more than two "rooted" terms The limit is
$$\lim_{x\to\infty} \left(\sqrt{x^2+5x-2}-\sqrt{4x^2-3x+7}+\sqrt{x^2+7x+5}\right)$$
which has a value of $\dfrac{27}{4}$.
Normally, I would know how to approach a limit of the form
$$\lim_{x\to\infty}\left( \sqrt{a_1x^2+b_1x+c_1}\pm\sqrt{a_2x^2+b_2... | Rewrite it as:
$$\left(\sqrt{x^2+5x-2} - \left(x+\frac{5}{2}\right)\right) -\\ \left(\sqrt{4x^2-3x+7}-\left(2x-\frac{3}{4}
\right)\right) +\\
\left(\sqrt{x^2+7x+5}-\left(x+\frac{7}{2}\right)\right)+\left(\frac{5}{2}+\frac 34+\frac{7}2\right)$$
Or something like that. Each of the first three terms has limit zero...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Why are there only limits and colimits? Part of my intuition about the construction of limits and colimits is based on the idea that they are initial and terminal objects in the appropriate category: The limit of a diagram $D$ is of course a/the final object in the category of cones over $D$, and similarly for colimits... | Here's one of the several equivalent ways of thinking about limits and colimits in which this question doesn't arise: for a diagram $J$ and a category $C$, there is a natural diagonal functor $C \to C^J$. If limits of shape $J$ always exist in $C$, they organize themselves into a right adjoint for this functor, and if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
How many $4\times 3$ matrices of rank three are there over a finite field of three elements?
Let $M$ be the space of all $4\times 3$ matrices with entries in the finite field of three elements. Then the number of matrices of rank three in $M$ is
A. $(3^4 - 3)(3^4 - 3^2)(3^4-3^3)$
B. $(3^4 - 1)(3^4 - 2)(3^4 - 3)$
C. $(... | Sorry for the previous post.the answer I am getting is c
$(3^4-1)(3^4-3)(3^4-3^2)$
For the $1$st column we have $4$ places and $3$ elements to fill it.So $3^4$ choices but the elements can't be all zero.S0 we have $3^4-1$ choices .For the second column we have $3^4$ choices but the second column cant be linearly depen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Factorise a matrix using the factor theorem Can someone check this please?
$$
\begin{vmatrix}
x&y&z\\
x^2&y^2&z^2\\
x^3&y^3&z^3\\
\end{vmatrix}$$
$$C_2=C_2-C_1\implies\quad
\begin{vmatrix}
x&y-x&z\\
x^2&y^2-x^2&z^2\\
x^3&y^3-x^3&z^3\\
\end{vmatrix}$$
$$(y-x)
\begin{vmatrix}
x&1&z\\
x^2&y+x&z^2\\
x^3&y^2+xy+x^2&z^3\\
\... | What you did is correct. But there is an easier way. Remember that for polynomial $p(x)$, if $p(a)=0$ then $(x-a)$ is a factor of $p(x)$.
Denote the determinant by $\Delta$. It is obviously a polynomial in $x,\ y$ and $z$. Now, note that:
*
*$x=0\implies \Delta = 0$, so $x$ is a factor of $\Delta$. Same for $y = 0$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solving $2x^2 \equiv 7 \pmod{11}$ I'm doing some previous exams sets whilst preparing for an exam in Algebra.
I'm stuck with doing the below question in a trial-and-error manner:
Find all $ x \in \mathbb{Z}$ where $ 0 \le x \lt 11$ that satisfy $2x^2 \equiv 7 \pmod{11}$
Since 11 is prime (and therefore not composite),... | $$2x^2\equiv 7\pmod{11}\\2x^2\equiv -4\pmod{11}\\x^2\equiv -2\pmod{11}\\x^2\equiv 9\pmod{11}\\x^2\equiv\pm3\pmod{11}$$
As HSN mentioned since $11$ is prime than there are at most 2 solutions,
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
"Bizarre" continued fraction of Ramanujan! But where's the proof?
$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$
"Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued fractions so i have no idea how to attempt to prove it.
| The left hand side is $ \tanh \frac{\pi}{2} $ so it may be worth looking at a continued fraction for $ \tanh x $. It looks like a special case of Gauss's continued fraction (I can't link to it directly but there is a Wikipedia page on it). Also, Ramanujan is not mentioned.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Natural Lie algebra representation on function space
There is natural Lie group representation of $GL(n)$ on $C^\infty(\mathbb{R}^n)$ given by
\begin{align}
\rho: GL(n) & \rightarrow \text{End}(C^\infty(\mathbb{R}^n)) \\
A & \rightarrow \left( \rho(A):f \rightarrow f\circ A^{-1} \right)
\end{align}
What is the as... | Your computation is correct, but you were too pessimistic about the continuation: the second derivative terms in the bracket of $\rho(X)$ and $\rho(Y)$ cancel.
It becomes less reasonable to ask for a Lie algebra repn on not-smooth functions, since natural Lie algebra repns are differential operators. Nevertheless, once... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Given $a+b+c$, Can I calculate $a^2+b^2+c^2$? I want to calculate $a^2 + b^2 + c^2$ when I am given $a+b+c$.
It is known that a,b,c are positive integers.
Is there any way to find that.
| If $a+b+c = k$ then $(a+b+c)^2 = k^2 \implies a^2+b^2+c^2 = k^2 - 2(ab + ac + bc)$, hence you would have to know the value of $ab + ac + bc$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Partial derivative notation I still have a little problem with notation for partial derivatives.
Let
$$
f(x,y) = x^2y
$$
What do you think that this should equal to?
$$
\frac{\partial f}{\partial x}(y,x) =\, ?
$$
There are two options $2yx$ or $y^2$.
Do you think that following is the same?
$$
\frac{\partial f(y,x)}{\p... | If $f$ is defined by $f(x,y)$, the operation $\frac{\partial f}{\partial x}$ means that you differentiate $f$ with respect to $x$. With your example this will become $2xy$. Now, $\frac{\partial f}{\partial x}(y,x)$ means, that you FIRST differentiate with respect to $x$ (which was $2xy$) and AFTERWARDS plug in $(x,y)=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 3
} |
Is there a generalization of the Lagrange polynomial to 3D? What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image interpolation.
| If you fix the value $x$ one variable you have the Lagrange interpolation formula
$$ f(x, y) = \sum_{k=1}^N f(x, y_k)\prod_{i \neq k} \frac{y_i - y}{y_i - y_j} $$
For each fixed value $y = y_k$ you can construct a Lagrange polynomial for the function $f(x, y_k)$.
$$ f(x, y_k) = \sum_{k=1}^N f(x_k, y_k)\prod_{i \neq k} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
What does "decidability" of a Model mean exactly? I'm looking at the theorem concerning the Model of Arithmetic:
M arith = (Integers, +, *, <) is undecidable.
What does the "decidability" of a model mean exactly? Does that mean that "the problem of determining if the given model satisfies any FOL statement" is undeci... | In this case, the undecidability means there is no algorithm for deciding in general whether a sentence in our language is true in $\mathbb{Z}$.
To prove the undecidability, it is easiest to use the fact that there is no algorithm for deciding whether a sentence over the the usual language for arithmetic is true in the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Do you decline a multiplier in reading a mathematical formula in Russian? How do you read "Порядок определителя равен $2n$"? Is it "двум эн" or is it "два эн"?
And in a sum, do you read $c = a_5 + a_6$ as "це равно а пятому плюс а шестому"? Or does the plus sign interfere with the declension in some way?
| Since Grigory M won't be posting an answer, I will post the answer I've received here, for future reference.
For the question about $2n$, it seems that the multiplying numeral $2$ can either be left invariable as "два" or put in the dative as "двум," with the latter option considered more correct by some people who hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
When the arguments of two roots of a quadratic equation are equal? Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$
How can I obtain the condition for $$\arg z_1=\arg z_2$$ containing $a,b,c?$
At present I have, Since $$z_1z_2=\dfrac{c}{a}$$ If $\arg z_1=\arg z_2,$
... | The roots $(-b\pm\sqrt{b^2-4ac})/2a$ are a real positive multiples of each other.
For some positive $\mu$,
$$-b+\sqrt{b^2-4ac}=\mu(-b-\sqrt{b^2-4ac})$$
$$(\mu-1)^2b^2=(\mu+1)^2(b^2-4ac)$$
$$b^2=\frac{(\mu+1)^2}{\mu}ac$$
The latter function of $\mu$ can take any value not smaller than $4$, and the condition is
$$b^2=\la... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Laplace tranform of $t^{5/2}$ It is asked to transform $t^{5/2}$.
I did $t^{5/2}=t^3\cdot t^{-1/2}$. Then followed the table result $$L\{{t^nf(t)}\}=(-1)^n\cdot\frac{d^n}{ds^n}F(s)$$
However i got $\frac{1}{2} \cdot\sqrt\pi \cdot s^{-7/2}$ instead of $\frac{15}{8} \cdot\sqrt\pi \cdot s^{-7/2}$.
Can you help me with the... | For every real number $r>-1$, we have,
$$L(t^r)(s)=\int_0^\infty e^{-st}t^rdt\\\hspace{60mm}=\int_0^\infty e^{-x}(\frac{x}{s})^r\frac{dx}{s}\hspace{10mm}\text{(Putting $x=st$)}\\\hspace{20mm}=\frac{1}{s^{r+1}}\int_0^\infty e^{-x}x^rdt\\\hspace{5mm}=\frac{\Gamma(r+1)}{s^{r+1}}.$$
So for $r=\frac{5}{2}$, we have, $$L(t^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
how to find number of sides of an irregular polygon? An irregular polygon has one angle 126 degrees and the rest 162 degrees. how many sides are there in this irregular polygon?
I have tried to find our the sides with the formula of a regular polygon; obviously it doesn't work.
| Using the formula of the sum of interior angles
$$162(n-1) + 126 = 180(n-2) \Rightarrow n=18. $$
Remark: I assume a simple and plane polygon.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
What is the terminology for "lemma of lemma" Let's say I need to prove a main theorem, to prove which I need three lemmas. Thus in writing the structure is as follows:
Lemma 1
Proof
Lemma 2
Proof
Lemma 3
Proof
Theorem
Proof
But if when proving Lemma 1, I need to prove another two results ("lemmas for the proof of a l... | You can use Claim to set it on a lower footing than a lemma, but it's really not a big deal. One lemma can build on other lemmas.
The biggest distinction you want to make is between the results you consider broad and important (theorems), narrow but important for the proof of a theorem (lemmas), and immediate conseque... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Trace of a matrix $A$ with $A^2=I$ Let $A$ be a complex-value square matrix with $A^2=I$ identity.
Then is the trace of $A$ a real value?
| Another way.
$$A^2-I=0,$$
Since minimal polynomial divide this polynomial (why) then it must be either
$(x-1)(x+1)$ or $(x-1)$ or $(x+1)$.
hence all the eigenvalues are reals (even in set $\{1,-1\}$) (why) and hence the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1091929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
How to define an affine transformation using 2 triangles? I have $2$ triangles ($6$ dots) on a $2D$ plane.
The points of the triangles are: a, b, c and x, y, z
I would like to find a matrix, using I can transform every point in the 2D space.
If I transform a, then the result is x. For b the result is y, and for c the r... | There is a neat formula for your case
$$
\vec{P}(p_1; p_2) = (-1)
\frac{
\det
\begin{pmatrix}
0 & \vec{x} & \vec{y} & \vec{z} \\
p_1 & a_1 & b_1 & c_1 \\
p_2 & a_2 & b_2 & c_2 \\
1 & 1 & 1 & 1 \\
\end{pmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Derivative of a continuous function. Let $g:R\to R$ be a continuous function with $g(x+y)=g(x)+g(y), \forall x,y\in R.$ Find $\frac{dg}{dx},$ if it exist.
| First we have $g(0) = 2g(0)\to g(0)=0$. Then it follows that
$$g'(x) = \lim_{y\to 0}\frac{g(x+y)-g(x)}{y} = \lim_{y\to 0}\frac{g(y)}{y} = \lim_{y\to 0}\frac{g(y)-g(0)}{y} = g'(0)$$
As noted below in the comments, this only shows that if $g'(x)$ exist for one $x$ then it exist for all $x$ and is indeed a constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Calculate the limit of $(1+x2^x)/(1+x3^x)$ to the power $1/x^2$ when $x\to 0$ I have a problem with this: $\displaystyle \lim_{x \rightarrow 0}{\left(\frac{1+x2^x}{1+x3^x}\right)^\frac{1}{x^2}}$.
I have tried to modify it like this: $\displaystyle\lim_{x\rightarrow 0}{e^{\frac{1}{x^2}\ln{\frac{1+x2^x}{1+x3^x}}}}$ and ... | As usual we take logs here because we have an expression of the form $\{f(x) \} ^{g(x)} $. We can proceed as follows
\begin{align}
\log L&=\log\left\{\lim_{x\to 0}\left(\frac{1+x2^{x}}{1+x3^{x}}\right)^{1/x^{2}}\right\}\notag\\
&=\lim_{x\to 0}\log\left(\frac{1+x2^{x}}{1+x3^{x}}\right)^{1/x^{2}}\text{ (via continuity of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Computing $\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$ I want to compute
$\int_0^\infty u^{-1}(1-e^{\frac{-u^2 t}{2}})\sin(u(|x|-r))\,du$ and so ,as shown below, I want to compute
$$\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$$
Attempt
We split the first integral into two:
*
*$\displaystyle \int_0^\infty u... | Are you wanting just an answer or the full solution. Assuming $b$ is positive and Real, Mathematica says:
$$
\int_0^\infty \frac{\sin{u}}{u}e^{-u^2 b}du\quad= \quad\frac{1}{2}\pi \,Erf(\frac{1}{2\sqrt{b}})
$$
Where $Erf$ is the error function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Find $\sin^3 a + \cos^3 a$, if $\sin a + \cos a$ is known
Given that $\sin \phi +\cos \phi =1.2$, find $\sin^3\phi + \cos^3\phi$.
My work so far:
(I am replacing $\phi$ with the variable a for this)
$\sin^3 a + 3\sin^2 a *\cos a + 3\sin a *\cos^2 a + \cos^3 a = 1.728$. (This comes from cubing the already given state... | Squaring
$\sin a + \cos a = b$,
$b^2
=\sin^2a+2\sin a \cos a + \cos^2 a
= 1+2\sin a \cos a
$,
so
$\sin a \cos a
=(b^2-1)/2
$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 1
} |
The n-th prime is less than $n^2$? Let $p_n$ be the n-th prime number, e.g. $p_1=2,p_2=3,p_3=5$. How do I show that for all $n>1$, $p_n<n^2$?
| In Zagier's the first 50 million prime numbers a very elementary proof is given that for $n > 200$ we have
$$\pi(n) \ge \frac23 \frac{n}{\log n}$$
where $\pi(x)$ is the number of primes below $x$ (as well as a bound in the other direction). In fact, it already holds for $n \ge 3$, as can be directly checked.
Suppose t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 2,
"answer_id": 0
} |
Solve $x^5 - x = 0$ mod $4$ and mod $5$ I'm trying to solve $$x^5-x=0$$ in $\mathbb{Z/5Z}$ and $\mathbb{Z/4Z}$
I don't see how to proceed, could you tell me how ?
Thank you
| In $\mathbb{Z}/5\mathbb{Z}$, the answer follows quickly from Fermat's Little Theorem.
In $\mathbb{Z}/4\mathbb{Z}$, by the Euler Totient Theorem, we have
$$x^{\varphi(4)} \equiv x^2 \equiv 0 \mod 4$$
for $x$ coprime to $4$. The only remaining cases are $x=0$ and $x=2$, which can easily be checked.
I observe that the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why does it seem to decrease probability to roll more times? I am looking how to find the probability of rolling 10 sixes in 60 rolls. After reading Calculate probabilities. 40 sixes out of a 100 fair die rolls & 20 sixes out of a 100 fair die roll, it seems that I can use the following formula: $60 \choose 10$$(\frac{... | The formula you quote is for getting exactly (not at least) $10\ 6$'s in $60$ rolls. The factor $(1-\frac 16)^{60-10}$ ensures that the other dice roll something other than $6$. Your extension is correct for getting exactly (not at least) $10\ 6$'s in $120$ rolls. The expected number is $20$, so it is not surprising... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Equivalent definitions of a surface do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post.
Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or regular submanifold of $\mathbb{R}^3$ of codimension 1, namely a subset $S\subset\ma... | The answer to my question is yes, and is given by Theorem 5-2 in Spivak Calculus on Manifolds combined with Theorem 5.8 in Lee Introduction to Smooth Manifolds. See also here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Prim , Kruskal or Dijkstra I've a lot of doubts on these three algorithm , I can't understand when I've to use one or the other in the exercise , because the problem of minimum spanning tree and shortest path are very similar . Someone can explain me the difference and give me some advice on how I can solve the exercis... | Prim and Kruskal are for spanning trees, and they are most commonly used when the problem is to connect all vertices, in the cheapest way possible.
For Example, designing routes between cities.
Dijkstra, is for finding the cheapest route between two vertices.
For Example, GPS navigation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How many ways to tie $2$ ropes so that we do not have a loop BdMO 2014 Higher Secondary:
Avik is holding six identical ropes in his hand where the mid portion of the rope is in his
fist. The first end of the ropes is lying in one side, and the other ends of the rope are lying
on another side. Kamrul randomly choos... | After doing what he did he has three ropes: $1,2,3$. Suppose he puts them on the table, He begins tying rope $1$, the probability he ties it with another rope is $\frac{4}{5}$ since there are $5$ other rope ends and only $1$ is the same rope. If it ties with another rope there are now two ropes, proceed to tie the rope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
uniform continuity in $(0,\infty)$ Prove uniform continuity over $(0,\infty)$ using epsilon-delta of the function
$$f(x)=\sqrt{x}\sin(1/x)$$
I know that I start with: For every $\varepsilon>0$ there is a $\delta>0$ such that for
any $x_1$, $x_2$ where $$|x_1-x_2|<\delta$$ then $$|f(x_1)-f(x_2)|<\varepsilon$$
But I can'... | Note that
$$\left| \sqrt{x} \sin (1/x)- \sqrt{y} \sin (1/y) \right|\\ \leqslant \sqrt{x}\left| \sin (1/x)- \sin (1/y) \right|+|\sqrt{x}-\sqrt{y}||\sin(1/y)| \\\leqslant \sqrt{x}\left| \sin (1/x)- \sin (1/y) \right|+|\sqrt{x}-\sqrt{y}|.$$
Using $\displaystyle \sin a - \sin b = 2 \sin \left(\frac{a-b}{2}\right)\cos \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding value of 1 variable in a 3-variable $2^{nd}$ degree equation The question is: If $a,b,\space (a^2+b^2)/(ab-1)=q$ are positive integers, then prove that $q=5$. Also prove that for $q=5$ there are infinitely many solutions in $\mathbf N$ for $a$ and $b$. I simplified the equation as follows:-$$\frac {a^2+b^2}{ab-... | For such equations:
$$\frac{x^2+y^2}{xy-1}=-t^2$$
Using the solutions of the Pell equation. $$p^2-(t^4-4)s^2=1$$
You can write the solution.
$$x=-4tps$$ $$y=t(p^2+2t^2ps+(t^4-4)s^2)$$
It all comes down to the Pell equation - as I said. Considering specifically the equation:
$$\frac{x^2+y^2}{xy-1}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Solve the following functional equation $f(xf(y))+f(yf(x))=2xy$ Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $f(xf(y))+f(yf(x)=2xy$.
By putting $x=y=0$ we get $f(0)=0$ and by putting $x=y=1$ we get $f(f(1))=1$.
Let $y=f(1)\Rightarrow f(x)+f(f(x)f(1))=2x$, which tells us that $f$ is an injective functi... | This is all I have found so far:
Just keep feeding the snake with its own tail. As you noted $f(f(1))=1$. So with $x=y=f(1)$ we then see that $1=f(f(1))=(f(1))^2$ so that $f(1)=\pm 1$. As you almost correctly noted, we have for $y=f(1)$ that
$$
f(x)+f(f(x)f(1))=2xf(1)
$$
which indeed shows that $f$ is injective. With t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$ Given: Let $G$ be a group, and let $\mathcal{S}$ be the set of subgroups of $G$. For $g\in G$ and $H\in S$, let $g\cdot H=gHg^{-1}$
Question: Deduce that if $G$ is a finite $p$-group, for some prime $p$, the... | Let $X$ be the set of not-normal subgroups of $G$. A conjugate of a not-normal subgroup is not-normal again, so $G$ acts on $X$ by conjugation and (by construction) there are no trivial orbits, so the size of every orbit is a positive power of $p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Derangement, example, paradox? How can we explain that $!0 =1 $, but $!1=0$?
I understand the case of permutations. I get why $0! =1$, and that $1!$ is also $1$.
This result doesn't argue with my intuition.
But, when it comes to derangements, it's hard to understand for me. Please help me.
| $0!$ is one, since the number of bijections between $\emptyset$ and $\emptyset$ is $1$. moreover the one bijection that exists does not send any element of $\emptyset$ to itself, so it is also a derrangement and we conclude $!0=1$
The number of bijections from $\{1\}$ to $\{1\}$ is also $1$, so $1!=1$. However this bij... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Help on proving a trigonometric identity involving cot and half angles Prove: $\cot\frac{x+y}{2}=-\left(\frac{\sin x-\sin y}{\cos x-\cos y}\right)$.
My original idea was to do this:
$\cot\frac{x+y}{2}$ = $\frac{\cos\frac{x+y}{2}}{\sin\frac{x+y}{2}}$, then substitute in the formulas for $\cos\frac{x+y}{2}$ and $\sin\fra... | Try using
$ \displaystyle \sin{x} - \sin{y} = \sin{\frac{2x}{2}} - \sin{\frac{2y}{2}} =
2 \frac{\sin \left( {\frac{x+y}{2} + \frac{x-y}{2}} \right) - \sin \left( {\frac{x+y}{2} - \frac{x-y}{2}} \right)}{2}$
and then the product to sum-formula for sine, i.e.
$ \displaystyle \frac {\sin \left({x + y}\right) - \sin \left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A problem for laplace operator in Sobolev space Suppose $u\in L^2(\Omega)$, then for any $\phi\in C_c^\infty(\Omega)$ we have
$$ \int_\Omega v\,\phi\,dx=\int_\Omega u\Delta \phi\,dx $$
Then can I conclude that $u\in H_0^1\cap H^2(\Omega)$ and
$$\Delta u=v $$
Also assume that $\Omega\subset\mathbb R^N$ is open bounded... | Integrating by part, we can get
$$\int_\Omega u\frac{\partial^2\phi}{\partial x_i^2}\,dx=-\int_\Omega \frac{\partial u}{\partial x_i}\frac{\partial\phi}{\partial x_i}\,dx+\int_{\partial \Omega}u \frac{\partial\phi}{\partial x_i}\nu_i\,dS$$
and
$$\int_\Omega \frac{\partial u}{\partial x_i}\frac{\partial\phi}{\partial x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Riemann Hypothesis, is this statement equivalent to Mertens function statement? All:
I saw one form of Riemann Hypothesis, it says:
$$
\lim ∑(μ(n))/n^σ
$$
Converges for all σ > ½
Is this statement same as the order of Mertens function is less than square root of n ?
| Yes, since $\frac{1}{\zeta(\sigma)} = \sum{\frac{\mu(n)}{n^\sigma}}$, this is equivalent to the more canonical statement of RH that $\zeta$ has no zeroes to the right of the critical line.
You also mention $M(x) = O(x^{\frac{1}{2}+\epsilon})$, you can use the Mellin transform to show this is also equivalent to RH.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1094888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"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.