Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find eigenvalues and eigenvectors of the operator $A$ The question is: Find the eigenvalues and eigenvectors of the operator $A$ on $\Bbb{R}^3$ given by $A\mathbf{x}=|\mathbf{a}|^2 \mathbf{x}- (\mathbf{a} \cdot \mathbf{x}) \mathbf{a}$, where $\mathbf{a}$ is a given constant vector. How do you know without any calculati... | Notice that $A$ is a symmetric matrix. Because $A_{ij}=(Ae_{j}\cdot e_{i})=|a|^{2}(e_{j}\cdot e_{i})-(a\cdot e_{j})(a\cdot e_{i})=(Ae_{i}\cdot e_{j})=A_{ji}$. So it is diagonalizable and has an orthonormal eigen basis.
Intuition behind the eigen vectors: Notice that $A'x:=(a\cdot x)a$ is a projection matrix projecting ... | {
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How to do this limit: $\lim\limits_{n\to \infty}\large\frac{\sum_{k=1}^n k^p}{n^{p+1}}$? $$\large\lim_{n\to \infty}\large\frac{\sum_{k=1}^n k^p}{n^{p+1}}$$
I'm stuck here because the sum is like this: $1^p+2^p+3^p+4^p+\cdots+n^p$.
Any ideas?
| Hint: Recall that if $f$ is integrable on $[a,b]$, then:
$$
\int_a^b f(x)~dx = \lim_{n\to \infty} \dfrac{b-a}{n}\sum_{k=1}^n f \left(a + k \left(\dfrac{b-a}{n}\right) \right)
$$
Can you rewrite the given sum in the above form? What might be an appropriate choice for $a$, $b$, and $f(x)$?
| {
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Multiplication in symmetric product space STATEMENT: Let $V=\mathbb{R}^2$.Take $Y:=\left\{x\cdot y: x,y\in V\right\}$ where $S_2(V)$ is the symmetric product of $V$.
QUESTION: What is multiplication in the symmetric product space?
| The symmetric power $S^p(V)$ of a vector space $V$ is defined as the quotient $V^{\otimes p} / (v_1 \otimes \dotsc \otimes vp = v_{\sigma(1)} \otimes \dotsc \otimes v_{\sigma(p)} : v_i \in V, \sigma \in \Sigma_p)$. The natural isomorphism $V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes p+q}$ extends to a linear map... | {
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Sum of convergent and non-convergent series, does it converge? And how to prove? Series $a_n$ is convergent and $b_n$ is not-convergent. Will the sum $a_n + b_n$ converge? I think it will not converge, But how do I show it?
I believe I have to use the definition.
$|a_n - A| < \epsilon$
$|b_n - B| >= \epsilon$
Then choo... | It is not converge. because $$b_n=(b_n+a_n)-a_n,$$ so if $\sum b_n+a_n $ converges then $b_n$ so is.
| {
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Basic probabilities: one poisson then 2 binomial (Wasserman 2.14 - 11) I am self-refreshing some stats concepts by reading "All of Statistics" (Wasserman 2004) and am puzzled by the following problem (section 2.14, exercise 11):
Let N ~ Poisson($\lambda$) and suppose we toss a coin N times and let $p$ be the probabili... | Let us first partition the event $(Y=y,X=x)$ into disjoint events indexed by $N$:
$$\begin{align}
\mathbb{P}(X=x,Y=y)&=\sum_n \mathbb{P}(X=x,Y=y,N=n),\\
&=\sum_n \mathbb{P}(X=x,Y=y|N=n)\mathbb{P}(N=n),\\
&=\sum_n \mathbb{P}(Y=y|X=x,N=n)\mathbb{P}(X=x|N=n)\mathbb{P}(N=n).
\end{align}$$
Up to this point this is exactly w... | {
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A converse of schur's lemma Suppose $\rho: G \rightarrow GL(V)$ is a representation. and if $T: V \rightarrow V$ is a linear operator such that $T\circ \rho_g= \rho_g\circ T$ for all $g\in G$ implies $T=k\cdot Id$ for some number $k$. (i.e. $T$ is $G$-invariant/ $G$-intertwining implies $T$ is a homothety). Prove that ... | Hint: If there is a non-trivial proper subspace, find a $G$-complement. Think about the projection map to one of the factors.
| {
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Sum of all triangle numbers Does anyone know the sum of all triangle numbers? I.e
1+3+6+10+15+21...
I've tried everything, but it might help you if I tell you one useful discovery I've made:
I know that the sum of alternating triangle numbers, 1-3+6-10... Is equal to 1/8 and that to change
1+3+6... Into 1-3+6... You... | The $r$-th triangular number is
$$T_r=\frac {r(r+1)}2=\binom {r+1}2$$
i.e. $1, 3,6, 10, ...$ for $r=1, 2, 3, 4, ...$.
The sum of the first $n$ triangular numbers is
$$S_n=\sum_{r=1}^n T_r=\color{blue}{\sum_{r=1}^n \binom {r+1}2=\binom {n+2}3}=\frac {(n+2)(n+1)n}6$$
i.e. $1, 4, 10, 20, ...$ for $n=1, 2, 3, 4...$. This... | {
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Find flux using usual method and divergence theorem (results don't match) I'm trying to calculate complete flux through a pyramid formed by a plane and the axis planes. I can't to come to the same answer using usual method and the divergence theorem.
The plane is
$2x+y+z-2=0$
The vector field is
$\vec F=(-x+7z)\vec k$... | Since the divergence of the given vector field is constant, we can just multiply the volume of the pyramid with the divergence. The pyramid has ground surface (in the $xy$-plane) $1$, because it is a right triangle with base $1$ and height $2$. The height of the pyramid is $2$, so the pyramid volume is $\frac{1}{3}\cdo... | {
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Primes of the form $an^2+bn+c$? Wondering if this has been proven or disproven. Given:
$a,b,c$ integers
$a$, $b$, and $c$
coprime
$a+b$ and $c$ not both even
$b^2$-$4ac$
not a perfect square
are there infinite primes of the form $an^2 + bn + c$?
| Take $a = 1$, $b = 0$, $c = 1$. It is currently open whether there are infinitely many primes of the for $x^2 + 1$.
| {
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Is this theory complete? I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are:
$\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$,
$\psi \equiv \forall x \exists y \exists z(y\ne z \land P(x, y) \land P(x,z) \land \forall v (P(x,v) \to (v=y ... | This theory axiomatizes the class of graph of degree $2$ such that for all positive integers $n$ there exists a path of length $n$. Needless to say, such a graph has to be infinite to satisfy the last condition.
To prove that the theory is not contradictory, it suffices to come up with a model. An example of a model is... | {
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Converting expressions to polynomial form My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook.
Page 57. Exercise 12. Show that the following are polynomials by converting them to the form $\sum_{k=0}^{m}a_kx^k$ for a suitable $m$. In each case $n$ is a positive integ... | I think I understand all questions from this problem and I have the solutions, so since no one is posting good answer I will, hope its ok.
Answer to $a)$ part of the problem I already listed but here it is anyway.
$a)$ Using binomial theorem, we have $$(1+x)^{2n}=\sum_{k=0}^{2n}(^{2n}_{k})x^k.$$
For $b)$, @DiegoMath's ... | {
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show that the solution is a local martingale iff it has zero drift Most financial maths textbook state the following:
Given an $n$-dimensional Ito-process defined by
\begin{equation}
X_t = X_0 + \int_0^{t} \alpha_s \,d W_s + \int_0^{t} \beta_s \,d s,
\end{equation}
where $(\alpha_t)_{t \geq0}$ is a predictable process ... | Suppose that $(X_t)_{t \geq 0}$ is a local martingale. Since
$$X_0 + \int_0^t \alpha_s dW_s$$
is also a local martingale, this means that
$$M_t := X_t - \left( X_0 + \int_0^t \alpha_s \, dW_s \right) = \int_0^t \beta_s \, ds$$
is a local martingale. Moreover, $(M_t)_{t \geq 0}$ is of bounded variation and has continuou... | {
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Every ideal is contained in a prime ideal that is disjoint from a given multiplicative set Let $R$ be a ring $I\subset R$ an ideal and $S\subset R$ be a set for which holds:
$1)$ $1\in S$
2) $a,b \in S\Rightarrow a\cdot b\in S$
Show that there exists a prime ideal $P$ in $R$ containing $I$ with $P\cap S =\emptyset$
An... | It's a standard argument in commutative algebra. Consider the set of all ideals that contain $I$ and avoid $S$, ordered by inclusion. By Zorn's lemma there are maximal elements (since unions are upper bounds) and you can prove that an ideal maximal via inclusion among those avoiding $S$ and containing $I$ turns out to ... | {
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Arithmetic progressions. "Consider an 4 term arithmetic sequence. The difference is 4, and the product of all four terms is 585. Write the progression".
My way of finding the progression seems like it will take too long, but here it is, anyway:
$$a_1\cdot a_2\cdot a_3\cdot a_4=585$$
$$a_1\cdot (a_1+4)\cdot (a_1+8)\cdot... | Another way to do it is to factor $585$, which turns out to be
$$3^2\cdot 5\cdot 13$$
And using the factorization, find four factors which each differ by $4$.
$$1\times 5\times 9\times 13$$
| {
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If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$? If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$?
I know the proof that as rings, $\mathbb Z_{... | gcd$(a, bc) = 1 \Rightarrow$ there exists no prime integer $p$ such that $p|a$ and $p|bc \Rightarrow$ there exists no prime integers $r, s$ such that $r|a, r|b$ and $s|a, s|c \Rightarrow$ gcd$(a, b) = 1$ and gcd$(a, c) = 1.$
On the other hand, suppose gcd$(a, b) = 1$ and gcd$(a, c) = 1.$ Let $p$ be a prime integer such... | {
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Find and solve a recurrence relation for the number of n-digit ternary sequences in which no 1 appears to the right of any 2. Find and solve a recurrence relation for the number of n-digit ternary sequences in which no 1 appears to the right of any 2.
$a_1=3$ and $a_2=8$
I am having trouble creating the recurrence rela... | We interpret "to the right" as meaning immediately to the right.
Let $b_n$ be the number of "good" strings that do not end in $2$, and let $c_n$ be the number of good strings that end in $2$.
We can see that $b_{n+1}=2b_n+c_n$. For we append a $0$ or a $1$ to a good string that does not end in $2$, or a $0$ to a good ... | {
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Prove that $F(1) + F(3) + F(5) + ... + F(2n-1) = F(2n)$ (These are Fibonacci numbers; $f(1) = 0$, $f(3) = 1$, $f(5) = 5$, etc.) I'm having trouble proving this with induction, I know how to prove the base case and present the induction hypothesis but I'm unfamiliar with proving series such as this. Any help would be gr... | Hint: If you can use the that the sequence of Fibonacci numbers is defined by the recurrence relation $$F(n)=F(n-1)+F(n-2)$$ then you can prove it by induction, since $$\begin{align*}F(2(n+1))&=F(2n+2)\\&=F(2n+1)+F(2n)\\&=F(2n+1)+\underbrace{F(2n-1)+\ldots+F(5)+F(3)+F(1)}_{=F(2n) \text{ by induction hypothesis}}\end{al... | {
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Integral on sphere and ellipsoid Let $a,b,c \in \mathbb{R},$ $\mathbf{A}=\left[\begin{array}{*{20}{c}}
\mathbf{a}&{0}&{0}\\
{0}&\mathbf{b}&{0}\\
{0}&{0}&\mathbf{c}
\end{array}\right] , ~~\det A >1$
Let $~D = \{(x_1,x_2,x_3): x_1^2 + x_2^2 +x_3^2 \leq 1 \}~$ and
$~E = \left\{(x_1,x_2,x_3): \frac{x_1^2}{a^2} + \frac{x_2... | A function $~f~$ is said to be compactly supported if it is zero outside a compact set.
Let $~x=(x_1,x_2,x_3)\in\mathbb R^3~,$ be any arbitrary vector.
$$\therefore~~Ax=\left[\begin{array}{*{20}{c}}
\mathbf{a}&{0}&{0}\\
{0}&\mathbf{b}&{0}\\
{0}&{0}&\mathbf{c}
\end{array}\right]\left[\begin{array}{*{20}{c}}
{x_1}\\{x_2}... | {
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Definitions of Platonic and Archimedean Solids using Symmetry Groups? A Platonic Solid is defined to be a convex polyhedron where all the faces are congruent and regular, and the same number of faces meet at each vertex. An Archimedean Solid drops the requirement that all the faces have to be the same, but they must st... | The main observation here is the following:
Being vertex transitive (and considering bounded solids only) requires the vertices to lie on sphere. Faces on the other hand are bound to be planar by definition, thence those are contained within a supporting plane. The intersection of the plane and the sphere (ball) simply... | {
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Compute the map $H^*(CP^n; \mathbb{Z}) \rightarrow H^*(CP^n, \mathbb{Z})$ I'm trying to solve problem 3.2.6 in Hatcher. The problem is stated:
Use cup products to compute the map $H^*(CP^n; \mathbb{Z}) \rightarrow H^*(CP^n, \mathbb{Z})$ induced by the map $CP^n \rightarrow CP^n$ that is a quotient of the map $C^{n+1} ... | Le $f:P^n\to P^n$ be the map you describe and let $f^*:H^*(P^n)\to H^*(P^n)$ be the induced map on cohomology. Since $f^*$ is a map of rings and $H^*(P^n)$ is generated as a ring by its degree two component, it is enough to describe $f^2:H^2(P^n)\to H^2(P^n)$.
Moreover, $H^2(P^n)$ is a free $\mathbb Z$-module or rank $... | {
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Spectrum of Multiplication Operator $T$ in $C[0,1]$ "Let $X=C[0,1]$ and $v \in X$ be a fixed function. Let $T$ be the multiplication operator by $v$, i.e. $Tx(t)=v(t)x(t)$. Find the spectrum of $T$."
This is an exercise from a PDF of notes I found online. I'm trying to better understand Spectral Theory.
So $\lambda$ is... | The map $(T-\lambda I)^{-1}$ is defined as the solution mapping of the equation
$$
(T-\lambda I) y = z,
$$
i.e. $ y= (T-\lambda I)^{-1}z$. In case of the multiplication with $v$, this is equivalent to
$$
(v(t)-\lambda) y(t) = z(t).
$$
First consider the case that $\lambda\ne v(t)$ for all $t\in [0,1]$. Then the above ... | {
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Show sequence is convergent and the limit Given the sequence
$$\left\{a_n \right\}_{n=1}^\infty $$
which is defined by
$$a_1=1 \\
a_{n+1}=\sqrt{1+2a_n} \ \ \ \text{for} \ n\geq 1 $$
I have to show that the sequence is convergent and find the limit.
I am quite stuck on this, hope for some help.
| Monotonicity
$$a_2>a_1$$ $\boxed{\therefore\ a_{n+1}> a_n \implies 1+2a_{n+1}>1+2a_{n} \implies \sqrt{1+2a_{n+1}}>\sqrt{1+2a_{n}} \implies a_{n+2}>a_{n+1}}$
Boundedness
$$a_1<4$$ $$\boxed{\therefore\ a_n<4 \implies 1+2a_n<9 \implies \sqrt{1+2a_n}<3 \implies a_{n+1}<4}$$
Limit
$$\displaystyle\lim_{n\to \infty}a_{n}=l$$ ... | {
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Two roots of the polynomial $x^4+x^3-7x^2-x+6$ are $2$ and $-3$. Find two other roots. I have divided this polynomial first with $(x-2)$ and then divided with $(x+3)$ the quotient. The other quotient I have set equal to $0$ and have found the other two roots. Can you explain to me if these actions are correct and why?
| Yes, provided the computings are OK, what you have done is what must be done.
Given a polynomial $P(x)$, and a root of its, $a$, we have that $P(a)=0$. But if you divide $P(x)$ by $x-a$, you will get a quotient $Q(x)$ and a remainder $r$, which is a number because the degree of the divisor is $1$.
Then
$$P(x)=Q(x)(x-a)... | {
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Simplifying Double Integrals to Single-Variable Integrals Let D be a subset of $\mathbb{R}^2$ defined by $ |x| + |y| \leq 1$, and let $f$ be a continuous single-variable function on the interval $[-1,1]$. Show that
$$
\iint\limits_D \,f(x+y) \, \mathrm{d}x \, \mathrm{d}y = \int_{-1}^{-1} \, f(u) \, \mathrm{d}u
$$
Thi... | Write a change of variables, $u = x + y$, $v = x - y$. Then the Jacobian $J = \partial(x,y)/\partial(u,v) = 1/2$ and hence
$$\iint_D f(x+y) \ dx \ dy = \iint_D f(u) \ J \ du \ dv = \frac{1}{2} \int_{-1}^1\int_{-1}^1 f(u) \ du \ dv$$
Can you take it from here?
| {
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how many positive integer solutions to the following equation? $a^2 + b^2 + 25 = ab + 5a + 5b$
I have tried looking for a factorisation that could solve this question but couldn't find anything useful - found $(a+b+5)^2$ - don't know if this is useful
The equation does look similar to an equation of a circle - can you ... | The intuition is that the right-hand side is, with a few possible exceptins, smaller than the left-hand side.
Note that $(a-b)^2\ge 0$, so $ab\le \frac{a^2+b^2}{2}$. Thus the right-hand side is $\le \frac{a^2+b^2}{2}+5a+5b$.
It follows that
$$(a^2+b^2+25)-(ab+5a+5b)\ge \frac{a^2+b^2}{2}+25-5a-5b.$$
The right=hand sid... | {
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$I$ is an ideal in $R$ implies that $I[x]$ is an ideal in $R[x]$. Is the following statement right?
If $I$ is an ideal in the ring $R$, then $I[x]$ is an ideal in the polynomial ring $R[x]$.
If so, how can I prove it?
| Suppose $I$ is an ideal in $R$. Clearly $I[x]$ is a subring of $R[x]$ since $I$ must be a subring of $R$. By definition, $\forall r\in R\;\forall x\in I\;rx,xr\in I$. Then since the coefficients of the product of two polynomials result from products of the coefficients, it is clear that $\forall p\in R[x]\;\forall q\in... | {
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Irrational number inequality : $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}>\sqrt{3}$ it is easy and simple I know but still do not know how to show it (obviously without simply calculating the sum but manipulation on numbers is key here.
$$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}>\sqrt{3}$$
| $2 \lt 3 \lt 4$ so $\sqrt{2} \lt \sqrt{3} \lt 2$ and
$$1+\frac1{\sqrt{2}}+\frac1{\
\sqrt{3}} \gt 1+\frac12+\frac12 = 2 \gt \sqrt{3}.$$
Of course that does not generalise in any nice way.
However $2 \lt \frac{100}{49}$ with $3 \lt \frac{49}{16}$ and $5 \lt \frac{81}{16}$
does yield $$1+\frac1{\sqrt{2}}+\frac1{\
\sq... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expressing $\sin\theta$ and $\cos\theta$ in terms of $\tan(\theta/2)$ This is the question:
Let $t = \tan(\theta/2)$. Express the following functions in terms of $t$:
*
*$\tan \theta$
*$\sin\theta$
*$\cos\theta$
I know that for part (1),
$$\tan\theta = \frac{2t}{1-t^2}$$
How do I get parts (2) and (... | $$z_{\frac{\theta}2} = \cos\frac{\theta}2 + i \sin \frac{\theta}2 = (1+t^2)^{-\frac12}(1+it)
$$
by deMoivre's theorem
$$
z_{\theta} = z_{\frac{\theta}2}^2=(1+t^2)^{-1}\left((1-t^2)+2it\right) = \cos\theta+i\sin\theta
$$
hence, equating real parts:
$$
\cos \theta = \frac{1-t^2}{1+t^2}
$$
and imaginary parts:
$$
\sin \th... | {
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"source": "stackexchange",
"question_score": "8",
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Proof that the harmonic series is < $\infty$ for a special set.. In one of my books i found a very interesting task, i am really curios about the solution:
Let $M = \{2,3,4,5,6,7,8,9,20,22,...\} \subseteq \mathbb{N}$ be a set that contains all natural numbers, that don't contain a "1" in their depiction. Show that:
$$... | Begin by defining
$$S(1;9) := \sum_{i=2}^9 \frac{1}{i}$$
Let $a = S(1;9)$. Then $a < 2$. Let $S(10;99)$ be the sum over the fractions of all numbers between $10$ and $99$ that don't contain a $1$ in their depiction.
Prove that $S(10;99) < \frac{9}{10}a$. Similarly, define $S(100;999)$ to be the sum over the fractions ... | {
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"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Real analysis book suggestion I am searching for a real analysis book (instead of Rudin's) which satisfies the following requirements:
*
*clear, motivated (but not chatty), clean exposition in definition-theorem-proof style;
*complete (and possibly elegant and explicative) proofs of every theorem;
*examples and so... | Analysis 1 and Analysis 2 by Terence Tao.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 11,
"answer_id": 7
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Proof that $\sin10^\circ$ is irrational Today I was thinking about proving this statement, but I really could not come up with an idea at all. I want to prove that $\sin10^\circ$ is irrational. Any ideas?
| Suppose that $x=\sin(10^\circ)$ and we want to prove the irrationality of $x$. Then we can use the Triple Angle Formula for $\sin$ to get $-4x^3+3x = \sin(30^\circ)=\frac12$; in other words, $x$ is a solution of the equation $-8x^3+6x-1=0$.
But now that we have this equation we can use another tool, the Rational Root ... | {
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"source": "stackexchange",
"question_score": "13",
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"answer_id": 1
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Counting Stones If you have a bucket of stones and remove two at a time, one will be left. If you remove three at a time, two will be left. If they're removed four, five, or six at at time, then three, four, and five stones will remain. If they're removed seven at a time, no stones will be left over.
What is the smalle... | Based on the comment below, I am adding a bit more honesty to this response:
The conditions of the problem are to find $n$ such that $n \equiv 1 \text{ (mod 2)}$, $n \equiv 2 \text{ (mod 3)}$, $n \equiv 3 \text{ (mod 4)}$, $n \equiv 4 \text{ (mod 5)}$, $n \equiv 5 \text{ (mod 6)}$, $n \equiv 0 \text{ (mod 7)}$.
$n \equ... | {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Last 2 digits of $\displaystyle 2014^{2001}$ How to find the last 2 digits of $2014^{2001}$? What about the last 2 digits of $9^{(9^{16})}$?
| Hint:
For finding last two digits you need to reduce this modulo $100$. That is you ne need to find
$$2014^{2001} \equiv ? \pmod{100}.$$
This is the same as asking
$$14^{2001} \equiv ? \pmod{100}.$$
Now in order to facilitate computation, you need to use Euler's Theorem. But keep in mind that $\gcd(14,100) = 2.$ So yo... | {
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"answer_id": 1
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Proof that an involutory matrix has eigenvalues 1,-1 I'm trying to prove that an involutory matrix (a matrix where $A=A^{-1}$) has only eigenvalues $\pm 1$.
I've been able to prove that $det(A) = \pm 1$, but that only shows that the product of the eigenvalues is equal to $\pm 1$, not the eigenvalues themselves.
Does an... | You can easily prove the following statement:
Let $f: V\to V$ be an endomorphism. If $\lambda$ is an eigenvalue of $f$, then $\lambda^k$ is an engeinvalue of $\underbrace {f\ \circ\ ...\ \circ f}_{k \text{ times}}$
In this case, let $A$ be a matrix of an endomorphism $f$ such that $f\circ f = I$. This means that $A$ ... | {
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"source": "stackexchange",
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"answer_id": 4
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Probability that last child is a boy Johnny has 4 children. It is known that he has more daughters
than sons. Find the probability that the last child is a boy.
I let A be the event that the last child is a boy, P(A) = $\frac{1}{2}$.
and B be the event that he as more daughters than sons. But im not sure how to calcula... | The number of girls in the family would have a binomial distribution, so the prior probability that there are 3 or 4 girls in the family would be:
$$\begin{align}
\mathsf P(B) & = {4\choose 3}(\tfrac 1 2)^3(\tfrac 1 2)+{4\choose 4}(\tfrac 1 2)^4
\\ & = \frac 5{16}
\end{align}$$
Now for the probability that the last chi... | {
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"source": "stackexchange",
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"answer_count": 6,
"answer_id": 2
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$(-3)^{3/2} \neq (-3)^{6/4}$ $(-3)^{\frac{3}{2}}=-3\sqrt{3}i$
$(-3)^{\frac{6}{4}}=\sqrt{27}$
(not the same thing).
What's the deal? It's interesting because people work with fractional exponents all the time and I've never seen someone bother to check whether the top and bottom maintain their parity when canceling, but... | You have to be careful with these kind of things if your base is not a non-negative real number. For example,
$$1=1^{1/2}=[(-1)^2]^{1/2}=-1.$$
The reason for this can be found when you look at the "true" definition of $x\mapsto a^x$ when $a\in\mathbb{C}\setminus[0,\infty)$. We define this as:
$$a^x:=e^{x\log a}$$
which... | {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb{C}$? I'm trying to see if for several cases changing the ring in a tensor product affects the result or doesn't. Now I'm trying to prove $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb... | There is an isomorphism of rings $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \times \mathbb{C}$ (Hint: Use $\mathbb{C}=\mathbb{R}[x]/(x^2+1)$ and then CRT), but $\mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} = \mathbb{C}$. So these are not isomorphic rings, since $\mathbb{C} \times \mathbb{C}$ has zero di... | {
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"source": "stackexchange",
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Any object in a locally noetherian Grothendieck category has a noetherian subobject If $\mathcal{A}$ is a locally noetherian Grothendieck category, is that straightforward the fact that any object $M$ in $\mathcal{A}$ has a noetherian subobject?
| Yes: Suppose $\{M_i\}_{i\in I}$ is a generating set of Noetherian objects in the given locally Noetherian Grothendieck category ${\mathscr A}$. Then for any nonzero $X\in{\mathscr A}$ there exists some $i\in I$ and a non-zero morphism $\varphi: M_i\to X$. The image of this morphism is a nonzero Noetherian subobject of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1038107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Secret Santa Perfect Loop problem
*
*(n) people put their name in a hat.
*Each person picks a name out of the hat to buy a gift for.
*If a person picks out themselves they put the name back into the hat.
*If the last person can only pick themselves then the loop is invalid and either
. start again
. or step ba... | You are asking for the chance of a single cycle given that you have a derangement. For $n$ people, the number of derangements is the closest integer to $\frac {n!}e$ To have a cycle, person $1$ has $n-1$ choices, then that person has $n-2$ choices, then that person has $n-3$, etc. So there are $(n-1)!$ cycles. The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1038219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Symbols for "odd" and "even" Let $A$ be a sequence of letters $\langle a,b,c,d,e,f \rangle$. I want to create two subsequences, one with the values with odd index and other with the values with even index: $A_\mathrm{odd} = \langle a,c,e \rangle$ and $A_\mathrm{even} = \langle b,d,f \rangle$.
My question is: is there a... | How about $A_\mathcal O$ and $A_\mathcal E$?
To produce these: A_\mathcal O and A_\mathcal E
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1038340",
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"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 0
} |
$A$-stability of Runge-Kutta methods I am studying Runge-Kutta methods, but I can't understand why explicit Runge-Kutta methods are not $A$-stable. Can someone please explain it to me?
| First, recall the definition of A-stability in the context of Dahlquist's test equation
\begin{align}
u(t_0 = 0) &= 1\\
u'(t) &= \lambda u(t) =: f\big(u(t)\big) \tag{1} \label{1}
\end{align}
which reads:
A method is called A-stable if $\forall z = \lambda \Delta t : \text{Re}(z) \leq 0$ it holds that
$$\vert u_{n+1} \v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1038403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
possible pizza orders You are ordering two pizzas. A pizza can be small, medium, large, or extra large, with any combination of 8 possible toppings (getting no toppings is allowed, as is gettting all 8). How many possibilities are there for your two pizzas?
Would it be ${\large[}4{\large[}{8\choose8}+{8\choose7}+{8\cho... | Note that this is problem 14 in chapter 1 of Introduction to Probability by Blitzstein and Hwang.
The problem amounts to sampling with replacement where the order doesn't matter, which is similar to problem 13 in and the hint there (use Bose Einstein) applies here too. This is also known as the "stars and bars" method... | {
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"source": "stackexchange",
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Eigendecomposition Parameterization of Real Matrix Given a set of distinct non-real eigenvalues $\lambda_1, \dots, \lambda_N$, so that $\lambda_{2n} = \overline{\lambda_{2n+1}}$. Accordingly given a set of non-real orthonormal eigenvectors $v_1, \dots, v_N$, so that $v_{2n} = \overline{v_{2n+1}}$. (N is even.)
We defin... | First, one minor observation. I believe you meant $v_{2n}=\overline{v_{2n-1}}$ (instead of $\overline{v_{2n+1}}$) otherwise $v_1$ is not conjugated of any vector.
Now, write $v_{2n-1}=a_{2n-1}-ia_{2_n}$ and $v_{2n}=a_{2n-1}+ia_{2_n}$, where $a_{2n-1},a_{2n}\in\mathbb{R}^n$.
Notice tha $a_{2n-1}=\dfrac{v_{2n-1}+v_{2n}}... | {
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Simple indefinite integral of a vector function I am having trouble with this simple integration. I am not sure of the process or steps to follow to solve this type of problem:
If $\mathbf{V}(t)$ is a vector function of $t$, find the indefinite integral:
$$\int \Big( \mathbf{V} \times \frac{d^2\mathbf{V}}{dt^2}\Big)\hs... | I am doubtful that this problem can be solved by integrating by parts. Even if possible, it would be tedious since you have to separate in terms of each directional components.
Step 1:
Remember what differentiation of a cross product looks like:
$\frac{d (V \times U)}{dt} = \frac{dV}{dt} \times U + V \times \frac{dU}{d... | {
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How to solve $\cos(5\alpha + \pi/2) = \cos(2\alpha + \pi/8)$ for $a$? I missed the lecture. I don't want you to solve my homework, I just want to learn how to solve equations like this one. Since I have no idea, I'll post the task I got for homework, rather than obfuscating it beyond recognition. Please give general di... | Use Cos(C) - Cos(D) formula.
U can also solve by graphical method. You know the graph of Cosx then apply transformation to draw LHS and RHS intersection point will be the solution
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basis and dimension of the subspace of solutions to $A\mathbf{x}=\mathbf{0}$
Consider
$$
A =\left( \begin{matrix}
1 & -1 & 0 & -2 \\
0 & 0 & 1 & -1 \\
\end{matrix}
\right)
$$
and find a basis and the dimension of $S(A,0)$, where $S(A,0)$ is the subspace of all the solutions $\mathbf{x}\in\mathbb{R}^n$ to the line... | It's worth mentioning that one can easily check what the dimension of $S$ (usually called the null space) is going to be.
For an $m \times n$ matrix $A$, the rank-nullity theorem states that $\operatorname{Rank}(A) + \operatorname{Null}(A) = n$.
In this case, $n = 4$, and $\operatorname{Rank}(A) = 2$ since it is al... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to simpify $\cos x - \sin x$ How does one simplify
$$\cos x - \sin x$$
I tried multiplying by $\cos x + \sin x$, but that just gets me $$\cos x - \sin x = \frac{\cos 2x}{\cos x + \sin x}$$ which is worse.
Yet wolframalpha gives me $\cos x - \sin x = \sqrt{2}\sin\left(\dfrac{\pi}{4}-x\right)$. How does one obta... | $$
s = \cos x - \sin x \\
s^2 = \cos^2 x - 2 \cos x \sin x + \sin^2 x = 1 - \sin 2x \\
= 1 - \cos (\frac{\pi}2 -2 x)\\
= 1 - \left(1 - 2 \sin^2(\frac{\pi}4 - x)\right)\\
=2 \sin^2(\frac{\pi}4 - x)
$$
so
$$
s = \pm \sqrt{2} \sin(\frac{\pi}4 - x)
$$
and evaluating at $x=0$ shows that the positive sign must be taken
| {
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Prove $8n^{3}$ $+$ $√n$ $∈$ $Θ$($n^{3})$ just wondering if I proved this question correctly. Any hints, help, or comments would be appreciated.
There are two cases to consider to prove $8n^{3}$ $+$ $√n$ $ϵ$ $Θ(n^{3})$
*
*$8n^{3}$ $+$ $√n$ $ϵ$ $O$$(n^{3})$
*$8n^{3}$ $+$ $√n$ $ϵ$ $Ω$$(n^{3})$
1.)
There should exis... | An alternative is to use the limit test. Consider $f(n) = 8n^{3} + \sqrt{n}$ and:
$$L = lim_{n \to \infty} \frac{f(n)}{n^{3}}$$
Note $f(n) \in o(n^{3})$ iff $L = 0$ (and little-o is the strict inequality, which implies Big-O).
Similarly, $0 < L < \infty \implies f(n) \in \Theta(n^{3})$.
And finally, $L = \infty \implie... | {
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Find sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly, but $f'_n$ converges only pointwise Question:
Find a sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly to a differentiable function $f$, such that $f'_n$ converges pointwise but not uniformly to $f... | Let $f_n(x)=0$ if $|x|\ge 1/n.$ For $|x|<1/n$ let $f_n(x)=n^3(x^2-1/n^2)^2.$
$|f_n(x)|\le 1/n$ for all $x$ so $f_n$ converges uniformly to $f=0.$ So $f'=0.$
It is easy to confirm that $f'_n(x)$ exists when $x=\pm 1/n.$
$-1=\frac {f_n(1/n)-f_n(0)}{1/n-0}=f'_n(y_n)$ for some $y_n\in (0,1/n)$ so $f'_n$ does not converge u... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Combinatorial identity with sum of binomial coefficients How to attack this kinds of problem? I am hoping that there will some kind of shortcuts to calculate this.
$$\sum_{k=0}^{38\,204\,629\,939\,869} \frac{\binom{38\,204\,629\,939\,869}{k}}{\binom{76\,409\,259\,879\,737}{k}}\,.$$
EDIT:
As I see, the numerator is $... | According to the Gosper's algorithm (Maxima command
AntiDifference(binomial(n,k)/binomial(2*n-1,k),k),
also implemented in Mathematica and Maple):
$$
{\frac {n \choose k} {{2n-1} \choose k}}
=
{{\left((k+1)-2n\right){{n}\choose{k+1}}}\over{n{{2n-1}\choose{k+1
}}}}
-{{\left(k-2n\right){{n}\choose{k}}}\over{n{{2n-1}\ch... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 3
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Resolvent matrix Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$?
Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the characteristic polynomial of $A$, which is easy to compute once we know an eigendecomposition of $A$. But wha... | Since $A$ is triangular, you may try to first diagonalize if, $A=PDP^{-1}$. You already know what the eigenvalues of $A$ are. Then, $$(xI-A)^{-1} = (P(xI-D)P^{-1})^{-1} = P(xI-D)^{-1}P^{-1}$$ and $(xI-D)^{-1}$ is trivial to calculate. Does this help?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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New proof about normal matrix is diagonalizable. I try to prove normal matrix is diagonalizable.
I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable.
I can not go more. I want to prove statement use only this fact.
I need you help.
(professor said that we can prove only use this fact... | This Wikipedia article contains a sketch of a proof. It has three steps.
*
*If a normal matrix is upper triangular, then it's diagonal. (Proof: show the upper left corner is the only nonzero entry in that row/column using a matrix-norm argument; then use induction.)
Details of proof: write $A$ as $Q T Q^{-1}$ for so... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Continuous functions satisfying $f(x)+f(2x)=0$? I have to find all the continuous functions from $\mathbb{R}$ to $\mathbb{R}$ such that for all real $x$,
$$f(x)+f(2x)=0$$
I have shown that $f(2x)=-f(x)=f(x/2)=-f(x/4)=\cdots$ etc. and I have also deduced from the definition of continuity that for any $e>0$, there exists... | first of all for $x=0$ we have
$$f(0)+f(2\cdot0)=0\Leftrightarrow f(0)=0$$
On the other hand
$$f(x)+f(2x)=0$$
$$-f(2x)-f(4x)=0$$
$$f(4x)+f(8x)=0$$
$$.......$$
$$f(2^nx)+(-1)^nf(2^{n+1}x)=0$$
Adding both sides respectively yields
$$f(x)+(-1)^nf(2^{n+1}x)=0\Rightarrow f(x)=(-1)^{n+1}f(2^{n+1}x)$$
The LHS is continuous b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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} |
Countable collection of countable sets and Axiom of choice Do we need Axiom of choice(or weaker version axiom of countable choice) to say countable Cartesian product of countable sets is nonempty? I think yes.
I read somewhere answer no giving argument: each countable set can be well ordered and after well ordering eac... | Even when free the sets have a natural well order to them, the countable union of countable sets is not necessarily countable.
For example, in some models of $\sf ZF$ the first uncountable ordinal, $\omega_1$ is the countable union of countable ordinals.
And no, the countable product of finite sets doesn't have to be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1039725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Connected sum of projective plane $\cong$ Klein bottle How can I see that the connected sum $\mathbb{P}^2 \# \mathbb{P}^2$ of the projective plane is homeomorphic to the Klein bottle?
I'm not necessarily looking for an explicit homeomorphism, just an intuitive argument of why this is the case. Can we see it using funda... | Here's an answer more in the spirit of the question. All figures should be read from upper left to upper right to lower left to lower right.
Fig 1: A Klein bottle...gets a yellow circle drawn on it; this splits it into two regions, which we reassemble, at which point they're obviously both Mobius bands:
To see that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1039819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 3,
"answer_id": 0
} |
Proof by Induction - Wrong common factor I'm trying to use mathematical induction to prove that $n^3+5n$ is divisible by $6$ for all $n \in \mathbb{Z}^+$.
I can only seem to show that it is divisible by $3$, and not by $6$. This is what I have done:
Let $f(n) = n^3+5n$.
Basis Step: When $n=1$, $f(n)= 6$ and clearly $6$... | From your proof...
Note that if k is odd, k^2 + k + 2 is even and hence divisible by 2, and that the same is true if k is even. Therefore k^2 + k + 2 is divisible by 2 for all k. this gives you the extra factor of 2 you need to get a factor of 6.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1039927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving of $\frac{\pi }{24}(\sqrt{6}-\sqrt{2})=\sum_{n=1}^{\infty }\frac{14}{576n^2-576n+95}-\frac{1}{144n^2-144n+35}$ This is a homework for my son, he needs the proving.I tried to solve it by residue theory but I couldn't.
$$\frac{\pi }{24}(\sqrt{6}-\sqrt{2})=\sum_{n=1}^{\infty }\frac{14}{576n^2-576n+95}-\frac{1}{14... | You can apply the residue theorem after a bit of playing with the sums:
\begin{align*}&\sum_{n=1}^\infty\frac{14}{576n^2-576n+95}-\sum_{n=1}^\infty\frac4{576n^2-576+140}=\\&\sum_{n=1}^\infty\left(\frac1{24n-19}-\frac1{24n-5}\right)-\sum_{n=1}^\infty\left(\frac1{24n-14}-\frac1{24n-10}\right)=\\&\sum_{n=1}^\infty\left(\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Formula for perfect squares spectrum. I have been working on exercises from "A first Course in Logic" by S. Hedman.
Exercise 2.3 (d) asks to find a first-order sentence $\varphi$ having the set of perfect squares as a finite spectrum. But I am not sure whether or not my understanding of concepts of model and spectrum i... | You can do this with the signature that has one unary relation $A(x)$ and one binary function $f(x,y)$. The sentence will say, essentially, that $f$ is a bijection between $A \times A$ and $M$.
It is much more difficult to do this sort of thing with the signature of arithmetic. You would need to include in $\phi$ seve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Evaluate $\int_0^1 \sec^3x \sin x \,dx $ Was working on some trig-based integration. I've been confident with easier ones, but can't seem to approach this one correctly.
Evaluate $\displaystyle\int_0^1 \sec^3x \sin x \,dx $
Which method of integration should I use to solve this integral?
| $$
\int_0^1 sec^3(x)sin(x)dx = \int_0^1tan(x)sec^2(x)
$$
setting $u = tan(x)$
$du = sec^2(x)dx$
$dx = cos^2(x) du \therefore$
$$
\int_{tan(0)}^{tan(1)} u du
$$
$$
= \frac{tan^2(1)}{2}-\frac{tan^2(0)}{2}
$$
$$= \frac{tan^2(1)}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Covariance of uniform distribution and it's square I have $X$ ~ $U(-1,1)$ and $Y = X^2$ random variables, I need to calculate their covariance.
My calculations are:
$$
Cov(X,Y) = Cov(X,X^2) = E((X-E(X))(X^2-E(X^2))) = E(X X^2) = E(X^3) = 0
$$
because
$$
E(X) = E(X^2) = 0
$$
I'm not sure about the $X^3$ part, are my cal... | We know it is $E(X^3)$ so:
$$E(X^3)=\int_{-1}^{1}x^3f(x)=0$$
So it is correct
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Counterexample for the infinitely many primes between two primes in a Noetherian ring Consider the following Proposition:
Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many distinct primes $q$ such that $p_0 \su... | Consider a non-noetherian valuation ring of rank two. For such an example you can take a look at Examples of Non-Noetherian Valuation Rings.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Graph theory proofs I am trying to prove that half of the vertex cover of graph is less than it's matching number. The problem is I don't know how to start and what the solution should be like, please help!
| Let $G=(V,E)$ be a graph. Denote by $F\subseteq E$ be a maximal matching of $G$ and by $U$ a minimal vertex cover of $G$. For each edge $e=(u,v)\in E$ we know that at least one of $u$ and $v$ is in $U$, so for each edge we have either one vertex in $U$ or two vertices, thus $|F|\leq 2\cdot |U|$. Hence, making it $|U|\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Visualizing Balls in Ultrametric Spaces I've been reading about ultrametric spaces, and I find that a lot of the results about balls in ultrametric spaces are very counter-intuitive. For example: if two balls share a common point, then one of the balls is contained in the other.
The reason I find these results so count... | Think of the Cantor set and its basic closed-and-open intervals, pictured below.
Note that for any two such intervals, they either do not intersect, or one is contained in the other.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
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What is the limit of this sequence? Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Let $s_1 \colon= \sqrt{2}$, and let
$$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for } n = 1, 2, 3, \ldots. $$
Then how to rigorously calculate $$\lim_{n\to\infty} s_n,$... | As you have mentioned already, $L = \lim_{n \rightarrow \infty} s_n$ exists in $\mathbb{R}$. Also $L \geq 0$. Then
$$ L = \sqrt{2 + \sqrt{L}} $$
$$ L^2 = 2 + \sqrt{L} $$
Let $k = \sqrt{L}$. We have
$$k^4 = 2 + k$$
$$(k + 1)(k^3 - k^2 + k - 2) = 0$$
Since $k \geq 0$, we have
$$ k^3 - k^2 + k - 2 = 0$$
which has one posi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can every positive integer be expressed as a difference between integer powers? In mathematical notation, I am asking if the following statement holds:
$$\forall\,n>0,\,\,\exists\,a,b,x,y>1\,\,\,\,\text{ such that }\,\,\,\,n=a^x-b^y$$
A few examples:
*
*$1=9-8=3^2-2^3$
*$2=27-25=3^3-5^2$
*$3=128-125=2^7-5^3$
I w... | See OEIS sequence A074981 and references there. $10$ does have a solution as $13^3-3^7$, but apparently no solutions are known for $6$ and $14$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$ Title says most of it. Could you help me find an example?
It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would be done for all $p$. The reason I'm asking... | Take
$$f(x) = a_n \quad\text{ if }\quad x\in\left(\frac{1}{2^n},\frac{1}{2^{n-1}}\right]$$
for $n = 1, 2, \dots$ and for some $a_n$.
You'll get
$$\int_0^1 f(x)\, dx = \sum_{n=1}^{\infty}\frac{a_n}{2^n}$$
$$\int_0^1 f^p(x)\, dx = \sum_{n=1}^{\infty}\frac{a_n^p}{2^n}$$
Then choose the sequance $a_n$ so that the first sum... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is a flat coherent sheaf over a connected noetherian scheme already a vector bundle? Let $A$ be a connected noetherian ring (not necessarily irreducible), $M$ be a finitely presented flat $A$-module. Then $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module for each $\mathfrak{p} \in \operatorname{Spec}(A)$.
Is it a... | It is a standard fact in commutative algebra that, for an arbitrary commutative ring $A$, an $A$-module $M$ is finitely presented flat if and only if $M$ is finitely genertated projective if and only if $M$ is locally free of finite rank (:= there are elements $f_1,\dotsc,f_n \in A$ generating the unit ideal such that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1040999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the empty set the only possible set for $A$ such that $A=\{x|x\not\in A\}$? Is the empty set the only possible set for $A$ such that $A=\{x|x\not\in A\}$?
| Not even the empty set has that property.
For any set $A$ we can show $A\ne\{x\mid x\notin A\}$. Namely, either $42\in A$ (in which case $42\notin \{x\mid x\notin A\}$), or $42\notin A$ (in which case $42\in \{x\mid x\notin A\}$). In both cases we have found something that is a member of exactly one of the collections,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find a closed form for the equations $1^3 = 1$, $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$ This is the assignment I have:
Find a closed form for the equations
$1^3 = 1$
$2^3 = 3+5$
$3^3 = 7+9+11$
$4^3 = 13+15+17+19$
$5^3 = 21+23+25+27+29$
$...$
Hints. The equations are of the form $n^3 = a1 +a2 +···+an$, where
$a_{i+1} = a_... | $$n^3=\sum_{k=0}^{n-1}(n^2-(n-1)+2k)$$
Since
$\sum_{k=0}^{n-1}(n^2-(n-1)+2k)=n^3-n(n-1)+2(\frac{n(n-1)}{2})=n^3$
So it is the summation of $n$ consequitive odd number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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prove linear independence of polynomials Let $f_1,...,f_k$ be polynomials at field $K$ not including the zero polynomial and $\deg f_i \neq \deg f_j$ for every $i \neq j$ Show that polynomials $f_1,...,f_k $ are linear independent.
I don't know how to use the information that $\deg f_i \neq \deg f_j$.
I will try to c... | Suppose that $f_1 ,f_2 ,...,f_k$ are linear dependent and let $f_j $ be the polynomial whose degree is the largest then $f_j = c_1f_1+ ...c_{j-1}f_{j-1} + c_{j+1}f_{j+1}+.... +c_{k}f_{k}$ but the degree of right hand side is less thar left hand side and this is contradiction
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Multiplying game strategy Given the following game, what is the strategy to win?
Given $X,N\in \mathbb{N}$ such that $N>X$ and $N>1000$, two players play against each other. Each player multiply $X$ by $2$ or by $3$ by his own choice. The player who reach $N$ or above- wins.
I realized that if it's my turn and my oppo... | The best method of attack for this is probably to work backwards. So, you see that if the number given to you is above $\frac{N} 3$, you win. What numbers, less than this, can you give to your opponent such that they have to give you a number at least $\frac{N}3$? Well, since they have to multiply by at least two, if y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Procedure to find a level curves I'm having trouble finding level curves. What's the procedure? In this case, for example:
$z=x^2+y^2=k$ $\hookrightarrow y=\sqrt(k-x^2)$
Then I sketch this based on knowing the formulas of circunference, parabola, ellipse, etc? Or do I sketch this based on first, second derivatives and ... | Either you use a computer, or you sketch the curves based on recognizing the equation as something you already know.
In the first case you should recognize $x^2+y^2=k$ as the equation for a circle with radius $\sqrt k$ rather than try to rewrite it to get $y$ as a function of $x$.
Similarly for $e^{x^2-y^2}=k$ you woul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Understanding a step in a double series proof I'm really confused, how do they get from the first line to the second line ?
$$\begin{align*}
S&=\frac12\left[\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2n}{3^m(n\cdot3^m+m\cdot3^n)}+\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{n^2m}{3^n(n\cdot3^m+m\cdot3^n)}\right]\\\\
&=\frac1... | Making a common denominator and factoring, observe that:
\begin{align*}
\frac{m^2n}{3^m(n \cdot 3^m + m \cdot 3^n)} + \frac{n^2m}{3^n(n \cdot 3^m + m \cdot 3^n)}
&= \frac{m^2n \cdot 3^n + n^2m \cdot 3^m}{3^m3^n(n \cdot 3^m + m \cdot 3^n)} \\
&= \frac{mn(m \cdot 3^n + n \cdot 3^m)}{3^{m+n}(n \cdot 3^m + m \cdot 3^n)} \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$. Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that
$$\inf_{f\in C}\int_0^1|f'(x)-f(x)|dx=\frac{1}{e}?$$
I have be... | Here is another method.
Let $f \in C^1([0,1])$ with $f(0)=0$ and $f(1)=1$, and $g=f'-f$.
then solving the ODE : $f' =f +g$ with $f(0)=0$ gives
$$f(x) = e^x \int_0^x e^{-t} g(t) dt.$$
Since $f(1)=1$, we get
$$ \int_0^1 e^{-t} g(t) dt = \frac{1}{e} \quad \quad (1). $$
By Holder we have : $| \int_0^1 e^{-t} g(t) dt| \leq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 2
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How to integrate $\int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x$? I encountered this integral in the quantum field theory calculation. Can I do this:
$$
\left. \int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x
=x\ln\left(\, x\,\right)\right\vert_{0}^{1}
-\int_{0}^{1}\,{\rm d}x
=\left. x\ln\left(\, x\,\right)\right\vert_{\, ... | Use L'Hopital's Rule to resolve the indeterminate form: $$\lim_{x\to 0^+}x\ln x=\lim_{x\to 0^+}{\ln x\over x^{-1}}=\lim_{x\to 0^+}{x^{-1}\over -x^{-2}}=\lim_{x\to 0^+}(-x)=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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$ (3+\sqrt{5})^n+(3-\sqrt{5})^n\equiv\; 0 \; [2^n] $ Proof that for all $n\in \mathbb{N}$ :
$$
(3+\sqrt{5})^n+(3-\sqrt{5})^n\equiv\; 0 \; [2^n]
$$
| HINT:
If $a_n=(3+\sqrt5)^n+(3-\sqrt5)^n$
$$a_{n+2}-6a_{n+1}+4a_n=0$$
Now use Strong induction like $2^m\mid a_m$ for $1\le m\le n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding polynomials with their values at points Is there any way I can find a polynomial given any $2$ points (with $x$ coordinate OF MY CHOICE): Let's say there's some polynomial I don't know $(p(x)=2x^3+x^2+3)$, but my machine will give me an output. I give one $x$ value of my choice, and it returns $p(x)$, where $p(... | No. You would a minimum of $n$ points, where $n$ is the dimension of the polynomial. For instance, let's try to find a cubic polynomial $p$ where $p(0)=0$ and $p(1)=1$. Notice that $p_1(x)=x^3-x^2+1$ and $p_2(x)=x^3-x+1$ both satisfy the given criteria.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
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Real subfield of cyclotomic field is generated by $\zeta+\zeta^{-1}$ Let $p\neq 2$ a prime number, $\zeta=e^{\frac{2i\pi}{p}}$ and $\alpha=2\cos\left(\frac{2\pi}{p}\right)$. We consider the field extensions $F=\mathbb Q(\zeta)$ and $E=F\cap \mathbb R$ of $\mathbb Q$. I have shown that $[F:\mathbb Q]=p-1$, that $\zeta+\... | Basically you have $Q(\xi):Q(\xi+\xi^{-1})=2$ because of the minimal polynomial you've found.
Then notice that $Q(\xi):E >1$, then since $Q(\xi+\xi^{-1}) \subset E$, we are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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boundedness of the sequence $a_n=\frac{\sin (n)}{8+\sqrt{n}}$ How can i prove boundedness of the sequence
$$a_n=\frac{\sin (n)}{8+\sqrt{n}}$$ without using its convergence to $0$? I know since it is convergent then it is bounded.
| $$\Big|\frac{sin (n)}{8+\sqrt{n}}\Big|\le\Big|\frac{1}{8+\sqrt{n}}\Big|\le\dfrac{1}{9}, \forall n\in\mathbb{N}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Bell number vs Factotial We have $B_n$ is Bell number and $n!$ - factorial.
So, what is greater: $n!$ or $B_n$ ?
How it can be proven?
| Factorials are bigger than Bell numbers, except for the initial cases when there is equality.
A comment from Emeric Deutsch on OEIS A048742 says that the difference counts
Number of permutations of $[n]$ which have at least one cycle that has
at least one inversion when written with its smallest element in the
fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Directional derivative of a function Feel like I may have gone wrong somewhere with this question:
Find the directional derivative of the function $f(x,y) = \displaystyle\dfrac{2x}{x-y}$ at the point $P(1, 0)$ in the direction of the vector $v=(4, 3)$.
I got: $f_x(x, y) = \dfrac{-2y}{(x-y)^2}$ $f_y(x, y) = \dfrac{2x}{(... | You made a mistake in the second-to-last step. You should get $-8y+6x$ instead of $-8+6x$ in the numerator.
Other than that your approach and calculations look fine to me.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
How to solve a recursively defined system It has been a while since I have tackled a problem like this and could use a refresher. I have a recursive system of equations that looks a little like:
$$x_{n}\left(\frac{1}{2}\right)+y_{n}\left(\frac{1}{4}\right)=x_{n+1} \\
x_{n}\left(\frac{1}{2}\right)+y_{n}\left(\frac{3}{4}... | Probably the best answer uses linear algebra. You can rewrite this system as:
$$\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & \frac{1}{4} \\ \frac{1}{2} & \frac{3}{4} \end{bmatrix} \begin{bmatrix} x_n \\ y_n \end{bmatrix}.$$
This implies that
$$\begin{bmatrix} x_n \\ y_n \end{bmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Convergence test of the series $\sum\sin100n$ I need to prove that $$\sum_{n=1}^{\infty} {\sin{100n}} \; \text{diverges}$$ I think the best way to do it is to show that $\lim_{n\to \infty}{\sin{100n}}\not=0$. But how do I prove it?
| We know that every subgroup of $\mathbb{R}$ is either discrete/cyclic or dense.
Consider the subgroup $G=\left\{100n+2m\pi:n,m\in\mathbb{Z}\right\}$. A simple argument shows that this subgroup is not cyclic (if $\gamma$ was a generator, then $\gamma$ would have to be rational because $100\in G$, but would also have to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
expected value of a random palindrome If you choose a 6-digit palindrome at random, what is the expected value for the number?
All possible palindromes are equally likely to be chosen. Beginning number must be NONZERO, so numbers like 012321 are NOT allowed.
I'm not sure where to start. What values and the probabiliti... | Well, first of all, you need to count all possible outcomes.
In other words, you need to count all possible palindromes. All your palindromes are uniquely defined by their first three digits. You have $9$ possibilities for the first digit (can't be zero) and $10$ possibilities for the second and third.
Secondly, since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1042903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Summation of trigonometric functions such as $\sin x$ I am currently studying Integration (a very basic introduction) and I have a question regarding the summation of trigonometric functions. Given $f(x) = \sin x$, determine the area under the curve between a and b. By definition of a definite integral (using sigma not... | What you really want for the Riemann sum of $\int_a^b \sin x \, dx$ is
to take $\sin x$ at $n$ uniform steps within the interval $[a,b].$
So you want $x_i$ to be something like $a + i\Delta x,$
or even better,
$$x_i = a + i\Delta x - \tfrac12\Delta x \quad \text{where}
\quad \Delta x = \frac{b-a}{n},$$
so that $x_1 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Multivariable Calculus Integral volume question
Volume is:
$$
\int_{x=0}^{1}\int_{z=4x^2}^{z=5-x^2}(1-x)\;dz\;dx
$$
The picture above is the solution for the question that I need to find the volume of the region bounded by $z=5-x^2, z=4x^2,$ and the planes $y=0, x+y=1$. Is the integral set up correctly?
I'm not sure w... | Those limits don't work unless your teacher meant something else or I have misunderstood the question.
$z=4x^2$ and $z=5-x^2$ define a pair of parabolic cylinders with the y-axis as the axis. These intersect at$ x=\pm 1, z=4$.
The lower face of this volume is capped by the xz plane ($y=0$).
If the upper face had bee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Countrymen seated around a round table probability question Seated around the table are:
- 2 Americans
- 2 Canadians
- 2 Mexicans
- 2 Jamaicans.
Each countryman is distinguishable.
How many ways possible can all 8 people be seated such that AT LEAST TWO men from the same country sit next to each other?
Rotations are... | Let us try to find those arrangements in which no two countrymen sit together.Let A1A2,B1B2,C1C2 and D1D2 represent same countrymen. Since we are talking about circular permutations,let A1 be reference point and let B1,C1 and D1 be seated at poition 3,5 and 7 respectively.
1
A1
8 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Condition for plane $ax+by+cz+d = 0$ to touch surface $px^2+qy^2+2z=0$ Q. Show that the plane
$$ax+by+cz+d = 0$$
touches the surface
$$px^2+qy^2+2z=0$$
if $a^2/p + b^2/q +2cd = 0$.
How to start to solve this problem?
| Using pole-and-polar relation
*
*Polar for a pole $(X,Y,Z)$
$$pXx+qYy+(z+Z)=0$$
*
*Identifying with the given plane:
$$\frac{pX}{a}=\frac{qY}{b}=\frac{1}{c}=\frac{Z}{d}$$
*
*For tangency, $(X,Y,Z)$ should be on the quadric, that is
\begin{align}
0 &= pX^2+qY^2+2Z \\
0 &=
p\left( \frac{a}{pc} \right)^2+
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to evaluate the following integral $\int_0^{\pi/2}\sin{x}\cos{x}\ln{(\sin{x})}\ln{(\cos{x})}\,dx$? How to evaluate the following integral
$$\int_0^{\pi/2}\sin{x}\cos{x}\ln{(\sin{x})}\ln{(\cos{x})}\,dx$$
It seems that it evaluates to$$\frac{1}{4}-\frac{\pi^2}{48}$$
Is this true? How would I prove it?
| Find this
$$I=\int_{0}^{\frac{\pi}{2}}\sin{x}\cos{x}\ln{(\cos{x})}\ln{(\sin{x})}dx$$
Solution
Since
$$\sin(2x) = 2\sin(x)\cos(x)$$
then
$$I=\dfrac{1}{8}\int_{0}^{\frac{\pi}{2}}\ln{(\sin^2{x})}
\ln{(\cos^2{x})}\sin{(2x)}dx$$
Let $\cos{(2x)}=y$, and since
$$\cos(2x) = 2\cos^2x - 1 = 1 - 2\sin^2x$$
we get
$$I=\dfrac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 1
} |
Use the definition of infinite limit to prove $\lim_{x \to 1+} \frac{x}{x^2-1}=\infty$
Prove
$$\lim_{x \to 1+} \frac{x}{x^2-1}=\infty$$
And I was given the solution like this: but I could not understand how it removes the complicated terms.
Let $\delta=\min(0.5,\frac{1}{5M})$.
$$\frac{x}{x^2-1}=\frac{x}{(x+1)(x-1... | When $M$ is large, it is enough to take $\delta=\frac{1}{2M}$. This is because if we have $\delta=\frac{1}{2M}$ then the numerator is larger than $1$ while the denominator is less than $\left ( 1 + \frac{1}{2M} \right )\frac{1}{2M}$. Hence the quotient is larger than $\frac{2M}{1+\frac{1}{2M}}$. To make this last expre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Help me to find $\frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{<(1,1,1),(1,3,2)>}$. I have previously asked this question. But now I'm stuck in finding $\frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{<(1,1,1),(1,3,2)>}$.
Please give me hints to find it.
| Since the matrix
$$
\begin{pmatrix}
1 & 1 & 1\\
1 & 3 & 0\\
1 & 2 & 1
\end{pmatrix}
$$
has determinant $1$, then $\{v_1, v_2, v_3\} = \{(1,1,1), (1,3,2), (1,0,1)\}$ is a basis for $\mathbb{Z}^3$. Then
\begin{align*}
\frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle(1,1,1),(1,3,2)\rangle} \cong \frac{\mathbb{Z}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Determine the minimal polynomial of $\sqrt 3+\sqrt 5$ I am struggling in finding the minimal polynomial of $\sqrt{3}+\sqrt{5}\in \mathbb C$ over $\mathbb Q$
Any ideas? I tried to consider its square but it did not helped..
| I propose the following way to prove the polynomial $\;x^4-16x^2+4\;$ is irreducible over $\;\Bbb Q\;$ . First, factor it over the reals:
$$x^4-16x^2+4=(x^2-2\sqrt5\,x+2)(x^2+2\sqrt5\,x+2)$$
(this is way easier than what can thought at first, at least in this an other similar cases).
From here, it's clear the polynomia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 2
} |
Basis for a vector space consisting of the set of linear combinations of these functions S is the vector space consisting of the set of all linear combinations of the functions $f_1(x)=e^x$, $f_2(x)=e^{-x}$, $f_3=sinh(x)$. Find a basis for S and find the dim[S].
First, I let
$c_1f_1$+$c_2f_2$+$c_3f_3$=$f$
Then, since... | $f_1$ and $f_2$ are linearly independent (check using the definition), while $f_3$ is a linear combination of the first two. So the basis (i.e. linearly independent set which spans the space) is just $\{f_1,f_2\}$. Since this set contains two elements, the space is two dimensional.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to show that $\lim_{n\to\infty} c^n\sqrt n = 0$, where $c\in(0,1)$?
How to show that $\lim_{n\to\infty} c^n\sqrt n = 0$, where $c\in(0,1)$?
We often claim that exponent is "stronger" than root (and we even proved it last semester) but I can't remember how.
| $c^n\sqrt n=e^{n\ln c}\sqrt n=e^{n\ln c+\ln\sqrt n}$ but
$n\ln c+\ln\sqrt n=n(\ln c+\dfrac{\ln \sqrt n}{n})\to -\infty$ ((because $\dfrac{\ln \sqrt n}{n}\to 0$ and $\ln c<0$)). Hence $c^n\sqrt n\to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1043982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Local tilt angle based on height-field I'm trying to implement a mathematical formula in a program I'm making, but while the programming is no problem I'm having trouble with some of the math. I need to calculate $\sin(\alpha(x,y))$ with $\alpha(x,y)$ the local tilt angle in $(x,y)$.
I have $2$-dimentional square grid,... | A helpful construct here would be the normal vector to our terrain.
Our terrain is modeled by the equation
$$
z = h(x,y)
$$
Or equivalently,
$$
z - h(x,y) = 0
$$
We can define $g(x,y,z) = z - h(x,y)$. It turns out that the vector normal to this level set is given by
$$
\operatorname{grad}(g) =
\newcommand{\pwrt}[2]{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The inverse of a fractional ideal is a fractional ideal
Let $A$ be an integral domain and $K$ its field of fractions. If $M$ is a non-zero fractional ideal of $A$, then $$N=\{x \in K : xM \subseteq A\}$$
is also a fractional ideal of $A$.
The proof I am trying to follow is given in Dummit and Foote.
I agree that i... | $M\ne 0$ and thus there is $x\in M$, $x\ne 0$. What about $dx$? (In fact, you don't need $dM\subset A$ in order to find a non-zero element in $M$ which is also in $A$: write $x=a/b$ with $a,b\ne 0$, and notice that $bx=a\in M\cap A$.)
Edit. In order to conclude that $N$ is a fractional ideal note that $(dx)N\subseteq A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
We're looking for a treasure Here's a problem my sister-in-law just sent me and she doesn't find the answer. It's to help her daughter.
We have a map of an island. On this island there's a palm tree P, a house M and a big rock R. The rock is 8 meters far from the palm tree, and 5 meters far from the house. The house is... | My answer is not much different than what @Nick, who was quicker, already posted. But here it is, there is one detail at the end that is a bit different.
See the picture below. Start with a circle centered at the rock, R, with radius $8$, I have labeled this circle 8R, meaning "8 from the rock". Pick a point P on that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Convergence of a function with $e$ in the denominator $$\int^{\infty}_1\frac{dx}{x^3(e^{1/x}-1)}$$
I'm given the hint that the function $y = e^x$ has a tangent $y=x+1$ when $x=0\land y=1$.
How do I prove its convergence and find a upper-limit for the improper integral's value?
What I've tried myself
Note that $e^{1/x}-... | What you tried concerns ''0''. The
provided hint concerns what about $+\infty .$ Note that the exponential
function is above its tangent at $x=0,$ that is,
$$
e^{x}\geq x+1,\ \ \ \ \ \ for\ all\ x\in
%TCIMACRO{\U{211d} }
%BeginExpansion
\mathbb{R}
%EndExpansion
$$
then, for $t=\frac{1}{x}>0$ one has
\begin{eqnarray*}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
What quantity does a line integral represent? I'm currently trying to wrap my head around line integrals, Green's theorem, and vector fields and I'm having a bit of difficulty understanding what a line integral represents geometrically.
Is it basically the arc length of a curve, for a scalar field?
And then when you br... | There are at least two worthwhile interpretations of a line integral in two-dimensions.
First, $\int_C (\vec{F} \cdot T)ds = \int_C Pdx+Qdy$ measures the work or circulation of the vector field along the oriented curve $C$. This integral is largest when the vector field aligns itself along the tangent direction of $C$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
For what values of $x$ does the series converge: $\sum \limits_{n=1}^{\infty} \frac{x^n}{n^n}$? For what values of $x$ do the following series converge or diverge
$$\sum \limits_{n=1}^{\infty} \frac{x^n}{n^n}$$
I tried to solve this using the ratio test where the series converge when
$$\lim \limits_{n \to \infty} \frac... | Note that
$$
n^n \ge n! \implies \frac{x^n}{n^n} \le \frac{x^n}{n!}
$$
and use the comparison test against the series formulation of $e^x$
Note the above only works for positive $x$, however consider showing that the series is absolutely convergent for any $x$ using the above inequalities and thus it is convergent for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
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