Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Order estimates QUESTION:
Suppose $y(x) = 3 + O (2x)$ and $g(x) = \cos(x) + O (x^3)$ for $x << 1$. Then, for $x << 1:$
(a) $y(x)g(x) = 3 + O (x^2)$
(b)$ y(x)g(x) = 3 + O (x^4)$
(c) $y(x)g(x) = 3 + O (x^6)$
(d) None of these
MY WORKINGS:
$y(x) = 3+O(2x) = 3 + O(x) \implies y(x)g(x) = 3(\cos(x)) + 3(O(x^2)) + O(x)\cos(... | Well, the fact is that $\forall \beta<\alpha,\,\ O(x^\alpha)\subset o(x^\beta)\subset O(x^\beta)$
Proof: Indeed, by definition $o(x^\beta)\subset O(x^\beta)$.
Moreover, let $\alpha > \beta$, $g(x)\in O(x^\alpha)$. In a neighborhood of $x_0=0$ it holds
$|g(x)|\le C|x^\alpha|=|x^\beta|\cdot C\left|(x^{\alpha-\beta})\rig... | {
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"url": "https://math.stackexchange.com/questions/1308962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding a variable substitution for a double integral I have the following double integral
$$
\iint_D xdxdy
$$
where D is given by the inequalities
$$
x^2+xy+y^2 \le 4, x\ge 0
$$
In the solution given to the problem they apply this variable substitution when solving the integral:
$$
u = \frac{\sqrt{3}}{2}x
$$
$$
v = \f... | You can transform the region D by completing the square
$$x^2+xy+y^2=\frac{3}{4}x^2+\frac{1}{4}x^2+xy+y^2=(\frac{1}{2}x+y)^2+(\frac{\sqrt{3}}{2}x)^2\leq 4$$
Let $u = \frac{\sqrt{3}}{2}x, v=\frac{x}{2}+y$. And notice that $x\geq 0$ gives $u\geq 0$ too.
| {
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"url": "https://math.stackexchange.com/questions/1309038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Why more than 3 dimensions in linear algebra? This might seem a silly question, but I was wondering why mathematicians came out with more than 3 dimensions when studying vector spaces, matrices, etc. I cannot visualise more than 3 dimensions, so I am not seeing the goal of having more than 3 dimensions.
Except from th... | You might want to look at applied sciences. Weather forecast seems to be a nice example which shows how a rather easy question leads to high dimensional vector spaces.
Suppose you want to predict the temperature for tomorrow. Obviously you need to take today's temperature into account so you start with a function $$f:\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1309149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 0
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Absolute value of vector not equal to magnitude of vector I've come accross the following inequality for a norm (where the norm defines the length of the vector):
$$\lvert x \rvert ≤ \lvert \lvert x \rvert \rvert \leq \sqrt{n} \lvert x \rvert$$
where $x$ is a vector. Firstly, what is this inequality called? Secondly, i... | This is an instance of norm equivalence, here between some norm $\lVert.\rVert$ and the Euclidean norm $\lVert.\rVert_2$. For any two norms $\lVert.\rVert_a$ and $\lVert.\rVert_b$ (of a finite-dimensional vector space) one can give such an equation
$$
m \lVert x \rVert_a \le \lVert x \rVert_b \le M \lVert x \rVert_a
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof of Number of: *permutations of ‘n’ things, taken ‘r’ at a time, when ‘m’ specified things always come together* I read below at many sources
Number of permutations of ‘n’ things, taken ‘r’ at a time, when ‘m’ specified things always come together =$ m! * (n-m+1) !$
However no one gave the proof.
I reached till... | This is permutation question, so objects can come in different order.
*
*Treat M objects as single entity/object. Remember, inside this entity, m objects can be
arranged in M! ways.
*Now, total count of objects is (n-m+1).
Ex. if, n = 3 and m=2 then if we treat 2 objects as single entity/object we will have 2 (3... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Proof that $f(x)=0 \forall x \in [a,b]$ Lemma:
If $f \in C([a,b])$ and $\int_a^b f(x) h(x) dx=0 \ \forall h \in C^2([a,b])$ with $h(a)=h(b)=0$ then $f(x)=0 \ \forall x \in [a,b]$.
Proof of lemma:
Suppose that there is a $x_0 \in (a,b)$ such that $f(x_0) \neq 0$, for example without loss of generality we suppose that $... | Short answer:
*
*Just a mater of taste.
*Because $g(x)$, and therefore $f(x)g(x)$, is zero outside $(x_1,x_2)$.
*Because the function must be twice continuously differentiable.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding $\lim\limits_{n\to\infty }\frac{1+\frac12+\frac13+\cdots+\frac1n}{1+\frac13+\frac15+\cdots+\frac1{2n+1}}$ I need to compute: $\displaystyle\lim_{n\rightarrow \infty }\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}}{1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots+\frac{1}{2n+1}}$.
My Attempt: $\dis... | Hint
The numerator is $H_n$ and the denominator is $H_{2n+1}-\frac12H_n$.
Also,
$$\frac{H_n}{H_{2n+1}-\frac12H_n}=\frac1{-\frac12+\frac{H_{2n+1}}{H_n}}$$
and
$$H_n\sim\ln n$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 1
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The perpendicular distance from the origin to point in the plane The plane $3x-2y-z=-4$ is passing through $A(1,2,3)$ and parallel to $u=2i+3j$ and $v=i+2j-k$.
The perpendicular distance from the origin to the plane is $r.n = d$ but how to determine the point (Call it N) on the plane and what's the coordinate of the po... | There is a general procedure & formula derived in Reflection formula by HCR to calculate the point of reflection $\color{blue}{P'(x', y', z')}$ of the any point $\color{blue}{P(x_{o}, y_{o}, z_{o})}$ about the plane: $\color{blue}{ax+by+cz+d=0}$ & hence the foot of perpendicular say point $N$ is determined as follows
... | {
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"timestamp": "2023-03-29T00:00:00",
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Why does $e^{-(x^2/2)} \approx \cos[\frac{x}{\sqrt{n}}]^n$ hold for large $n$? Why does this hold:
$$
e^{-x^2/2} = \lim_{n \to \infty} \cos^n \left( \frac{x}{\sqrt{n}} \right)
$$
I am not sure how to solve this using the limit theorem.
| In a neighbourhood of the origin,
$$\log\cos z = -\frac{z^2}{2}\left(1+O(z^2)\right)\tag{1} $$
hence for any $x$ and for any $n$ big enough:
$$ \log\left(\cos^n\frac{x}{\sqrt{n}}\right)=-\frac{x^2}{2}\left(1+O\left(\frac{1}{n}\right)\right)\tag{2}$$
and the claim follows by exponentiating $(2)$:
$$ \cos^n\frac{x}{\sqrt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Constructing a multiplication table for a finite field
Let $f(x)=x^3+x+1\in\mathbb{Z}_2[x]$ and let $F=\mathbb{Z}_2(\alpha)$, where $\alpha$ is a root of $f(x)$. Show that $F$ is a field and construct a multiplication table for $F$.
Can you please help me approach this problem? I've tried searching around, but I don'... | By the division algorithm, any polynomial $g\in\mathbb{Z}_2[x]$ can be uniquely written as
$$g=a_0+a_1x+a_2x^2+qf$$
for some $q\in\mathbb{Z}_2[x]$ and some $a_0,a_1,a_2\in\mathbb{Z}_2$ (depending on $g$, of course). Thus, the quotient ring $\mathbb{Z}_2[x]/(f)$ consists precisely these eight cosets (corresponding to ea... | {
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Prove the following limit using the N-delta definition $\lim\limits_{x \to 0^+} {\ln x} = -\infty$
My attempt:
$\forall\ N<0,\ \exists \ \delta>0 \ st. \forall x,\
c < x < \delta + c \implies f(x) < N$
Let $N$ be given.
Consider $\ \ln x < N$
$\; \; \; \; \; \; \; \; \; \; \; \; \; e^{\ln x} < e^N$
$\; \; \; \; ... | Your work is correct. But usually, the statement is:
$$\forall \ M > 0, \ \exists \ \delta > 0, \ 0 < x < \delta \implies f(x) < -M. $$
That is, $M$ is taken positive.
In this case, the choice would be $e^{-M}$, which is $1/e^{M}$.
Note that this is equivalent to what you have done:
You arrived at $\delta = e^N$, wh... | {
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Lower bounding the eigenvalue of a matrix Suppose I have the following symmetric matrix
$$
A'=\begin{bmatrix}
A + b b^T & b \\
b^T & 1
\end{bmatrix}
$$
where $A$ is positive definite $n \times n$ symmetric matrix and $b$ is a $n \times 1$ vector.
Suppose $\|b\|_2 \leq B_1$, and all eigenvalues of $A$ are between $[B... | Let $z=(x,x_{n+1})$, $x\in\mathbb R^n$. Then
$$
z^TA'z = x^TAx+ (b^Tx)^2 + 2 x_{n+1} (b^Tx) + x_{n+1}^2
\ge x^TAx+ (b^Tx)^2 -((b^Tx)^2+ x_{n+1}^2) + x_{n+1}^2 = x^TAx,
$$
which tells that $A'$ is positive semi-definite. The right-hand side does not depend on $x_{n+1}$, we do not get positive definiteness here. Using th... | {
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$a^2=b^3+bc^4$ has no solutions in non-zero integers this problem is from number theory book ,
$$a^2=b^3+bc^4$$
has no solutions in non-zero integers
This book hint ：First show that $b$ must be a perfect square.and how to do?
| It is clear that $b\ge 0$. Suppose $b\gt 1$, and let $p$ be a prime that divides $b$. Let $p^k$ be the highest power of $p$ that divides $b$.
There are $2$ cases. If $p$ does not divide $c$, then since $p^{3k}$ divides $b^3$, it follows that the highest power of $p$ that divides $a^2$ is $p^k$, so $k$ is even.
If $p$ ... | {
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"url": "https://math.stackexchange.com/questions/1310216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Let $ I $ be an ideal in $\mathbb Z [i]$. Show that $\mathbb Z[i] /I $ is finite.
Let $I$ be an ideal in $\mathbb Z[i]$. I want to show that $\mathbb Z[i]/I$ is finite.
I start with $Z[i]/I$ is isomorphic to $Z$.
$Z$ is ID then $I$ is prime .Here i get stuck.
Thanks for help.
| Let $I \subseteq \mathbb{Z}[i]$ be a nonzero ideal. Since $\mathbb{Z}[i]$ is a principal ideal domain, it follows that $I = (\alpha)$ for some $\alpha \in \mathbb{Z}[i].$ Let $a + I$ be a coset of $I$ with $a = k \alpha + \beta$ and $\delta(\beta) < \delta(\alpha).$ In particular, every coset in $\mathbb{Z}[i]$ is repr... | {
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Name of the highest power of 2 smaller than or equal to a given number For a number $x$, I would like to know whether there is a common name for the number $2^n$ such as $2^n \leq x < 2^{n+1}$ (e.g. If $x = 7$, then $2^n = 4$, $n = 2$).
I have some computer science related article where I extensively use such a number ... | How about truncate? That is, you truncate all but the most significant digit to 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1310371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 14,
"answer_id": 13
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Subsets of set satisfying open set condition Suppose an iterated function system of similarity transformations $S_1, S_2, \dotsc, S_k:\mathbb{R}^n\to\mathbb{R}^n$ (with unique invariant set $F$) satisfies the open set condition for some non-empty bounded open set $O\subset \mathbb{R}^n$, so that $$\bigcup_{i=1}^k S_i(O... | The statement is false. Consider the IFS on $\mathbb R$ consisting of the similarities $S_1(x)=x/2$ and $S_2(x)=x/2+1/2$. The open unit interval $(0,1)$ verifies OSC for this IFS. However, no interval of the form $(a,b)$ with $0<a<b<1$ will verify OSC.
| {
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Product topology - definition Can someone please give me a detailed explanation of the concept of product topologies? I just can't get it. I have looked in a number of decent textbooks(Munkres, Armstrong, Bredon, Wiki :P, Class notes, a youtube video).
This is what it seems like to me:
We have two topological spaces $... | The product topology (on a product of two spaces $(X,\tau_1)$ and $(Y,\tau_2)$ consists of all unions of sets of the form $U \times V$, where $U \in \tau_1$ and $V \in \tau_2$. On easily checks that this forms a topology.
A more general way of defining it, which works for products of any number of spaces $(X_i, \tau_i... | {
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Clarify: "$S^0$, $S^1$ and $S^3$ are the only spheres which are also groups" The zero, one, and three dimensional spheres $S^0$, $S^1$ and $S^3$ are in bijection with the sets $\{a\in \mathbb{K}:|a|=1\}$ for $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$ respectively.
The real, complex and quaternionic multiplicatio... | The spheres $S^0,S^1$ and $S^3$ are the only spheres that are lie groups (a group that is a differentiable manifold as well). The proof uses the group cohomology (ie, studying groups using cohomology theory) of spheres. Check out the De Rham cohomology of the $n-$dimensional sphere, which states that $H^1(S^n\times I)$... | {
"language": "en",
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Solving easy first-order linear differential question. Question
Solve $y'=2x(1+x^2-y)$.
My attempt
Rearranging gives $y'+2xy=2x(1+x^2)$. Thus, the integrating factor is $e^{\int2x\,dx}=e^{x^2}$ and multiplying the equation throughout by this gives
$e^{x^2}y'+2xe^{x^2}y=2xe^{x^2}(1+x^2)\Rightarrow\dfrac{d}{dx}{e^{x^2}y}... | make a change of variable $$1+x^2 - y = u, \quad y =1+x^2 - u, y' = 2x-u' $$ then the de $y' = 2x(1+x^2 - y)$ is turned into $$2x-u' = 2xu $$ multiplying by $e^{x^2},$ we get $$e^{x^2}(u'+2xu) = \left(e^{x^2}u\right)' = 2xe^{x^2}$$ on integration yields $$e^{x^2}u = e^{x^2} + c\to u = 1+ce^{-x^2},\quad y = x^2 -ce^{-x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\lim\limits_{x\to\infty}(\sin\sqrt{x+1}-\sin\sqrt{x})$ When using Maclaurin series, the limit is
$$\lim\limits_{x\to\infty}\frac{1}{\sqrt{x+1}+\sqrt{x}}=0$$
If we expand the expression with two limits
$$\lim\limits_{x\to\infty}\sin\sqrt{x+1}-\lim\limits_{x\to\infty}\sin\sqrt{x}$$
it diverges.
Which solution i... | Hint:
Mean value theorem implies there exist $c\in]x,x+1[$ such that
$$\sin(\sqrt{x+1})-\sin(\sqrt{x})=\frac{\sqrt{x+1}-\sqrt{x}}{2\sqrt{c}}\cos\sqrt{c}$$
Then $$\left|\sin(\sqrt{x+1})-\sin(\sqrt{x})\right|\le\left(\frac{\sqrt{x+1}}{2\sqrt{x}}-\frac{\sqrt{x}}{2\sqrt{x+1}}\right)|\cos\sqrt{c}|\le\frac{1}{2}\left(\sqrt{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1310875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 4
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Fourier transform of a radial function Consider a function $f \in L^2(\mathbb{R}^n)$ such that $f$ is radial. My question is, is the Fourier transform $\hat{f}(\xi)$ automatically radial (I can see it is even in each variable $x_i$), or we need some conditions on $f$? Thanks for your help.
| Preliminaries: i) $f$ is radial iff $f\circ T = f$ for every orthogonal transformation $T$ on ${R}^n.$ ii) An orthogonal transformation $T$ preserves the inner product: $\langle Tx,Ty \rangle = \langle x,y \rangle$ for all $x,y \in \mathbb {R}^n.$ iii) If $T$ is orthogonal, then $|\det J_T|=1,$ where $J_T$ is the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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$H_0^1(\Omega)$ space where $\Omega$ is an open bounded subset of $\mathbb R^N$. I have some trouble with proper understanding of $H_0^1$ space. I confess that I am now beginning to study functional analysis and maybe my question may seem rather trivial. However I would like to know if
$$H_0^1(\Omega)$$
where $\Omega$ ... | It is not finite dimensional.
One way of seeing this is if you consider the simple case $\Omega = ~ ]0,1[ \subset \Bbb R$. Then $H^1_0(\Omega)$ is the set of square integrable functions, whose derivative is also square integrable, and that the trace map on the boundary is $0$.
A subset of this is the set of continuous ... | {
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If $F$ a sheaf and $S\subset F$ a subfunctor, then $S$ is a subsheaf if and only if... This is Proposition 1 from Maclane & Moerdijk's Sheaves in Geometry and Logic, part II, section 1.
Proposition 1. Let $F$ be a sheaf on $X$ and $S\subset F$ a subfunctor. $S$ is a subsheaf if and only if, for every open set $U$ on $... | Having reduced the problem this far, the argument no longer needs any information about $S$ or $F$: such a diagram's upper left corner is an equalizer in any category. That said, I'll write a proof using the names you've provided.
Suppose $f$ is equalized by the two top-right maps. Then the composition of $f$ with the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the most unusual proof you know that $\sqrt{2}$ is irrational? What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a corollary of
this result:
Theorem:
If $n$ is a positive integer... | $$\boxed{\text{If the boxed statement is true, then the square root of two is irrational.}}$$
Lemma. The boxed statement is true.
Proof. Assume for a contradiction that the boxed statement is false. Then it has the form "if $S$ then $T$" where $S$ is false, but a conditional with a false antecedent is true.
Theorem. Th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_count": 19,
"answer_id": 14
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Prove $\sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$
I want to show that $u_k(x)= \sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$.
We know trivially that $0 \in L^2(0,1)$.
I need to show that $\langle u^*,\sin(kx) \rangle \to \langle u^*, 0 \rangle$ for each bounded linear functional $u^* \i... | Let $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \sin (2\pi i \,n x) + b_n \cos (2\pi i \,n x) \in L^2[0,1]$ then we have Bessel inequality:
$$ \frac{a_0}{2} + \sum_{n=1}^\infty (a_n^2 + b_n^2) < \bigg|\bigg|\int_0^1 f(x)^2 \, dx\bigg|\bigg|^2 < \infty$$
Then the Fourier coefficients tend to zero $|a_n| =|\langle f ,\... | {
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Why conjugate when switching order of inner product? There is an axiom of inner product spaces that states:
*
*$\overline{\langle x,y\rangle } = \langle y,x\rangle$
Basically (without any conceptual understanding) it seems like all you have to do when you swap the order of the arguments in an inner product space i... | Conjugation is there to make sure the signs work out. If you don't conjugate, then you'll find that $\langle ix, ix \rangle = -\langle x, x\rangle$.
| {
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"url": "https://math.stackexchange.com/questions/1311394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Proving a formula using another formula These questions are from the book "What is Mathematics":
Prove
formula 1: $$1 + 3^2 + \cdots + (2n+1)^2 = \frac{(n+1)(2n+1)(2n+3)}{3}$$
formula 2: $$1^3 + 3^3 + \cdots + (2n+1)^3 = (n+1)^2(2n^2+4n+1)$$
Using formulas 4 and 5;
formula 4: $$1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n... | I would go like this: $(2k+1)^2 = 4k^2+4k+1 \Rightarrow 1+3^2+5^2+\cdots + (2n+1)^2=\displaystyle \sum_{k=0}^n (2k+1)^2=4\displaystyle \sum_{k=0}^n k^2 + 4\displaystyle \sum_{k=0}^n k + (n+1)= 4\cdot\dfrac{n(n+1)(2n+1)}{6}+4\cdot\dfrac{n(n+1)}{2}+(n+1)=...$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $A$ is normal and $A$ and $B$ commute, then $A^*$ and $B$ commute Let $A$ is a normal matrix: $A^*\! A = A A^*\!\!$,$\,$ and $AB = BA$. Prove that $A^*\!B=BA^*\!\!$.
I can prove that if $\det A\ne 0$ by multiplication $AB=BA$ by $A^*$ left and right and using some manipulation. But I have no idea what to do if $\det... | We can use the fact that
$$\def\tr{\mathrm{tr}}
X=0\iff \tr(XX^*)=0.
$$
Since $A$ and $B$ commute, $A^*$ and $B^*$ commute as well. Together with the cyclic property of trace, $\mathrm{tr}(XY)=\mathrm{tr}(YX)$, we find that in each term of
$$
\begin{split}
\tr[(A^*B-BA^*)(A^*B-BA^*)^*]
&=
\tr(A^*BB^*A)+\tr(BA^*AB^*)
-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Simplify $\prod_{k=1}^5\tan\frac{k\pi}{11}$ and $\sum_{k=1}^5\tan^2\frac{k\pi}{11}$ My question is:
If $\tan\frac{\pi}{11}\cdot \tan\frac{2\pi}{11}\cdot \tan\frac{3\pi}{11}\cdot \tan\frac{4\pi}{11}\cdot \tan\frac{5\pi}{11} = X$ and $\tan^2\frac{\pi}{11}+\tan^2\frac{2\pi}{11}+\tan^2\frac{3\pi}{11}+\tan^2\frac{4\pi}{11}+... | The main building block of our solution will be the formula
\begin{align*}\prod_{k=0}^{N-1}\left(x-e^{\frac{2k i\pi}{N}}\right)=x^N-1.\tag{0}
\end{align*}
It will be convenient to rewrite (0) for odd $N=2n+1$ in the form
\begin{align*}
\prod_{k=1}^{n}\left[x^2+1-2x\cos\frac{\pi k}{2n+1}\right]=\frac{x^{2n+1}-1}{x-1}. \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Proving pseudo-hyperbolic distance is distance The pseudo-hyperbolic distance on the unit disk is defined as:
$$\rho(z,w)=\left|\dfrac{z-w}{1-\bar wz}\right|.$$
I'd like to prove it's a distance. The real problem is, as always, the triangle inequality, because the other properties are mostly obvious. That is, I need to... | It helps to know that $\rho$ is invariant under Möbius transformations. Indeed, $\rho(z,w)=|\phi(z)|$ where $\phi$ is any Möbius map such that $\phi(w)=0$ (they all agree up to rotation, which doesn't change the modulus). Since Möbius maps form a group, applying one of them to both $z$ and $w$ does not affect the above... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Surjectivity and injectivity I need to show the injective and surjective for
$f:\mathbb R^2 \longrightarrow \mathbb R$ where
$f(x,y)=5xe^y$
For injective $f(0,0)=f(0,1)$ but $(0,0) \neq (0,1)$.
For surjective i must show that the function covers the codomain so that every value from $\mathbb R^2$ must have an exit valu... | For surjectivity you have to show that for any $z \in \def\R{\mathbf R}\R$ there is $(x,y) \in \R^2$ such that $f(x,y) = z$.
Hint. To do so, use that you (hopefully) know that $\exp \colon \R \to (0,\infty)$ is bijective. If for example $z > 0$, then we can choose $x = \frac 15$. Which leaves us with
$$ f\left(\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1311935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number system and $\pi$ Ok, we all use the decimal system with numbers from 0 to 9. And we have $\pi$ with an infinite number of decimals.
We also have a boolean system or hexadecimal.
Is there any decimal system where $\pi$ has an ending number of numbers?
| If a number has a finite expansion, in a rational base, using rational digits, then the number is rational. This is because the sum and product of rational numbers is rational.
Note: Even some rational numbers have non-terminating expansions in base 10. For example, $1/3$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Changing variables in multiple integral (commutative of convolution operation) In the space $\mathbb{R}^n$, $n\geq 1$, the Lebesgue measure is denoted by $dx=dx_1\dots dx_n$ and $\int_{\mathbb{R}^n}f(x)dx$ stnads for $\int_{\mathbb{R}^n}f(x_1\dots x_n)dx_1\dots dx_n$.
I want to prove that convolution operation is comm... | The problem is how you're parameterizing the whole space. In the left hand side, your integrals are from $-\infty$ to $\infty$. In the integrals on the right hand side, it's the opposite parameterization.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1312085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof by counting two ways Proof by counting two ways:
\begin{equation}\sum_{k_1+k_2+...+k_m=n}{k_1\choose a_1}{k_2\choose a_2}...{k_m\choose a_m}={n+m-1\choose a_1+a_2+...+a_m+m-1}\end{equation}
I have a proof by algebra for it, but I want to seek a proof by counting it two ways.
Can you help me?
| Let $\ell=n-\sum_{i=1}^ma_i$; then
$$\dbinom{n+m-1}{a_1+\ldots+a_m+m-1}$$
is the number of ways to distribute $\ell$ indistinguishable balls amongst $$\sum_{i=1}^ma_i+m=\sum_{i=1}^m(a_i+1)$$ distinguishable boxes. For $i=1,\ldots,m$ let $A_i$ be a distinct set of $a_i+1$ distinguishable boxes, and let $A=\bigcup_{i=1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Markov chain - is my diagram/matrix correct? A boy goes to school on a bike or on foot.
If one day he goes on foot, then on the second day he takes a bike with probability $0.8$.
If he goes on a bike one day, then he falls off the bike with probability $0.3$ and goes on foot the next day.
What is the probability that o... | I think you can simplify things a lot by just using two states, foot (state $1$) and bicycle (state $2$). Then you have the transition matrix: $$\begin{pmatrix} 0.2 & 0.8 \\ 0.3 & 0.7 \end{pmatrix}$$
In fact, the way you've done it is a bit confusing because falling off a bicycle is not a state in the same sense that g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\mathbb{C}$ is a one-dimensional complex vector space. What is its dimension when regarded as a vector space over $\mathbb{R}$? $\mathbb{C}$ is a one-dimensional complex vector space. What is its dimension when regarded as a vector space over $\mathbb{R}$?
I don't understand how $\mathbb{C}$ is one-dimensional. Please... | The complex numbers as a vector space over the field of real numbers is of dimension $2$. The two vectors $1$ and $i$ form a basis and any complex vector $a+ib$ is a linear combination of the two vectors $1$ and $i$, multiplied by real scalars $a$ and $b$ and added.
The complex numbers as a vector space over the field ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1312320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
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Show that every subspace of $\mathbb{R}^n$ is closed Show that every subspace of $\mathbb{R^n}$ is closed.
I'm not sure how to do this or even what closed means. I don't even have a starting point. Any hints or solutions are greatly appreciated.
| Let $S$ be a linear subspace of $\mathbb{R}^n$. Consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of points in $S$ which converges to a point $y\in\mathbb{R}^n$. Show that in fact $y$ must lie in $S$. (Here I am using a characterization of "closed" which is equivalent to that described in the other answers).
You could use... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 5
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Find an inner product that makes a given set of linearly independent vectors orthogonal I need to find an inner product such that given a set $S$ of linearly independent vectors in a Hilbert space $H$, $S$ will be orthogonal with these product.
I thought Gram -Schmidt Process would help but it's not, because for the pr... | Start with a set $\{ x_1,\cdots,x_n\}$. Define
$$
U : \mathbb{C}^{n} \rightarrow H
$$
by
$$
U(\alpha_1,\cdots,\alpha_n) = \alpha_1 x_1+\cdots+\alpha_n x_n
$$
Let $P$ the be orthogonal projection of $H$ onto the closed subspace spanned by $\{x_1,\cdots,x_n\}$. Define
$$
(x,y)_{\mbox{new}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1312512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If all pairs of addends that sum up to $N$ are coprime, then $N$ is prime. I think this must be a known theorem, but I've tried searching for it on google without much luck. I would state it as follows:
If for all possible pairs of addends that sum to the same number N each of those pairs is comprised of two numbers t... | I do not think it has a name, but we can prove it right now. We will prove it by proving the contrapositive:
If $N$ is not prime, then there exists a pair of addends $X,Y$ summing to $N$ such that $X$ and $Y$ are not coprime.
Proof:
Let $N = ab$ where $a, b> 1$ (since $N$ is composite). Then we can take $X = a(b-1)$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Combination of trees If you have 12 trees with five of one kind, four of another and three of a third kind how many combinations of these trees can be planted in twelve holes?
| When you have an $n$ element set with all distinct elements, you can arrange the elements of the set into a line in $n!$ ways (i.e. $n\times(n-1)\times(n-2)\times...\times1$). The reason for this is that there are $n$ options for which element is placed in the first spot, then $n-1$ remaining options for which of the r... | {
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Transform a polynomial so that positive roots are shifted right and negative roots are shifted left I'm trying to figure out if it is possible to shift the roots of a polynomial outward, instead of to the left or right. Its relatively simple to shift all the solutions in one direction by substituting (x-k) or (x+k) fo... | Let $p(z) = a_0 + a_1z + a_2 z^2 + \dots + a_m z^m$ and let $z_1, \dots, z_m \in \mathbb{C}$ be the roots of $p$. For $\lambda \in \mathbb{C} \setminus \{0\}$, set $q(z) = b_0 + b_1z + \dots + b_mz^m$, where $b_j = \lambda^{-j} a_i$. Then the zeros of $q$ are $\lambda z_1, \dots, \lambda z_m$.
Therefore if $\lambda > ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Riemann zeta function, representation as a limit is it true that $$\displaystyle \zeta(s) = \ \lim_{\scriptstyle a \to 0^+}\ 1 + \sum_{m=1}^\infty e^{\textstyle -s m a } \left\lfloor e^{\textstyle(m+1)a} - e^{\textstyle m a} \right\rfloor$$ my proof :
\begin{equation}F(z) = \zeta(-\ln z) = \sum_{n=1}^\infty z^{\ln n}\e... | $$\begin{align}\left|e^{-nas}\left\lfloor e^{(n+1)a}-e^{na}\right\rfloor\right|&\leq e^{-nas}\left(e^{(n+1)a}-e^{na}\right)\\&=e^{-nas}e^{na}\left( e^{a}-1\right)\\&=\frac{e^a-1}{e^{na(s-1)}}
\end{align}$$
As $a\to 0$, $e^a-1\to 0$. and $e^{na(s-1)}\to 1$. So each term goes to zero as $a\to 0$.
That seems to contradict... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Roots of a power series in an interval Let $ a_0 + \frac{a_1}{2} + \frac{a_2}{3} + \cdots + \frac{a_n}{n+1} = 0 $
Prove that $ a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n = 0 $ has real roots into the interval $ (0,1) $
I found this problem in a real analysis course notes, but I don't even know how to attack the problem. ... | Consider the function $f(x)=a_0x+ \frac{a_1}{2}x^2+\cdots + \frac{a_n}{n+1}x^{n+1}$. Then $f(1)=0$ is given, and $f(0)=0$ is clear. Rolle's theorem now shows there is $x \in (0,1)$ such that $f'(x)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1312930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solving a Diophantine equation with LTE Show that only positive integer value of $a$ for which $$4(a^n+1)$$
is a perfect cube for all positive integers $n$, is $1$.
Rewriting the equation we obtain:
$$4(a^n+1)=k^3$$
It is obvious that $k$ is even and that in his prime factorization presents a prime factor $=2$ I proved... | No need of LTE. Suppose $$4(a^n+1)=k^3 \tag{$\star$}$$ for some $a>1$. Then, since $a^n+1>2$, we must have $$a^n+1=16b$$ $$ a^n=16b-1 \tag{P(n)}$$ for some $b\ge1$ (in fact $b$ is a cube, but it is an unnecessary information for our purposes). However, if $P(n)$ holds, then multiplying both sides by $a$ we see that $P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1313139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show determinant of $\left[\begin{matrix} A & 0 \\ C & D\end{matrix}\right] = \det{A}\cdot \det{D}$ Let $A \in \mathbb{R}^{n, n}$, $B \in \mathbb{R}^{n, m}$, $C \in \mathbb{R}^{m, n}$ and $D \in \mathbb{R}^{m, m}$ be matrices.
Now, I have seen on Wikipedia the explanation of why determinant of $\left[\begin{matrix} A &... | The first thing to note is of course the fact that $\det (AB)=\det A \cdot\det B$. This is well known - if you search you will find several proofs - in some texts this condition is used as an axiom when defining the determinant. So this allows you to assert
\begin{equation} \det \begin{bmatrix} A & 0 \\ C & D\end{bmat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$p$(ain)-adic number sequence I am trying to figure out how $p$-adic numbers work and currently am having trouble wrapping my head around how they work, so I made a pun! HAH!
Jokes aside, I am working on this question
Show that the sequence $(3,34,334,3334,.....)$ is equal to $2/3$ in $\hat{\mathbb{Z}}_5$
I assume th... | Here, the notation is in the usual base $10$, so the number $a_n = 33\ldots34$ simply denotes
$$a_n = 3 \times 10^n + 3 \times 10^{n-1} + \ldots + 3 \times 10 + 4 = 1 + 3\sum_{k = 0}^n 10^k = 1 + \frac{10^{n+1}-1}{3}$$
Now, in the $5$-adic topology, the sequence $(10^n)_{n \ge 0}$ goes to $0$ at infinity because $\left... | {
"language": "en",
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Functions proof. Find all functions $$f: \mathbb{Z} \rightarrow \mathbb{Z}$$ such that
$$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$
for all integers $$a, b, c$$ satisfying $$a+b+c=0$$
I have no idea how to even begin this one? Any comments?
| There are three different families of solutions for this equation. For any constant $k$, we have the following possibilities:
$$\begin{align*}f(2n)=0, && f(2n+1)=k && \forall n \in \mathbb{Z}\end{align*}$$
or
$$\begin{align*}f(4n)=0, && f(4n+1)=f(4n-1)=k, && f(4n+2)=4k && \forall n \in \mathbb{Z}\end{align*}$$
or
$$f(n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Can I show $\int^\infty_0\frac{\cos^2x}{x^2}dx=\infty$ by saying $\int^\infty_0\frac{1-\sin^2x}{x^2}dx=\infty-\frac\pi2$?
$${I=\int^{\infty}_{0}\frac{\cos^{2} x}{x^2}\;dx=\infty}$$
Attempt:
$$\begin{align}&= \int^{\infty}_{0}\frac{1- \sin^{2} x}{x^2}\, dx \tag1 \\[8pt]
&= \infty-\int^{\infty}_{0}\frac{\sin^{2} x}{x^2... | This looks a little better, I'd say.
$$\int_0^\infty\frac{\cos^2x}{x^2}dx>\int_0^{\pi/4}\frac{\cos^2x}{x^2}dx\ge\frac12\int_0^{\pi/4}\frac{dx}{x^2}=\infty$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1313554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Sum with Generating Functions Find the sum
$$\sum_{n=2}^{\infty} \frac{\binom n2}{4^n} ~~=~~ \frac{\binom 22}{16}+\frac{\binom 32}{64}+\frac{\binom 42}{256}+\cdots$$
How can I use generating functions to solve this?
| If $f(z) = \sum\limits_{n=0}^\infty a_n z^n$ converges for $|z| < \rho$, then for any $m \ge 0$,
$$\frac{z^m}{m!} \frac{d^m}{dz^m} f(z) = \sum_{n=0}^\infty a_m\binom{n}{m} z^n
\quad\text{ for } |z| < \rho.$$
Apply this to $$f(z) = \frac{1}{1-z} = \sum_{n=0}^\infty z^n,$$ we get
$$\sum_{n=2}^\infty \binom{n}{2} z^n = \f... | {
"language": "en",
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Why can't we determine the limit of $\cos x$ and $\sin x$ at $x=\infty $ or $x=-\infty$? I'm confused about why we can't determine the limit of $\cos x$ and $\sin x$ as $x \to \infty$, even though they are defined over $\mathbb{R}.$
When I use Wolfram Alpha, I get the following result (link to page):
which shows o... | They can't have a limit because they're periodic functions. What Wolfram Alpha outputs are the limit inferior and the limit superior of these functions, which always exist as soon as the functions are bounded. In case you haven't
Similarly seen these notions yet, by definition:
$$\limsup_{x\to\infty}f(x)=\lim_{x\to\inf... | {
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Some questions about the cartesian product I understand that the cartesian product of $A \times B$ is a set with elements of the form $(a,b)$ where $a\in A$, $b\in B$.
My question arise from the fact that I was described $\Bbb{R}^3$ as $\Bbb{R} \times \Bbb{R} \times \Bbb{R}$, but elements of $\Bbb{R}^3$ have the form $... | $A \times A \times A$ is usually defined as $(A \times A) \times A$ when the Cartesian product of two sets has been defined. This corresponds to your first view of $\mathbb{R}^3$.
On the other hand, powers of sets are also defined, namely $A^B$ is defined as the set of all functions from $B$ to $A$. Defining the natura... | {
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Trace of AB = Trace of BA We can define trace if $A =\sum_{i} \langle e_i, Ae_i\rangle$ where $e_i$'s are standard column vectors, and $\langle x, y\rangle =x^t y$ for suitable column vectors $x, y$. With this set up, I want to prove trace of AB and BA are same, so it's enough to prove that $$\sum_{i} \langle e_i, ABe_... | by definition $$\begin{align}trace(AB) &= (AB)_{11}+(AB)_{22}+\cdots+(AB)_{nn}\\
&=a_{11}b_{11}+a_{12}b_{21}+\cdots + a_{1k}b_{k1} \\
&+ a_{21}b_{12}+a_{22}b_{22}+\cdots + a_{2k}b_{k2}\\
&+\vdots \\
&+a_{n1}b_{1n}+a_{n2}b_{2n}+\cdots + a_{nk}b_{kn}\end{align}$$ if you view the sum according to the columns, then you se... | {
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Solve the following PDE using Fourier transform Solve the following 3-D wave equation using Fourier transform $$PDE: u_{tt}=C^2[u_{xx}+u_{yy}+u_{zz}],\qquad-\infty<x,y,z<\infty,\qquad t>0$$ $$BC: u(x,y,z,t)\rightarrow 0\qquad as \qquad r^2=x^2+y^2+z^2\rightarrow \infty \qquad $$ $$IC: u(x,y,z,0)=f(r),\qquad r=\sqrt{x^... | Just take 3 dimensional Fourier transform on the equation, and then it will be solved. Let $\hat{u}$ be the 3-D Fourier transform of $u$ and the variables $x,y,z$ will transform to $s_1,s_2,s_3$. Take Fourier on the original equation, we have
$$
\hat{u}_{tt}=-C^2(s_1^2+s_2^2+s_3^2)\hat{u}\\
\hat{u}(s_1,s_2,s_3,0)=\hat{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Can a polynomial equation always be manipulated to give a recurrence formula? Let $p(x)$ be a real (or maybe complex) polynomial. Suppose we wish to (numerically) solve $p(x) = 0$. This can be done for example with Newton's method of course, but I was thinking about if you "solve $x$" from the equation somehow and then... | it depends on the right hand side of x=g(x). If g(x) is a contraction mapping or if the initial iteration is within a compact set where g sends to itself or at least if futher iterations lie within such a compact set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Laurent expansion of $\frac{2}{(z-1)(3-z)}$ The question asks me to find all the possible Laurent series expansions of $$f(z)=\frac{2}{(z-1)(3-z)}$$ about the origin so $$z_0 =0 $$
First I convert $f(z)$ into partial fractions to get $$f(z)=\frac{1}{z-1}+\frac{1}{3-z}$$
We can see there are three domains
$$D_1: |z|<1$... | Hint: $$\frac{1}{z-1}=-\frac{1}{1-z}=\frac{z^{-1}}{1-z^{-1}}$$ and $$\frac{1}{3-z}=\frac{1}{3}\frac{1}{1-\frac{z}{3}} = -\frac{z^{-1}}{1-3z^{-1}}$$
Can you see how to get a Laurent series for $\frac{z^{-1}}{1-z^{-1}}$ so that it converges when $|z|>1$ - that is, when $|z^{-1}|<1$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Uniform convergence implies continuity and differentiability? For example: Suppose I have the following series:
$$\sum_{k=0}^{\infty}e^{-k}\sin(kt)$$
The Weierstrass-M-Test shows that the series is uniformly convergent on $\mathbb R$. Does this imply differentiability and continuity on $\mathbb R$ aswell?
| Yes, since each finite sum is continuous, then the uniform convergence of the series implies the continuity of the limit on any compact of $\mathbb{R}$, thus the continuity of the limit sum on $\mathbb{R}$.
For the differentiability, you can check that the series
$$
\left|\sum_{k=0}^{\infty}k\:e^{-k}\cos(kt)\right|\leq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does notation $(a_1, a_2,\cdots, a_n)$ mean in book "The Classical Introduction to Modern Number Theory"? I am reading the book on number theory and I have problems understanding this definition:
DEFINITION: Let $a_1, a_2, \cdots, a_n\in\mathbb{Z}$; we define $$\left(a_1, a_2, \cdots, a_n\right):=\left\{a_1x_1+\c... | It is correct that $(5,7)$ is the set of all integers.
However, this does not make the definition meaningless.
For example, if all $a_i$ are even, clearly each element in $(a_1, \dots, a_n)$ will be even.
So the set $(a_1, \dots, a_n)$ is not always the set of all integers.
Presumably, the book will proceed to show... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate the residue of this function
Find the residue at $z=0$ of the function $f(z)=\frac{\cot z}{z^4}$
I know that $z_0=0$ is a pole of order $k=5$, and $$Res(f;z_0)=\frac{\phi(z_0)^{(k-1)}}{(k-1)!}$$
but I cannot get the right answer that is $-\frac{1}{45}$
| Laurent series approach It is easy to see that the Laurent series of $\cot(z)$ around $z=0$ is
$$
\cot(z)=\frac 1z - \frac z3 - \frac{z^3}{45} - \cdots
$$
Thus
$$
\frac{\cot(z)}{z^4}=\frac{1}{z^5} - \frac{1}{3z^3} - \frac{1}{45z} - \cdots
$$
Hence, as pointed out in the comments the order of the pole is $5$ and ind... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Assume that f is a one to one function: If $f(x) = x^5 + x^3 +x$ , find $f^{-1}(3)$ and $f(f^{-1}(2))$ If $f(x) = x^5 + x^3 +x$ , find $f^{-1}(3)$ and $f(f^{-1}(2))$
How do I go about solving this?
For example, since I am giving f inverse should $I = x^5 +x^3 + x = 3$ ?
| In this case, you don't have to do very much. You can see just by examining the coefficients that $f(1) = 1^5 + 1^3 + 1 = 3$, so $f^{-1}(3) = 1$.
Since we're assuming that $f$ is one-to-one, that means precisely that $f(f^{-1}(x)) = x$ for all $x$. So $f(f^{-1}(2)) = 2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $A$ and $A^T$ do not have the same eigenvectors in general I understood that $A$ and $A^T$ have the same eigenvalues, since
$$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$
The problem is to show that $A$ and $A^T$ do not have the same eigenvectors. I have seen around some posts, but I ... | The matrix $A=\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}$, and its transpose $A^T$, have only one eigenvalue, namely $1$. However, the eigenvectors of $A$ are of the form $\begin{bmatrix}
a\\
0
\end{bmatrix}$, whereas the eigenvectors of $A^T$ are of the form $\begin{bmatrix}
0\\
a
\end{bmatrix}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1314980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 2
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Visual representation of matrices I am used to seeing most basic mathematical objects being visually represented (for instance, a curve in the plane divided by the xy axis; the same goes for complex numbers, vectors, and so on....), However, I never saw a visual representation of a matrix. I do not mean the disposition... | You can represent a $2 \times 2$ matrices $A = \left[\begin{smallmatrix}a & b\\c & d\end{smallmatrix}\right]$ as a parallelogram $Q_A \subset \mathbb{R}^2$ with
vertices $(0,0), (a,c), (b,d)$ and $(a+b,c+d)$. If one identify the plane $\mathbb{R}^2$ with $M^{2\times 1}(\mathbb{R})$, the space of $2 \times 1$ column mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Ordering relations smallest/minimal elements definitions In How to Prove It: A Structured Approach, 2nd Edition, page 192, the author introduces the following definitions of smallest and minimal elements of partial orders:
Definition 4.4.4. Suppose R is a partial order on a set A, B $\subseteq$ A, and b $\in$ B. Then ... | There is no real number which can serve as a minimal element. 8 is the smallest natural number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there infinite number of postive integer pairs $(p>q)$ ,such $3(2^p+1)=(2^q+1)^2$ Is there infinite number of postive integer pairs $(p>q)$
$$3(2^p+1)=(2^q+1)^2$$
I add my some approach
$$3\cdot 2^p+3=4^q+2^{q+1}+1$$
Give by
$$2^{2q-1}+2^q=3\cdot 2^{p-1}+1$$ I don't see how to proceed from this point
| Look at the binary digits of both sides.
Or consider the remainders $\bmod 4$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Principal ideals containing an ideal in a Noetherian integral domain Let $R$ be a Noetherian integral domain and $I$ a nonzero ideal consisting only of zero divisors on $R/(x)$, where $x$ is a nonzero element of $I$. Could we always find an element $y\notin (x)$ such that $yI\subseteq (x)$?
Thanks for any help!
| Isn't this obvious?
$I$ is contained in the union of associated primes of $R/(x)$, so there is such a prime $\mathfrak p$ with $I\subset\mathfrak p$. Now write $\mathfrak p=\operatorname{Ann}(\hat y)$ for some non-zero $\hat y\in R/(x)$, and you are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis I've been trying to solve this linear algebra problem:
You are given $n^2 > 1$ pairwise different real numbers. Show that it's always possible to construct with them a basis for $\mathbb{R^n}$.
The problem seems to be intuitive enough, but I coul... | We may prove the statement by mathematical induction. The base case $n=2$ is easy and we shall omit its proof. Suppose $n>2$. We call the target matrix $A$ and we partition it in the following way:
\begin{align*}
A=\left[
\begin{array}{ccccc}
\pmatrix{|\\ |\\ \mathbf v_1\\ | \\ |}
&\pmatrix{|\\ |\\ \mathbf v_2\\ | \\ |... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Linear Algebra: orientations of vector spaces (problem) This is an exercise from J.Munkres's Analysis on Manifolds:
Consider the vectors $\mathbf{a_i}$ in $\mathbb{R}^3$ such that:$$[\mathbf{a_1},\mathbf{a_2},\mathbf{a_3},\mathbf{a_4}]=\begin{bmatrix} 1 &0&1&1 \\ 1&0&1&1\\1&1&2&0\end{bmatrix}$$
Let $V$ the subspace of ... | I'll only explain part 2.
$\{a_1,a_2\}$ forms a basis for $V$. I want to express $a_3, a_4$ as linear combination of the basis. $a_3=a_1+a_2$, $a_4=a_1-a_2$. Therefore, in the basis $\{a_1,a_2\}$, $a_3, a_4$ can be expressed in a component form: $(1,1), (1,-1)$.
Observe that $\det \begin{pmatrix} 1 & 1 \\ 1 & -1 \end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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In $\Bbb R^3$, is there a general principle governing these "visual" angles? I believe most of you have drawn the xyz coordinate system hundreds of times and so have I. You may have drawn it like these, on various occasions:
(the reverse directions of the axis are not shown.)
All look ok, don't they?
But if you draw l... | I don't think there is some general principle for this. If anything, this is entirely a social construct. All of the pictures that you have drawn above are valid, and we could probably come up with various surfaces such that each is most easily visualized using each of these "visual angles".
The only thing that might... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is every finite field a quotient ring of ${Z}[x]$? Is every finite field a quotient ring of ${Z}[x]$? For example, how a field with 27 elements can be written as a quotient ring of ${Z}[x]$?
| Every finite field has an order which is the power of a prime.
Every finite field of order $p$ is isomorphic to integers modulo $p$. Every finite field of order $p^k$ is isomorphic to polynomials over the field with p elements; modulo an irreducible polynomial of degree $k$. There are no other fields.
So yes. Taking $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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An inequality with $\sum_{k=2}^{n}\left(\frac{2}{k}+\frac{H_{k}-\frac{2}{k}}{2^{k-1}}\right)$
show that
$$\sum_{k=2}^{n}\left(\dfrac{2}{k}+\dfrac{H_{k}-\frac{2}{k}}{2^{k-1}}\right)\le 1+2\ln{n}$$where $ n\ge 2,H_{k}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{k}$
Maybe this $\ln{k}<H_{k}<1+\ln{k}$?
| We have, by partial summation:
$$\begin{eqnarray*} \sum_{k=2}^{n}\frac{H_k}{2^{k-1}}&=&H_n\left(1-\frac{1}{2^{n-1}}\right)-\sum_{k=2}^{n-1}\left(1-\frac{1}{2^{k-1}}\right)\frac{1}{k+1}\\&=&1-\frac{H_n}{2^{n-1}}+\sum_{k=2}^{n}\frac{2}{k\, 2^{k-1}}\tag{1}\end{eqnarray*}$$
hence:
$$ \sum_{k=2}^{n}\frac{H_k-\frac{2}{k}}{2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1315922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Critical point of a function - $\Bbb R^n$ Analysis Consider $f=(f_1,f_2,f_3): U \rightarrow \mathbb{R}^3$ a function not identically null, $f\in C^1$ and rank $3$ at every point of the open $U \subset \mathbb{R}^n$, $n \geq 3$. Show that $g(x)= f_1^2(x)+f_2^2(x)+f_3^2(x)$, $x \in U$, hasn't maximum in $U$.
Suggestion: ... | There is a 'geometry' intuition behind your question. Observe that $g$ is the composition of $f$ and the square of the distance to the origen in $\mathbb{R}^3$. Namely, let $s : \mathbb{R}^3 \to \mathbb{R}$ be $s(x,y,z) := x^2 + y^2 + z^2$ be the square of the distance to $(0,0,0)$. Then $g = s \circ f$. Since $f$ has ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Euclidean domains with multiplicative and super triangular norms
I want to prove that if the norm function $N$ of a Euclidean domain $R$ satisfies the conditions
*
*$N(ab)=N(a)N(b)$
*$N(a+b) \le max\{N(a),N(b)\}$
then $R$ is a field or R is a homomorphic image of a polynomial ring $F[x]$ where $F$ is any field.
I... | This is a sketch of the proof. Nice exercise from {Algebra , N.Jacobson , V.1 , p149}.
From (1) we can deduce that $a$ is invertible iff $N(a)=1$.
From (2) we can deduce that the sum of two invertible elements is again invertible or $0$. Hence, the set of invertible elements together with $0$ is a field we call $F$.
if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expected number of dice rolls for a sequence of dice rolls ending at snake eyes If I roll a pair of dice repeatedly and stop only when I get snake eyes (both dice show 1), what is the expected number of dice rolls that will occur? I know the answer is 36, but I'm having trouble understanding why that is the answer.
| That happens because the mean of a geometric distribution with $p=\frac{1}{36}$ is exactly $\frac{1}{p}=36$.
The probability that a double one occurs at the $k$-th throw is given by:
$$ \mathbb{P}[X=k] = \frac{1}{36}\left(1-\frac{1}{36}\right)^{k-1},\tag{1}$$
hence:
$$ \mathbb{E}[X]=\sum_{k\geq 1}k\cdot\mathbb{P}[X=k]=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Regarding chains and antichains in a partially ordered infinite space I've been given this as an exercise.
If P is a partially ordered infinite space, there exists an infinite subset S of P that is either chain or antichain.
This exercise was given in the Axiom of Choice section of the class. I answered it using Zo... | HINT: The easiest argument is to use the infinite Ramsey theorem; you need just two colors, one for pairs that are related in $P$, and one for pairs that are incomparable in $P$. There is a fairly easy proof of the theorem at the link.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding Function's Extension and Its Unique Existence. Let $$A= \left\{\frac j{2^n}\in [0,1] \mid n = 1,2,3,\ldots,\;j=0,1,2,\ldots,2^n\right\} $$ and let $$ f:A\rightarrow R $$ satisfy the following condition:
There is a sequence $ \epsilon_n \gt 0 $ with $\sum_{n=1}^\infty \epsilon_n \lt \infty $ and $$\left|f\lef... | An alternate answer that uses some of Alex Ravsky's method, but sticks to analysis methods, and in my opinion provides a more constructive demonstration:
For $x \in [0, 1]$, let $$x = \sum_{k = 1}^{\infty} \omega_{k}(x) 2^{-k},\; \omega_{k} \in \{0, 1 \}$$ i.e. suppose $(\omega_{k}(x))_{k \in \mathbb{N}}$ is the binary... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Why does an integral change signs when flipping the boundaries? Let us define a very simple integral:
*
*$f(x) = \int_{a}^{b}{x}$
for $a,b\ge 0$.
Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$?
I drew the graphs and thought about it but to me integration, at least in two-dimensions, is just taki... | There is nothing to prove here. It is just a definition (more precisely, just a notaion). Note that for $b \geq a$, from the point of view of Lebesgue integration, the value of the integral only depends on the domain $[a,b]$ (and does not depend on the fact that if we consider the function from $a$ to $b$ or $b$ to $a$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
"answer_count": 10,
"answer_id": 9
} |
Fourier transform of the principal value distribution I would like to compute the Fourier transform of the principal value distribution. Let $H$ denote the Heaviside function.
Begin with the fact that
$$2\widehat{H} =\delta(x) - \frac{i}{\pi} p.v\left(\frac{1}{x}\right).$$
Rearranging gives that the principal value di... | Another solution
The distribution $\mathrm{pv} \frac{1}{x}$ satisfies $x \, \mathrm{pv} \frac{1}{x} = 1.$ Therefore,
$$
2\pi \, \delta(\xi)
= \mathcal{F} \{ 1 \}
= \mathcal{F} \{ x \, \mathrm{pv} \frac{1}{x} \}
= i \frac{d}{d\xi} \mathcal{F} \{ \mathrm{pv} \frac{1}{x} \}
$$
Thus,
$
\mathcal{F} \{ \mathrm{pv} \frac{1}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What exactly is a trivial module? Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion.
Searching on the web, I managed to found two possible definition of trivial modules, referring actually to two different mathematical objects.
The first one is just the singleton set wit... | You seem to be just confusing different notions of module.
*
*$G$-module would be an abelian group $M$ with the action of a group $G$ compatible with addition. (This can well be the trivial action.)
*$R$-module, or just module, where $M$ is an abelian group and one has scalar multiplication with the elements from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Solving for n in the equation $\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$
Solving for $n$ in the equation $$\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$$
Can anyone show me a numerical method step-by-st... | The function :
$$y=0.25^x+0.5^x+0.75^x-1$$
is decreasing.
For example $y(1)=2$ and $y(2)=-\frac{1}{8}$. So, the root for $y=0$ is between $x=1$ and $x=2$.
In this case, among many numerical methods, the dichotomic method is very simple. The successives values $x_k$ are :
$$x_{k+1}=x_{k}+\frac{\delta_k}{2^k}$$
where $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1316957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why the lens space L(2,1) is homeomorphic to $\mathbb{R}P^3$? According to one definition of lens space $L(p,q)$, which is gluing two solid tori with a map $h:T^2_1 \rightarrow T^2_2$. And $h(m_1)=pl_2+qm_2$, $l_i$ means longitude and $m_i$ means meridian of the boundary torus. I cannot understand why $L(2,1)$ is homeo... | Here's a sort of diagrammatic argument from an old homework assignment:
Construct $\mathbb{R}P^3$ as the quotient of $B^3 \subset \mathbb{R}^3$ under the antipodal map $a: \partial B^3 \to \partial B^3$. Let $K$ be the knot in $\mathbb{R}P^3$ obtained as the quotient of the vertical segment $V=\{(0,0,z) \in \mathbb{R}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Special properties in the direct solving of sparse symmetric linear systems In the area of computational solving of large sparse linear systems, some solvers specialize only on symmetric sparse matrices, be it positive definite or indefinite as compared to general (non-symmetric) sparse systems solver.
What mathematic... | A great advantage of the sparse Cholesky factorization for symmetric and positive definite matrices is that you do not need to do pivoting for numerical stability but only focus on the symbolic diagonal pivoting to minimize fill-in. So you can completely separate the symbolic and numeric factorization and reuse the str... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to find $\lim_{x \to 0}\frac{\cos(ax)-\cos(bx) \cos(cx)}{\sin(bx) \sin(cx)}$ How to find $$\lim_\limits{x \to 0}\frac{\cos (ax)-\cos (bx) \cos(cx)}{\sin(bx) \sin(cx)}$$
I tried using L Hospital's rule but its not working!Help please!
| Before using L'Hospital, turn the products to sums
$$\frac{\cos(ax)-\cos(bx)\cos(cx)}{\sin(bx)\sin(cx)}=\frac{2\cos(ax)-\cos((b-c)x)-\cos((b+c)x)}{\cos((b-c)x)-\cos((b+c)x)}.$$
Then by repeated application
$$\frac{2a\sin(ax)-(b-c)\sin((b-c)x)-(b+c)\sin((b+c)x)}{(b-c)\sin((b-c)x)-(b+c)\sin((b+c)x)},$$
and
$$\frac{2a^2\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
Laplace equation on a disk I have the Laplace equation $$\Delta u=\frac{1}{r} \frac{\partial}{\partial r } \left(r \frac{\partial u}{\partial r} \right)+\frac{1}{r^2} \frac{\partial u }{\partial \theta^2}=0$$ on a unit disk $$0<r \leq 1$$
We note that we have the boundary condition for R $$|u(0,\theta)|<\infty \rightar... |
Why does this give me no solution?
You have $ae^{p\pi}+be^{-p\pi}=ae^{-p\pi}+be^{p\pi}$. That is equivalent to $a(e^{p\pi}-e^{-p\pi})=b(e^{p\pi}-e^{-p\pi})$, which implies either $a=b$ or $e^{p\pi}=e^{-p\pi}\iff p=ki$, but in our case $p^2=k>0$, so that $k$ must be 0. If $p\neq0$ you have no solution, with $p=0$ you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Which mathematics theory studies structures like these? Let $A_p$ be the set of all numbers whose prime factors are all in first $p$ prime numbers.
example: $A_2= \{2,3,4,6,9,12,16,18,\ldots \}$ (all of these numbers can be generated by repeatedly multiplying only $2$ and $3$ the first two prime numbers).
as $p \to \in... | You haven't really defined a meaning for your $A_\infty$, unless you think "the first infinity prime numbers" make sense.
If you choose a definition for $A_\infty$ -- such as the union of all $A_p$ for finite $p$,
$$ A_\infty = \bigcup_{p\in\mathbb N} A_p $$
then it will be easy to show that $A_\infty=\mathbb N$. (Note... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
When does the cardinality disappear? Pardon me, if this question sounds stupid. I am learning real analysis on my own and stumbled on this contradiction while reading this -- http://math.kennesaw.edu/~plaval/math4381/setseq.pdf. I appreciate any help or pointers.
Consider this example: We choose an open interval of $(... | In essence this boils down to a question about the interchange of the limit and the cardinality function. In general it is not true that
$$
\lim_{n\to\infty} card(C_n) = card\left(\lim_{n\to\infty} C_n\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
What is the definition of differentiability? Some places define it as:
If the Left hand derivative and the Right hand derivative at a point are equal then the function is said to be differentiable at that point.
Others define it based on the condition of the existence of a unique tangent at that point.
Which one of t... | A function is differentiable (has a derivative) at point x if the following limit exists:
$$
\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}
$$
The first definition is equivalent to this one (because for this limit to exist, the two limits from left and right should exist and should be equal). But I would say stick to this defini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
A finite set is closed Question:
Prove that a finite subset in a metric space is closed.
My proof-sketch: Let $A$ be finite set. Then $A=\{x_1, x_2,\dots, x_n\}.$ We know that $A$ has no limits points. What's next?
Definition: Set $E$ is called closed set if $E$ contains all his limits points.
Context: Principles of... | If $M$ is a metric space then every subset $A =\{x_1, \ldots, x_n\} \subseteq M$ is closed. In fact, if $a \notin A$ then $d(a,A)$ is the least of the numbers $d(a,x_1) ,\ldots, d(a,x_n)$ thus, $d(a,A) > 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 7,
"answer_id": 2
} |
Prove that if A is singular, then adj(A) is also singular Prove that if A is singular, then adj(A) is also singular.
How do you prove this without proving by contradiction?
| From the basic equation: $A\,\text{adj}(A)=\det(A)I$ we have:
$$
\det(A\,\text{adj}A)=\det(\det(A)I)\\
\det(A)\det(\text{adj}(A))=\det(A)^n \det(I)
$$
$\det(\text{adj}(A))=\det(A)^{n-1}$
Since $A$ is singular,
$$
\det(\text{adj}(A))=0
$$
and then $\text{adj}(A)$ is also singular
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
The Standarization of Matrix by Vector Multiplication I apologize for the trivialness of my question but it has been bugging me as to why the standard for multiplying a matrix by a vector that will give a column matrix mean that the vector has to be a column matrix? To me it seems more natural to write the vector horiz... | Let $u$ be one of those dreaded column vectors. Then
$$
(A u)^\top = u^\top A^\top \Rightarrow A u = (u^\top A^\top)^\top
$$
This means one can get the same result by left multiplying the transposed vector $u^\top$ (now a row vector) with the transposed matrix $A^\top$, getting a row vector result and transposing the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $\lim_{n \to \infty} \frac{3^n-1}{2 \cdot 3^n} = \frac{1}{2}$? I used this limit as an argument in a proof I wrote (Proof by induction that $\sum\limits_{k=1}^n \frac{1}{3^k}$ converges to $\frac{1}{2}$).
I was told I should "prove" the limit but given no indication as to how to go about it. I didn't... | $$\lim_{n\rightarrow\infty}\frac{3^n-1}{2\cdot 3^n}=\frac{1}{2}\lim_{n\rightarrow\infty}\frac{3^n}{3^n}-\frac{1}{3^n}=\frac{1}{2}(1-0)=\frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
How prove this $\cot(\pi/15)-4\sin(\pi/15)=\sqrt{15}$ I need some help with this demonstration, please
I have tried with some identities but nothing.
I wanted to use this $$\sin(\pi/15)\cdot \sin(2\pi/15)\cdots\sin(7\pi/15)=\sqrt{15}$$
| We may prove:
$$ \cos\frac{\pi}{15}-4\sin^2\frac{\pi}{15}=\sqrt{15}\sin\frac{\pi}{15} $$
by squaring both sides. By setting $\theta=\frac{\pi}{15}$, that leads to:
$$ \frac{13}{2}-2\cos(\theta)-\frac{15}{2}\cos(2\theta)+2\cos(3\theta)+2\cos(4\theta) = \frac{15}{2}-\frac{15}{2}\cos(2\theta)$$
or to:
$$ -\cos(\theta)+\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1317960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Binary tree bijection I've been studying for an up coming exam in combinatorics and I came across something interesting by accident.
We have the two combinatorial constructions:
$$\mathbb{U}\cong SEQ(\mathbb{ZU})$$
And
$$\mathbb{T}\cong \mathbb{Z}*SEQ_2(\mathbb{T})+\epsilon$$
The first one I interpret as planar trees w... | I agree with your interpretation of the second construction. The most natural interpretation of the first, however, seems to me to be ordered plane forests of rooted trees, where size is determined by the number of nodes. Ah, I see: that’s the same as rooted plane trees with size determined by the number of nodes not c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Convergence/divergence of the series $\sum_{n=1}^{\infty}\frac{1}{n^\sqrt{n}}$ $$\sum_{n=1}^{\infty}\frac{1}{n^\sqrt{n}}$$
Determine whether this series is convergent or not, with explanation.
Each element is positive, so I've tried bounding it by another convergent series, but couldn't see how.
I couldn't apply integr... | $1\lt \sqrt2\lt\sqrt{n}\,\,\,\,\,\,\,\,\,\,\forall n\ge3\implies0\lt\dfrac{1}{n^{\sqrt{n}}}\lt\dfrac{1}{n^{\sqrt2}}\,\,\,\,\,\,\,\,\,\,\forall n\ge3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
A topological group is embeddble in a product of a family of second-countable topological groups if and only if it is $\omega$-narrow How to prove the following property: a topological group is topologically isomorphic to a subgroup of the product of some family of second-countable topological groups if and only if it ... | This is a result of my first supervisor Igor Y. Guran. Its proof it rather long and can be found, for instance, in a book “Topological groups and related structures” by his supervisor Alexander V. Arhangel'skii and co-student Mikhail G. Tkachenko (Atlantis Press, Paris; World Sci. Publ., NJ, 2008), where this results i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Difference between Gentzen and Hilbert Calculi What is the difference between Gentzen and Hilbert Calculi?
From my understanding of Rautenberg's Concise Introduction to Mathematical Logic, Gentzen calculus is based on sequents and Hilbert calculus, on tautologies.
But isn't every Gentzen sequent a tautological modus p... | You can see a very detailed overview into : Francis Pelletier & Allen Hazen, Natural Deduction :
Sequent Calculus was invented by Gerhard Gentzen (1934), who used it as
a stepping-stone in his characterization of natural deduction [...]. It is a very general characterization of a proof; the basic notation being
$ϕ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Is there a name of such functions? Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions:
*
*Is there any name for such functions in Mathematics Literature?
*Given any set $U$ as above... | There's an old theorem (due to Whitney, I think) that says the following:
Given any smooth manifold $M$ and any closed subset $K\subseteq M$, there exists a smooth function $f\colon M\to \mathbb R$ that satisfies $f=0$ on $K$ and $f>0$ on $M\smallsetminus K$.
You can find a proof in my Introduction to Smooth Manifold... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Express 99 2/3% as a fraction? No calculator My 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is appreciated:
The possible answers are:
$1 \frac{29}{300}$
$\frac{269}{300}$... | $99 \frac{2}{3} \% = 99 \frac{2}{3} \cdot \frac{1}{100} = \frac{299}{3} \cdot \frac{1}{100} = \frac{299}{300}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
"answer_count": 14,
"answer_id": 7
} |
Finding incomplete geodesics I have a problem with the notion of incomplete geodesics. Can someone give me a minimal example for such a geodesic?
In particular, I am trying to solve the following exercise:
Consider the upper half plane $\mathbb{H}:= \{(x,y):y>0\}\subset \mathbb{R}$ equiped with the metric: $g_{q}:=\fra... | Hint: Try integrating up (if $q > 2$) or down (if $q < 2$) the $y$-axis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Explain a couple steps in proof that ${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r}$ Show ${n \choose r}$ = ${n-1 \choose r-1}$ + ${n-1 \choose r}$
I found a similar question on here but I am looking for a little bit more of an explanation on how they simplified
Right Hand Side $$= \frac{(n-1)!}{(r-1)!(n-r)!} ... | For a combinatorial proof, $\binom nr$ is the number of $r$-element subsets of $\{1,2,\ldots,n\}$. Each such subset either contains $n$ or does not. $\binom {n-1}{r-1}$ counts the former, $\binom {n-1}r$ counts the latter. Therefore the quantities $\binom nr$ and $\binom {n-1}{r-1} + \binom {n-1}r$ are equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
When does $(x^x)^x=x^{(x^x)}$ in Real numbers? I have tried to solve this equation:$(x^x)^x=x^{(x^x)}$ in real numbers I
got only $x=1,x=-1,x=2$ , are there others solutions ?
Note: $x$ is real number .
Thank you for your help .
| Just a sketch for $x>0$:
Let $y=x^x$ and rewrite $x^y=y^x$ by taking $log$'s as $\log (x)/x = \log(y)/y$.
For each $x \in (1, e)$ there is a unique $y(x)$ such that $f(x) = f(y(x))$.
As a function of $y(x)$, you can show that it is increasing. So we are looking for solutions of $y(x) = x^x$. Because $g(x) = y(x)-x^x$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1318919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Do $\Bbb Q (\sqrt 2)$ and $\Bbb Q [\sqrt 2]$ mean the same? Do $\Bbb Q (\sqrt 2)$ and $\Bbb Q [\sqrt 2]$ mean the same?
I'm trying to refer to the field of the real numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rationals.
E: I'm sorry, my question was unclear, I was using $\sqrt 2$ as an example number, bu... | Yes, they are same, but it's because $\mathbb{Q}$ is a field, and they are not same if you replace it by an arbitrary ring, which is not a field. In general, bracket gives the meaning "smallest ring containing the element in the bracket and the given ring", while paranthesis means "smallest field containing the element... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1319000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 4
} |
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