Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to solve $-20=15 \sin \theta- 30.98 \cos \theta$? How can I solve the following equation? $$-20=15 \sin \theta- 30.98 \cos \theta$$
I can't think of any way to solve it. You can't factor out cosine because of the annoying little negative twenty, and if you divide by cosine you also get nowhere.
| If you're familiar with basic vectors, you may reason the $a\cos\theta + b\sin\theta $ formula like this
$$20 = 30.98 \cos \theta - 15\sin \theta = \langle 30.98, -15 \rangle \bullet \langle \cos \theta, \sin\theta \rangle = C \cos(\theta - \alpha) $$
$C = \sqrt{30.98^2 + 15^2}$
$\alpha = \arctan\left(- \dfrac{30.98}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/929344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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A Definite Integral Simplification There was this question on integration that I am correct with the answers section of my textbook upto a point. So I will only ask this last step of evaluation.
This is what the book says -
And this is how my simplication ended up in -
As you can see both the book and my first part ... | $$\left(\frac{1}{\sqrt{2}}\right)^{2n+1}=\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right)^n\not=\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/929421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how to show that $a+c>2b$ do not guarantee that $ac>b^2$? HW problem. I solved it by giving counter example. But it will be nice if there is an algebraic way to show this.
Given both $a,c$ are positive, and given that $a+c>2b$, the question asks if this guarantee that $ac>b^2$. The answer is no. Counter example is $a... | Edit: perhaps this is what you are thinking of?
Fix $S>0$ and suppose that $2b<S$. With $b>0$, this is equivalent to $4b^2<S^2$ or $b^2\in(0,\frac{S^2}{4})$. Then a counter example is guaranteed if we can show that for any $P\in(0,\frac{S^2}{4})$, there exist $a>0,c>0$ such that $a+c=S$ and $ac=P$. Such $a$ and $c$ are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/929614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivative of a contravariant tensor Let $T$ be a contravariant tensor so it transforms under change of coordinates like
$$
T^{i'} = T^i\ \frac{\partial x^{i'} }{\partial x^i}
$$
In this it seems $T^{i'}$ is a function of the "primed" coordinates, i.e. $T^{i'}(x')$, so it should be possible to calculate the partial der... | No, actually $T^{i'}$ is a function in not primed coordinates. It's components show which value the primed coordinate will have depending of not primed one. Example: $$x=r \cos(\phi), y = r \sin(\phi)$$
Here primed coordinates are x and y. If you will take contravariant tensor of form $T=(f^1(\phi, r), f^2(\phi, r))^T$... | {
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Derived categories as homotopy categories of model categories Given an abelian category A, is there a model structure on the category of complexes C(A) (or K(A) ("classical" homotopy category)) such that its homotopy category "is" the derived category D(A)?
| Let $\mathbf{A}$ be an abelian (or Grothendieck) category, and consider the injective model structure on the category of cochain complexes $\mathrm{Ch}(\mathbf{A})$. Then the derived category $\mathcal{D}(\mathbf{A})$ is the homotopy category $\mathrm{Ho}(\mathrm{Ch}(\mathbf{A}))$ by inverting the weak equivalences.
| {
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Let $a$ and $b$ be two elements in a commutative ring $R$ and $(a, b) = R$, show that $(a^m, b^n) = R$ for any positive integers $m$ and $n$. I stumbled across a question that I have no idea how to start.
I know the questions asking to show that the multiples of $a$ and $b$ as an ordered pair make still make the whole ... | Write $1$ as a linear combination of $a$ and $b$. Then raise both sides to the $m+n$ power.
| {
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Convergence of multiple integral in $\mathbb R^4$ Denote $(x,y,z,w)$ the euclidean coordinates in $\mathbb R^4$. I am trying to study the convergence of the integral $$\int \frac{1}{(x^2+y^2)^a}\frac{1}{(x^2+y^2+z^2+w^2)^b} dx\,dy\, dz\, dw$$
over a disk (or cube, or any open set) containing the origin
in terms of the ... | As DiegoMath suggests, using polar coordinates for $(x,y)$ and $(z,w)$ is efficient, more precisely, define
$$(x,y,z,w):=(r\cos\theta,r\sin\theta,s\cos\phi,s\sin\phi).$$
Then the integral
$$I(a,b):=\int_{B(0,1)} \frac{1}{(x^2+y^2)^a}\frac{1}{(x^2+y^2+z^2+w^2)^b} dx\,dy\, dz\, dw$$
converges if and only if the integral
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Jordan form of a Matrix with Ones over a Finite Field Question: Find the Jordan Form of $n\times n$ matrix whose elements are all one, over the field $\Bbb Z_p$.
I have found out that this matrix has a characteristic polynomial $x^{(n-1)}(x-n)$ and minimal polynomial $x(x-n)$, for every $n$ and $p$.
Here I have two cas... | Your matrix always has rank$~1$ (if $n>0$). This means (rank-nullity) that the eigenspace for $\lambda=0$ has dimension $n-1$; this is the geometric multiplicity of that eigenvalue, and the algebraic multiplicity of $\lambda=0$ is either $n-1$ or $n$. The trace$~n$ of the matrix is the remaining eigenvalue (as the sum ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to explain Borel sets and Stieltjes integral to beginner maths student? The problem is that I know by the definition what Borel sets and Stieltjes integral are but I'm not good to explain in layman terms what they are. Is there easier answer that "write down the definitions until you have reduced everything to the ... | A Borel set is a set that we can create by taking open or closed sets, and repeatedly taking countable unions and intersections. Any measure defined on a Borel set is called a Borel measure.
The Riemann integral is a definite integral that was the foundation of integration over a given interval (before the far superio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/930129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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System of Linear Equations - how many solutions? For which real values of t does the following system of linear equations:
$$
\left\{
\begin{array}{c}
tx_1 + x_2 + x_3 = 1 \\
x_1 + tx_2 + x_3 = 1 \\
x_1 + x_2 + tx_3 = 1
\end{array}
\right.
$$
Have: a) a unique solution?
b) infinitely many solutions?
c) no soluti... | $\text{I have used the Gauss elimination and then studied the rank of the coefficient matrix:}$
$\text{1)If}\ t=1 \text{ the system reduces to just one equation, and it has}\ \infty^2 \text{solutions.}$
$\text{2)If }\ t=-2 \text{ there are no solutions.}$
$\text{3)If}\ t≠1,-2 \text{ there is a unique solution, dependin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/930252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Are there PL-exotic $\mathbb{R}^4$s? The title may or may not say it all. I know that there are examples of topological 4-manifolds with nonequivalent PL structures. In some lecture notes, Jacob Lurie mentions that not every PL manifold is smoothable, and that while smoothings exist in dimension 7 they may not be uniqu... | In this survey article on differential topology, Milnor outlines a proof that every PL manifold of dimension $n \leq 7$ possesses a compatible differential structure, and whenever $n<7$ this structure is unique up to isomorphism. He includes references for the various facts he uses.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/930349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $a+b+c=1$ and $abc>0$, then $ab+bc+ac<\frac{\sqrt{abc}}{2}+\frac{1}{4}.$ Question:
For any $a,b,c\in \mathbb{R}$ such that $a+b+c=1$ and $abc>0$, show that
$$ab+bc+ac<\dfrac{\sqrt{abc}}{2}+\dfrac{1}{4}.$$
My idea: let
$$a+b+c=p=1, \quad ab+bc+ac=q,\quad abc=r$$
so that
$$\Longleftrightarrow q<\dfrac{\sqrt{r}}{2... | Edit: Incomplete approach. Only works if $a,b,c\geq 0$.
By replacing $\frac{1}{4}$ on the RHS with $\frac{(a+b+c)^2}{4}$, the inequality you seek is equivalent to
$$
a^2+b^2+c^2+2\sqrt{abc}>2(ab+bc+ca).\tag{I}
$$
To prove (I), we use the following result
$$
a^2+b^2+c^2+3(abc)^{2/3}\geq 2(ab+bc+ca)\tag{II}
$$
the proof ... | {
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Can the logic associative law be applied here? $\big(p \rightarrow (q \rightarrow r)\big)$ is logically equivalent to $\big(q \rightarrow (p \rightarrow r)\big)$
I am a little confused when dealing with the 'implies' operator $\rightarrow$ and the logic laws.
To prove that these are logically equivalent, can I just a... |
$\big(p \rightarrow (q \rightarrow r)\big)$ is logically equivalent to $\big(q \rightarrow (p \rightarrow r)\big)$
Hint for alternative proof suggestion:
'$\to$'
*
*$p \rightarrow (q \rightarrow r)$ (Premise)
*$q$ (Premise)
*$p$ (Premise)
*$q \rightarrow r$ (Detachment, 1, 3)
etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/930494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculus limit epsilon delta Prove using only the epsilon , delta - definition
$\displaystyle\lim_{x\to 2}\dfrac{1}{x} = 0.5$
Given $\epsilon > 0 $, there exists a delta such that
$ 0<|x-2|< \delta$ implies $|(1/x) – 0.5| < \epsilon$.
Therefore, $$\begin{align}|(1/x) – 0.5| & = |(2-x)/2x| \\
& =|\frac{-(x-2)}{2(x-2+... | This is the jist of it. You are required to bound the value of $ | \dfrac 1 x - \dfrac 1 2 | $. But you are only allowed to bound $ |x - 2| $. So by putting in a restriction on $ |x - 2| \lt \delta$ you need to achieve the restriction $ | \dfrac 1 x - \dfrac 1 2 | \lt \epsilon$. And more importantly you need to pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/930576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\lim_{n\to\infty}\frac{a^n}{n!}=0$ and that $\sqrt[n]{n!}$ diverges.
Let $a\in\mathbb{R}$. Show that
$$
\lim_{n\to\infty}\frac{a^n}{n!}=0.
$$
Then use this result to prove that $(b_n)_{n\in\mathbb{N}}$ with
$$
b_n:=\sqrt[n]{n!}
$$
diverges.
Okay, I think that's not too bad.
I write
$$
\fr... | For the first limit, hint: Let $a$ be any real number. As $n$ tends to $+\infty,$
$$
\left|\dfrac{\dfrac{a^{n+1}}{(n+1)!}}{\dfrac{a^n}{n!}}\right|=\left|\dfrac{a}{n+1}\right| < \frac{1}2, \quad n\geq2|a|,
$$ and, by an elementary recursion,
$$
\left|\dfrac{a^n}{n!}\right|< \frac{C(a)}{2^n},
$$ as $n$ is great, where $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/930778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Show that lim inf Bn and lim sup Bn equals to a null set Suppose that ${B_n: n \geq 1}$ is a sequence of disjoint set.
Show that
$$\begin{align}\limsup_{n\rightarrow \infty}B_n &= \emptyset \text{ and}\\
\liminf_{n \rightarrow \infty}B_n&= \emptyset \end{align}$$
Where $$\begin{align}\limsup_{n\rightarrow \infty}B_n &... | As a given element $x$ cannot be in more than one $B_n$, in particular it can't be in infinitely many, giving both results as the lim sup requires $B_n$ to be in infinitely many, and the lim inf is even stronger requiring it to be in all but finitely many.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\arctan a+\arctan(\frac{1}{a})=\frac{\pi}{2}$ $$\arctan a+\arctan\left(\frac 1 a \right)=\frac \pi 2$$
I have the mark scheme in front of me, and I understand where the numbers come from, but I don't understand why they do what they do.
You need this part in to show it:
The markscheme says:
Again, I unders... | Differentiate with respect to $a$, to prove that it has to be constant. Then taking the value at $a=1$ gives you the result. That's probably the easiest way to prove it.
| {
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"timestamp": "2023-03-29T00:00:00",
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How do I find $\lim\limits_{x \to 0} x\cot(6x) $ without using L'Hôpital's rule? How do I find $\lim\limits_{x \to 0} x\cot(6x) $ without using L'Hôpital's rule?
I'm a freshman in college in the 3rd week of a calculus 1 class. I know the $\lim\limits_{x \to 0} x\cot(6x) = \frac{1}{6}$ by looking at the graph, but I'm n... | $$
x\cot (6x) = \frac 1 6 \cdot \frac{6x}{\sin(6x)}\cdot\cos(6x) = \frac 1 6 \cdot \frac{u}{\sin u}\cdot\cos(6x)
$$
and as $x$ approaches $0$, so does $u$. The fraction $u/\sin u$ has both the numerator and denominator approaching $0$, and its limit is well known to be $1$ (presumably you've seen that one before).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/931212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$? Let's say that I have a vector $\mathbf{w}$.
How can I calculate the derivative in the following expression?
$\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$
Update: found these useful definitions
| It is easier to see it in component form.
Let $\hat{x_i}$ be the unit vector in the $i$-th direction, then we can express a vector as
$$\mathbf{w}=\sum_{i=1}^{n}w_i \hat{x_i} \tag{1}$$
And
$$\frac{d}{d\mathbf{w}}=\sum_{i=1}^{n}\hat{x}^T_i \frac{d}{dw_i} \tag{2}$$
So
$$\mathbf{w}^T \mathbf{w}=\sum_{i=1}^{n}w_i^2 \tag{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/931432",
"timestamp": "2023-03-29T00:00:00",
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How can I determine the base of the following numbers for the operations to be correct? Given:
$24_A + 17_A = 40_A$
How can I find the base of the following number $A$ so the operations are correct?
| Hint to get you started: By definition, in base b, the first digit from the right is representative of the $b^0$ place, the second from the right is $b^1$'s, just like in base 10 the number 12 is $2 * 10^0 + 1* 10^1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find a basis of $M_2(F)$ so that every member of the basis is idempotent Let $V=M_{2\times 2}(F)$ (the space of 2x2 matrices with coefficients in a field $F$). Find a basis $\{A_1,A_2,A_3,A_4\}$ of $V$ so that $A_j^2=A_j$ for all $j$.
My attempt. Let $A_j$ be $$\begin{pmatrix} a_1 & a_2\\ a_3 & a_4\\ \end{pmatrix}. $$
... | An idempotent basis for V is:
$\{A_1, A_2, A_3, A_4\} =\{[1, 0; 0, 0], [1, 0; 1, 0], [0, 1; 0, 1], [0, 0; 0, 1]\}$
To see this, first of all notice that A1^2 = A1, A2^2 = A2, A3^2 = A3, and A4^2 = A4.
Next, to show that $\mathrm{ span}\{A_1, A_2, A_3, A_4\} = V$, notice that A2- A1 = [0, 0; 1, 0] and A3 - A4 = [0, 1; ... | {
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"timestamp": "2023-03-29T00:00:00",
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List of unsatisfiable cores? Is there a place I can find a list of known unsatisfiable cores for X variables [no more then 10] in CNF format?
Or is there an 'easy' way to find out, say I have 7 variables how many clauses [of the 7 variables] do I need for an unsatisfiable cores.
[I calculated 2-5 by hand, I am just try... | The question you are asking has been dealt with in the literature, and there is even a system that does it. Here is an example: https://sun.iwu.edu/~mliffito/publications/jar_liffiton_CAMUS.pdf
| {
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"timestamp": "2023-03-29T00:00:00",
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Using Big-O to analyze an algorithm's effectiveness I am in three Computer Science/Math classes that are all dealing with algorithms, Big-O, that jazz. After listening, taking notes, and doing some of my own online searching, I'm pretty damn sure I understand the concept and reason behind Big-O, and what it means when ... |
Give a big-O estimate for this function. For the function $g$ in your estimate $f(x)$ is $O(g(x))$, use a simple function $g$ of smallest order.
$$x_n=(n^3 + n^2\log n)(\log n + 1) + (17\log n + 19)(n^3 +2)$$
To solve this, one first spots the dominant term in each part, here $n^2\log n\ll n^3$, $1\ll\log n$ and $1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Two definitions of connectedness: are they equivalent? A topological space $(X, \tau)$ is connected if $X$ is not the union of two nonempty, open, disjoint sets. A subset $Y \subseteq X$ is connected if it is connected in the subspace topology.
In detail, the latter means that there exist no (open) sets $A$ and $B$ in ... | When dealing with the subspace topology, one really shouldn't bother what happens outside the subspace in consideration.
Nevertheless, assume that $Y$ is a subset of $\mathbb R^n$ such that there exist open sets $A,B$ such that $Y\cap A$ and $Y\cap B$ are disjoint and nonempty, but necessarily $A\cap B\ne \emptyset$ fo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The area not covered by six pointed star In a circle with radius $r$, two equi triangles overlapping each other in the form of a six pointed star touching the circumference is inscribed! What is the area that is not covered by the star?
Progress
By subtracting area of the star from area of circle , the area of the surf... | The length of a side of an equilateral triangle is
$$\sqrt{r^2+r^2-2\cdot r\cdot r\cdot \cos (120^\circ)}=\sqrt 3r.$$
The distance between the center of the circle and each side of an equilateral triangle is $$\sqrt 3r\cdot \frac{\sqrt 3}{2}\cdot \frac{1}{3}=\frac 12r.$$
Hence, the length of a side of the smaller equil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/932172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can we composite two piecewise functions? Let say
$$f(x) := \left\{\begin{array}{l l}x+1 & \text{ if } -2\leq x<0 \\ x-1 & \text{ if }0\leq x\leq 2\end{array}\right.$$
How can we find out $f\circ f$?
| Hint: The first thing to do is to write down the domain and codomain of your function. Here we have:
\begin{equation}
f:[-2,2]\longrightarrow[-1,1]
\end{equation}
Next, since $[-1,1]\subset[-2,2]$, it is OK to compose $f$ with itself. Try to figure out yourself what $f\circ f$ does.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_count": 2,
"answer_id": 1
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I want to form a group of order one, is it possible? Can a set with one element be a group, or does a group need to have at least two elements?
| It is possible to have an element with just one element. Take $G=\{e\}$ with the group law defined as $e\circ e=e$. Then it is easy to verify that:
*
*The group law is associative: $(e\circ e)\circ e=e\circ(e\circ e)=e$
*There is an identity $e$ such that $e\circ g=g = g \circ e$ for all $g\in G$.
*Every element o... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\frac{1}{1^4}+\frac{1}{2^4}+\cdots+\frac{1}{n^4} \le 2-\frac{1}{\sqrt{n}}$ I have to show: $\displaystyle\frac{1}{1^4}+\frac{1}{2^4}+...+\frac{1}{n^4} \le 2-\frac{1}{\sqrt{n}}$ for natural $n$
I tried to show it by induction (but I think it could be possible to show it using some ineqaulity of means) so for... | You should show its:
$\displaystyle2-\frac{1}{\sqrt{n}}+\frac{1}{(n+1)^4}\le2-\frac{1}{\sqrt{n+1}}$
The other ways does not help your proof (think about it for a while)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/932427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Why does the product topology allow proper subsets for only finitely many elements? Consider Theorem 19.1 from Munkres' topology:
The box topology on $\prod X_\alpha$ has as basis all sets of the form $\prod U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for each $\alpha$. The product topology on $\prod X_\alpha$ h... | There are two ways to view the generation of a topology from a subbasis $\mathcal S$.
*
*The topology generated by $\mathcal S$ is the smallest (coarsest) topology in which all the sets in $\mathcal S$ are open.
*We first transform $\mathcal S$ into a basis $\mathcal B$ consisting of all (nonempty) finite intersect... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/932506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Calculating $\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - x^2 $ I want to calculate the following limit:
$\displaystyle\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - x^2 $
I tried the following:
$\displaystyle\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - x^2 = $
$\displaystyle\lim_{x\rightarrow +\infty} x^2... | Hint: $-\frac{1}{2}x^2\geq x+1-x^2\geq (x\sin x+\cos x-x^2)$ for $x\,$ large enough. Now let $x$ go to infinity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/932589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove some properties of the $p$-adic norm I need to prove that the $p$-adic norm is an absolute value in the rational numbers, by an absolute value in a field $K$ I mean a function $|\cdot|:K \to \mathbb{R}_{\ge 0}$ such that:
$$\begin{align}
\text{I)}&~~~~ |x|=0 \implies x=0\\
\text{II)}&~~~~|x\cdot y|=|x|\cdot|y|\\... | For (II), write
$q_1=p^mr, q_2=p^ns$ where $r,s\in\Bbb Q$ has numerator and denominator prime to $p$ and $m,n\in\Bbb Z$
Then
$$|q_1q_2|_p=p^{-(n+m)}=p^{-n}p^{-m}=|q_1|_p\cdot|q_2|_p$$
For (III) assume, that $n> m$, then
$$|q_1+q_2|_p=p^{-m}|p^{n-m}s+r|_p=p^{-m}\le p^{-m}+p^{-n}$$
because
$$p|(p^{n-m}s+r)\implies p\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/932763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof that $S_3$ isomorphic to $D_3$ So I'm asked to prove that $$S_{3}\cong D_{3}$$ where $D_3$ is the dihedral group of order $6$. I know how to exhibit the isomorphism and verify every one of the $6^{2}$ pairs, but that seems so long and tedious, I'm not sure my fingers can withstand the brute force. Is there a bett... | Every symmetry of the triangle permutes its vertices, so there is an embedding $D_3\to S_3$ already just from geometry. Then it suffices to check they have the same order.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/932843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 3,
"answer_id": 0
} |
What does it really mean when we say that the probability of something is zero? Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible.
But when we look at the probability from a mathematics perspective, probability is defined as the frequency of occurrence over the h... | One way of making this rigorous is to make an analogue between probability and area. The comparison here is made precise via measure theory, but can be explained without recourse to technical definitions.
Consider two "shapes": a point and the empty shape consisting of no points. Think of these geometric entities as si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/932928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
computing the Haar measure for O(n) and U(n) groups My question is about how to compute the Haar measure for O(n) and U(n) groups.
For example, for the conventional parametrization of SO(3) with 3 angels, the Haar measure is
$ dO= cos(\theta_{13})*d\theta_{12}*d\theta_{13}*d\theta_{23}$ .
How one derives this formula?... | You might like these lecture notes. In particular, the second page discusses such calculations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/932985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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There is a unique polynomial interpolating $f$ and its derivatives I have questions on a similar topic here, here, and here, but this is a different question.
It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives at certain points) exists uniquely. That means that th... | One can prove that derivatives of the form we mention of $x_1, \dotsc, x_k \mapsto \prod\limits_{1\leq i<j\leq n}(x_i-x_j)$ are nonzero. Note that the operation of taking the derivative naturally divides the $x_i$ into groups, e.g. in the above matrix there are two groups, $x_1, x_2, x_3$ and $x_4, x_5$. So it helps t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Second difference $ \to 0$ everywhere $ \implies f $ linear Exercise 20-27 in Spivak's Calculus, 4th ed., asks us to show that if $f$ is a continuous function on $[a,b]$ that has
$$
\lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=0\,\,\,\text{for all }x,
$$
then $f$ is linear. We do this by first treating the case $f(a)... | counterexample :set $f(x)=-1$ for $0\leq x<1$ , $f(x)=1$ for $1<x\leq 2$ and $f(1)=0$. Clearly that limit is zero on for $x\not=1$ and $\frac{f(1+h)+f(1-h)-2f(1)}{h^2}=0. $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/933188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that $GL_n(F)$ is non-abelian for $n \geq 2$ and for any field $F$ I'm trying to show that $GL_n(F)$ is non-abelian for any field $F$ and $n \geq 2$. I'm doing so by constructing two $2 \times 2$ matrices that do not commute and "extending" them to $n \times n$ matrices with zeros in every other entry. We defin... | As explained in the comments, the problem is one of extending the two matrices in such a way that 1) the same non-commutativity can be easily verified and 2) the matrices are non-singular.
A standard recipe is to extend by ones along the diagonal and zeros elsewhere.
Why is this "standard"? The way I think about it is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
What is the universal cover of a discrete set? Just curious, what is the universal covering space of a discrete set of points? (Finite or infinite, I'd be happy to hear either/or.)
If there is just a single point, I think it is its own universal covering space, since it is trivial simply connected. At two points or mor... | I don't think there's a widely accepted definition of what the universal cover of a disconnected space is; the standard definition, as the maximal connected cover, only applies to (sufficiently nice) connected spaces, since a disconnected space has no connected covers.
One candidate is "the disjoint union of the unive... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Is this set neccesarily to be a vector space? Suppose $F$ be a field and $S$ be a non empty set such that
1) $a+b \in S $
2) $ \alpha a \in S$
for all $a,b \in S$ and $ \alpha \in F.$
Is always $S$ to be a Vector space?
| No, for we could take the multiplication map $\cdot: F \times S$ to be zero map---the resulting structure fails to be a vector space only because it does not have a multiplicative identity, i.e. an element $1 \in F$ such that $1 \cdot s = s$ for all $s \in S$.
Ferra's nice example shows that preventing this degenerate ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inequality $\int^{1}_{0}(u(x))^2\,\mathrm{d}x \leq \frac{1}{6}\int^{1}_{0} (u'(x))^2\,\mathrm{d}x+\left(\int_{0}^{1} u(x)\,\mathrm{d}x \right)^2$ I've been scratching my brain on this one for about a week now, and still don't really have a clue how to approach it.
Show that for $u \in C^1[0, 1]$ the following inequalit... | The constant $\frac{1}{6}$ can be slightly improved without using the full strength of Wirtinger Inequality.
$$\begin{align} \int_0^1 u^2(t)\,dt - \left(\int_0^1 u(t)\,dt\right)^2 & = \frac{1}{2}\int_0^1\int_0^1 \left(u(x) - u(y)\right)^2\,dx\,dy \\ &= \int_0^1 \int_x^1 \left(\int_x^y u'(t)\,dt\right)^2 \,dy\,dx \\ & \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
A special case: determinant of a $n\times n$ matrix I would like to solve for the determinant of a $n\times n$ matrix $V$ defined as:
$$
V_{i,j}=
\begin{cases}
v_{i}+v_{j} & \text{if} & i \neq j \\[2mm]
(2-\beta_{i}) v_{i} & \text{if} & i = j \\
\end{cases}
$$
Here, $v_i, v_j \in(0,1)$, and $\beta_i>0$. Any suggestion... | Let me consider $W:=-V$; we have
$$
\det(V)=(-1)^n\det(W).
$$
As already noted in the comments, $W=D-(ve^T+ev^T)$, where $v:=[v_1,\ldots,v_n]^T$, $e:=[1,\ldots,1]^T$, and $D:=\mathrm{diag}(\beta_i v_i)_{i=1}^n$.
Since $D>0$, we can take
$$
D^{-1/2}WD^{-1/2}=I-(\tilde{v}\tilde{e}^T+\tilde{e}\tilde{v}^T),
$$
where $\ti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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Define $A = \{ 1,2,A \}$, $A$ can not be a set (Axiom of regularity). Can $A$ be a "class" or a "collection" of elements. This question is probably very naive but it does bother me and I am not sure where to look for an answer.
Define $A = \{ 1,2,A \}$, $A$ can not be a set (Axiom of regularity). Can $A$ be a "class" o... | A class in standard (ZF) set theory generally coincides with a property that any set may or may not have. For instance, the property might be "$\phi(x) \equiv x \not\in x$". Then $\{x \;:\; x\not\in x\}$, while not necessarily a set, is a well-defined class. But note that the elements of a class are sets, not classe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Gosper's Identity $\sum_{k=0}^n{n+k\choose k}[x^{n+1}(1-x)^k+(1-x)^{n+1}x^k]=1 $ The page on Binomial Sums in Wolfram Mathworld http://mathworld.wolfram.com/BinomialSums.html (Equation 69) gives this neat-looking identity due to Gosper (1972):
$$\sum_{k=0}^n{n+k\choose k}[x^{n+1}(1-x)^k+(1-x)^{n+1}x^k]=1 $$
Would anyon... | Let $m>0$ a natural number and let's consider a set of polynomial functions:
$$A_{0}\left(x\right)=1$$
$$A_{n}\left(x\right)=\sum_{k=0}^{n}\binom{n+m}{k}x^{k}\left(1-x\right)^{n-k},\,\,\,\,n=1,\,2,\,3,\,\ldots$$
We have
$$A_{l}\left(x\right)=\sum_{k=0}^{l}\binom{l-1+m}{k}x^{k}\left(1-x\right)^{l-k}+\sum_{k=1}^{l}\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Can someone help me to find a counter example that shows that $a \equiv b \mod m$ does not imply $(a+b)^m \equiv a^m +b^m \mod m$ Can someone help me to find a counter example that shows that $a \equiv b \mod m$ does not imply $(a+b)^m \equiv a^m +b^m \mod m$. I have tried many different values but I can't seem to find... | Take $a=1$, $b=2$ and $m=6$.
$(a+b)^6 = a^6 + 6 a^5 b + 15 a^4 b^2 + 20 a^3 b^3 + 15 a^2 b^4 + 6 a b^5 + b^6$
$(a+b)^6 \equiv a^6 - a^3 b^3 + b^6 \bmod 3$
If $(a+b)^6 \equiv a^6 + b^6 \bmod 6$, then $3$ must divide $a$ or $b$.
It is enough to take $a=1$ and $b=2$ to have $(a+b)^6 \not\equiv a^6 + b^6 \bmod 6$.
In ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/933915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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A question involving the probability generating function I'm stumped by the following exercise: The probability generating function for X is given by $g_{X}(t)$. What is the probability generating function for $X+1$ and $2X$, in terms of $g_{X}(t)$?
For the first part I think I've got an answer down although I can't sa... | WLOG, consider that $X$ is a discrete random variable, which assumes values in the set $\mathcal{X}$. Using the definitions....
$$ g_{2X}(t) = \mathbb{E}[t^{2X}] = \sum_{x \in \mathcal{X}} p(x)t^{2x} = \sum_{x \in \mathcal{X}} p(x)(t^{2})^x = g_X(t^2).$$
In general:
$$ g_{aX+b}(t) = \mathbb{E}[t^{aX+b}] = \sum_{x \in \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Maximal commutative subring of the ring of $2 \times 2$ matrices over the reals Motivated by complex numbers, I noticed that the set of all elements of the following forms a commutative sub-ring of $M_2(\mathbb{R})$:
\begin{pmatrix}
x & y\\
-y & x
\end{pmatrix}
Is this sub-ring maximal w.r.t commutativity? If this sub-... | Maximal dimension of a commutative subalgebra of $n$ by $n$ matrices is
$$ 1 + \left\lfloor \frac{n^2}{4} \right\rfloor. $$
In even dimension $2m,$ this is realized by
$$
\left(
\begin{array}{rr}
\alpha I & A \\
0 & \alpha I
\end{array}
\right),
$$
where $A$ is any $m$ by $m$ matrix and $I$ is the $m$ by $m$ iden... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Compositions of filters on finite unions of Cartesian products Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set $\Gamma$: http://www.mathematics21.org/binaries/funcoids-... | Let $f$, $g$ be filters on $Γ$.
"define $g∘f$ as the filter on $Γ$ defined by the base $\{G∘F∣F∈f,G∈g\}$."
So as claimed, $g∘f$ is the filter generated by (=smallest filter containing) the base $\{G∘F∣F∈f,G∈g\}$.
This means that
$\{G∘F∣F∈f,G∈g\}$ is a base for the filter $g\circ f$.
Which, by definition of a base ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Can a game with negative expectation still have a positive utility? Intuitively, I think not. But I can't clearly prove why. Specifically, I've been thinking about lottery games, where the expectation is obviously negative. But can the utility of hitting the jackpot ever be enough to justify playing?
| A common assumption about a utility function is that it is concave down: $U(tx + (1-t)y) \geq tU(x) + (1-t)U(y)$ for any $x,y$ and $0 \leq t \leq 1$. One can prove that any concave down function also satisfies the condition that, if $p_i$, $1 \leq i \leq n$ are nonnegative numbers with $\sum_i p_i = 1$, then for any $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Why can't a set have two elements of the same value? Suppose I have two sets, $A$ and $B$:
$$A = \{1, 2, 3, 4, 5\} \\ B = \{1, 1, 2, 3, 4\}$$
Set $A$ is valid, but set $B$ isn't because not all of its elements are unique. My question is, why can't sets contain duplicate elements?
| Short Answer:
The axioms of the Set Theory do not allow us to distinguish between two sets like $A = \{1,2\}$ and $B = \{1,1,2\}$. And every sentence valid about sets should be derived in some way from the axioms.
Explained:
The logic of the set theory is extensional, that means that doesn't matter the nature of a set,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "55",
"answer_count": 9,
"answer_id": 3
} |
the shortest path between two points and the unit sphere and the arc of the great circle Prove that the shortest path between two points on the unit sphere is an arc of a great circle connecting them
Great Circle: the equator or any circle obtained from the equator by rotating
further: latitude lines are not the great ... | HINT: (Edited 8/27/2021) Start with two points on the equator. Every great circle (except one) meets the shorter great circle arc joining them in at most one point. Let $\Sigma$ be the set of great circles meeting it in one point. Show that for any other curve $C$ joining the points, there must be an open set containin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Does $\vec{a} \times \vec b=\vec a \times\vec c$ and $\vec{a} \cdot \vec b=\vec a \cdot\vec c$ imply that $\vec b = \vec c$? Does $\vec{a} \times \vec b=\vec a \times\vec c$ and $\vec{a} \cdot \vec b= \vec a \cdot\vec c$ imply that $\vec b = \vec c$ if $\vec a \not=0$ ?
My attempt-
$\vec{a} \times {(\vec b- \vec c)}... | Yes you are right. Both the vector equations to be true vector b and c must be the same.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/934604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid(5x+3y)\}$ is an equivalence relation.
Let $R$ be a relation on $\mathbb{Z}$ defined by $$ R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid (5x+3y)\}.$$ Show that $R$ is an equivalence relation.
I'm having a bit of trouble with this exercise in... | Reflexive: $xRx = 5x + 3x = 8x$ which is clearly divisible by $4$ if $x\in\mathbb Z.$
Symmetry: Assume $xRy$, that implies that $5x + 3y$ is divisible by $4$.
Then $yRx = 5y + 3x = (9y - 4y) + (15x - 12x) = 9y + 15x - 4y - 12x $$= (3y + 5x)- 4(y + 3x).$ Since $3y + 5x$ is divisible by $4$ and $4(y + 3x)$ is divisible... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Why this gamma function reduces to the factorial? $$\Gamma(m+1) = \frac{1\cdot2^m}{1+m}\frac{2^{1-m}\cdot3^m}{2+m}\frac{3^{1-m}\cdot4^m}{3+m}\frac{4^{1-m}\cdot5^m}{4+m}\cdots$$
My books says that in a letter from Euler to Goldbach, this expression reduces to $m!$ when $m$ is a positive integer, but that Euler verified ... | This involves the fact that $\Gamma(1)=1$ and $\Gamma(n+1)=n\Gamma(n)$ based on the definition of Gamma, $\Gamma(t)=\int_0^\infty x^{t-1}e^{-x}dx$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/934767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
differentiable as an R2 function v.s. as a complex function Let $f: \mathbb C \to \mathbb C$ be a complex function. Denote its real and imaginary part by $u: \mathbb C \to \mathbb R$ and $v: \mathbb C \to \mathbb R$ respectively.
Consider the function $\widetilde f : \mathbb R^2 \to \mathbb R^2$ defined by $\widetilde ... | You can sort this thing one point in the domain at a time.
Fact:
$f\colon U \to \mathbb{C}$ is complex differentiable at $z$
if and only if
$f$ is differentiable at $z$ ( as a function from $U$ to $\mathbb{R}^2$ ) and
the partial derivatives at $z$ ( which exist since $f$ is differentiable at $z$) satisfy the Cauc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/934845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How to use Newton's method to find the roots of an oscillating polynomial?
Use Newton’s method to find the roots of $32x^6 − 48x^4 + 18x^2 − 1 = 0$ accurate to within $10^{-5}$.
Newton's method requires the derivative of this function, which is easy to find. Problem is, there are several roots near each other:
Thes... | Here you go:
alpha = 0.05;
NM = @(x,f)x-alpha*prod(([1,0]*f([x,1;0,x])).^[1,-1]);
f = @(x)32*x^6-48*x^4+18*x^2-x^0;
x = 0.5;
eps = 1;
while eps > 1e-5
xn = NM(x,f);
eps = abs(xn-x);
x = xn;
end
[x f(x)]
I caution you to not turn in this code to your professor. He will probably ask you how it works.
I provi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/935144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Proofs about Matrix Rank I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the properties of matrices.
First, I want to prove that similar matrices have the same rank. Thi... | If $n$ vectors $v_1,..v_n$ are linearly independent(dependent) then for non singular matrix $P$ the vectors $Pv_1,...Pv_n$ also will be independent(dependent). From this fact and from the definition of the rank as a number of linearly independent columns (rows) we immediately can conclude that similar matrix have the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/935253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$ Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that
${\bf u} {\bf v}$ is an $n \times n$ matrix such that $\det({\bf uv})=... | This question can be easily answered by recalling determinant properties. Specifically:
$det(A_1,A_2,\dots,cA_j,\dots,A_n)$ = $c$$(det(A_1,A_2,\dots,A_j,\dots,A_n))$
Therefore if $u = \begin{bmatrix}u_1\\ u_2\\ \vdots \\ u_n \end{bmatrix}$ and $v = \begin{bmatrix} v_1,& v_2,& \dots,& v_n\end{bmatrix}$
Then:
$uv$ = $\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/935319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 9,
"answer_id": 4
} |
Series expansions and perturbation My professor said that $ f \left( y_1(x)+ \epsilon y_2(x)+... \right)= f(y_1(x)) +f'(y_1(x))\> (\epsilon y_2(x)+...) + ...$ but I have no idea how the series continues. Has anyone seen this before? Can you please tell me how the series progresses?
| Taylor expansion:
$$+\frac{1}{2!}f''(y_1(x)) (\epsilon y_2(x)+\ldots)^2+\frac{1}{3!}f'''(y_1(x)) (\epsilon y_2(x)+\ldots)^3+\ldots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/935364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Kernel of an integral operator with Gaussian kernel function Suppose we have the integral operator $T$ defined by
$$Tf(y) = \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}f(xy)\,dx,$$
where $f$ is assumed to be continuous and of polynomial growth at most (just to guarantee the integral is well-defined). If we are to inspect... | If you are able to prove that, given your constraints, $Tf$ is an analytic function, we have:
$$(Tf)^{(2n+1)}(0) = 0,\qquad (Tf)^{(2n)}(0) = 2^{n+1/2}\cdot\Gamma(n+1/2)\cdot f^{(2n)}(0)$$
hence $Tf\equiv 0$ implies that all the derivatives of $f$ of even order in the origin must vanish. Assuming that $f$ can be well-ap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/935454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Evaluating an indefinite integral $\int\sqrt {x^2 + a^2} dx$ indefinite integral $$\int\sqrt {x^2 + a^2} dx$$
After some transformations and different substitution, I got stuck at this
$$a^2\ln|x+(x^2+a^2)| + \int\sec\theta\tan^2\theta d\theta$$
I am not sure I am getting the first step correct. Tried substituting $ x=... | Here we have another way to see this:
$$
\int \sqrt{x^2+a^2} dx
$$
using the substitution
$$
t=x+\sqrt{x^2+a^2}\\
\sqrt{x^2+a^2}=t-x
$$
and squaring we have
$$
a^2 =t^2-2tx\\
x=\frac{t^2-a^2}{2t}.
$$
Finally we can use:
$$
dx=\frac{2t(t)-(t^2-a^2)(1)}{2t^2}dt = \frac{t^2+a^2}{2t^2}dt\\
\sqrt{x^2+a^2}=t-\frac{t^2-a^2}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/935519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
Evaluating modulos with large powers I need some help evaluating: $$13^{200} (mod \ 6)$$
What I've been trying to do:
$$13^1 \equiv 1 (mod \ 6)$$
$$13^2 \equiv 1 (mod \ 6)$$
Can I just say that: $$13^{200} = 13^2 * 13^2 * ... * 13^2 \equiv 1^{200}$$
Or is this incorrect?
Thanks in advance.
| Since $gcd(13,6)=1$ we apply Euler's Theorem:
$$13^{\phi{(6)}} \equiv 1 \pmod 6 \Rightarrow 13^2 \equiv 1 \pmod 6$$
$$13^{200} \equiv (13^2)^{100} \equiv 1^{100} \equiv 1 \pmod 6$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/935649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Find the average acceleration Find the average acceleration of the tip of the 2.4-cm long hour hand in the interval noon to 6pm.
I found the average velocity is -2.2x10^6 but I'm not sure how to go about finding acceleration. If someone has a few minutes can we chat and you can explain the set up to me?
| Average velocity is displacement over time: in this case, $4.8$ cm in $6$ hours.
That won't help much with computing average acceleration.
Average acceleration is change in velocity over time. The speed of the tip of the hand doesn't
change, but between noon and $6$ pm its direction reverses. So the change in velocity ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/935753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there any book/resource which explain the general idea of the proof of Fermat's last theorem? I look for a book/resource which display the general idea of the proof of Fermat last theorem in a simple manner for the public.
I mean, books which is not for mathematicians but for the general public. Books like:
Gödel's... | Perhaps the closest thing is this article:
"A marvelous proof", by Fernando Gouvêa,
The American Mathematical Monthly, 101 (3), March 1994, pp. 203–222.
This article got the MAA Lester R. Ford Award in 1995.
This and other papers (with various degrees of difficulty) can be found at Bluff your way in Fermat's Last Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/935833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 7,
"answer_id": 1
} |
Probability: Distinguishable vs Indistinguishable So there are 5 red balls and 4 blue balls in an urn. We select two at random by putting them in a line and selecting the two leftmost. What's the probability both are different colors if the balls are numbered, and if the balls of the same color are indistinguishable? A... | Distinguishability is not a property of the balls. It is the property of the observer. If there are balls of different colors arranged in boxes then certain arrangements will be distinguishable for those with a good eyesight and some arrangements will not be distinguishable for the colorblind.
In probability theory ob... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What to do *rigorously* when the second derivative test is inconclusive? How do you rigorously check if a point is a local minimum when the second derivative test is inconclusive?
Does there exist a way to do this in general for arbitrary smooth (or analytic...) functions?
I know I can graph the function, plug in a few... | This answer is for functions of one variable. First, a comment. The second derivative test is often not the best approach. For one thing, it involves computing and evaluating the second derivative. That involves work, and carries a non-zero probability of error.
In many of the usual calculus problems, the right approa... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question of trigonometry If $\cos^2 A=\dfrac{a^2-1}{3}$ and $\tan^2\left(\dfrac{A}{2}\right)=\tan^{2/3} B$. Then find $\cos^{2/3}B+\sin^{2/3}B $.
I tried componendo and dividendo to write the second statement as cos A but i couldnt simplify it
| $$\cos^{2/3}B+\sin^{2/3}B=\cos^{2/3}B\left(1+\tan^{2/3}B\right)=\cos^{2/3}B\left(1+\tan^2\frac{A}{2}\right)=\left(\cos^2B\right)^{1/3}\left(1+\tan^2\frac{A}{2}\right)=\left(\frac{1}{1+\tan^6\left(\frac{A}{2}\right)}\right)^{1/3}\left(1+\tan^2\frac{A}{2}\right)=\left(1+\frac{3\tan^2 \left(\frac{A}{2}\right)\left(1+\tan^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/936253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is functions of cauchy sequences is also Cauchy? Recently i saw in some book that if a sequence is Cauchy then function of that sequence is also Cauchy.I have confusion about this. Please help me.
| Let $\{a_n\}$ be a Cauchy sequence in R. $ \,\,\, $ Then $\{a_n\}$ has a limit $a$.
If $f:R→R$ is continuous on $x=a$, then the sequence $\,\,\{f(a_n)\}\,\,$ converges to $\,\,f(a)$, thus it is Cauchy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/936321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Can anyone tell me how $\frac{\pi}{\sqrt 2} = \frac{\pi + i\pi}{2\sqrt i}$ I was working out a problem last night and got the result
$\frac{\pi + i\pi}{2\sqrt i}$
However, WolframAlpha gave the result
$\frac{\pi}{\sqrt 2}$
Upon closer inspection I found out that
$\frac{\pi}{\sqrt 2} = \frac{\pi + i\pi}{2\sqrt i}$
But I... | Rewriting everything in polar form, the numerator is $\sqrt{2} \pi e^{i \pi/4}$ and the denominator is, for the choice of branch of $\sqrt{}$ that Wolfram is using, $2 e^{i \pi/4}$. So the $e^{i \pi/4}$ terms cancel and you're left with a real number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/936415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer. How do we prove that $\dfrac{\phi^{400}+1}{\phi^{200}}$ is an integer, where $\phi$ is the golden ratio?
This appeared in an answer to a question I asked previously, but I do not see how to prove this..
| More generally,
$\dfrac{\phi^{2n}+1}{\phi^{n}}$ is an integer for $n$ even.
Indeed, let $n=2m$ and $\alpha=\phi^2$. Then
$$
\dfrac{\phi^{2n}+1}{\phi^{n}}
= \phi^{n}+\dfrac{1}{\phi^{n}}
= \alpha^{m}+\dfrac{1}{\alpha^{m}}
=: y_m
$$
Since $\alpha$ and $\dfrac{1}{\alpha}$ are the roots of $x^2=3x-1$, we have
$ y_{k+2} = 3y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/936479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 4
} |
Use of $\mapsto$ and $\to$ I'm confused as to when one uses $\mapsto$ and when one uses $\to$. From what I understand, we use $\to$ when dealing with sets and $\mapsto$ when dealing with elements but I'm not entirely sure.
For example which of the two is used for the following? $$\begin{pmatrix} x \\ y \\ z \end{pmatr... | The difference is that $\mapsto$ denotes the function itself. Thus you need not name the function. $a\mapsto b$ fully describes the action of the function. $\to$, on the other hand, describes only the domain and codomain. Thus one might say $x\mapsto x+1$ is equivalent to $f(x)=x+1$, in which case we would say $f:\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/936558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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The Number of Binary Vectors Whose Sum Is Greater Than $k$ I want to determine the number of vectors $(x_1,\ldots,x_n)$, such that each $x_i$ is either $0$ or $1$ and $$\sum\limits_{i=1}^n x_i \geq k$$
*
*My Approach
The number of $1$'s range from a minimum of $k$ to a maximum of $n$. Thus I must count all the ways... | If an approximation is good enough for your needs, you can always use the central limit theorem. Just think of each $x_i$ as a random variable, whose mean and standard deviation would each be $\frac12$. Then, by the central limit theorem, your sum $\sum_{i=1}^nx_i$ approximately has a normal distribution with mean $\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/936677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Probability of impossible event. There is question in my book: Probability of impossible event is?
After reading the question my instant answer was $0$ and that was the answer given.
But then i thought other way, question is probability of impossible event, so there are two outcomes possible or impossible (event can be... | Your first answer is right. Your second argument only works to compute the probabilities of equally likely outcomes. So it's perfectly good to figure out the chances of heads coming up from a coin flip. But if one outcome is specified to be impossible, then by definition you are not working with equally likely outcomes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/936763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does the first player never lose this numbers game? There is an even number of numbers in a row. Two players cross out numbers one by one from left or right. It is not allowed to cross out a number in the middle. Only left or right. After all numbers are crossed out they find the sum of numbers they crossed. The winner... | Here is a simple strategy that guarantees a tie for the first player:
Colour the numbers alternately blue and yellow. Let $B$ be the sum of the blue numbers, and $Y$ the sum of the yellow numbers.
If $B \ge Y$: Always pick the blue number. (When it is your turn to play, the two numbers at the end will always be coloure... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/936846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Associative law is not self evident The statement:
"It is important to understand that the associative law is not
self-evident; indeed, if $a*b=a/b$ for positive numbers $a$ and $b$
then, in general, $(a*b)*c\ne a*(b*c)$." - p. 3, A. Beardon, Algebra
and Geometry.
I am unclear as to how to take the assumption $a*b=a/b$... | You are basically right in your guesses. What is being emphasized is that not all binary operations (i.e. things that take two inputs from your set and return a third element of the set) that one can define are necessarily associative just because they are binary operations. In this example they are saying that if you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/936954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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How do I solve $\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$? I'm having trouble finding this limit:
$$\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$$
I tried multiplying by the conjugate:
$$\lim_{x\to -\infty}(\frac{\sqrt{x^2 + x + 1} + x}{1} \times \frac{\sqrt{x^2 + x + 1} - x}{\sqrt{x^2 + x + 1} - x}) = \lim_{x\to -\i... | Little mistake:
$$
\lim_{x\to -\infty}\frac{x + 1}{\sqrt{x^2 + x + 1} - x} \times \frac{\frac{1}{x}}{\frac{1}{x}} = \lim_{x\to -\infty}\frac{1 + \frac{1}{x}}{{\color{red}-}\sqrt{1 + \frac{1}{x} + \frac{1}{x^2}} - 1}=-\frac{1}{2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/937182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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derivate of a piecewise function $f(x)$ at$ x=0$. There is a piecewise function $f(x)$
$$f(x)= \begin{cases} 1 ,\ \ \text{if}\ \ x \geq \ 0 \\ 0,\ \ \text{if}\ \ x<0 \end{cases}$$
what is the derivative of the $f(x)$ at $x=0$?
Is it $0$? Or since it is not continuous, the derivative does not exists?
| Hint: Prove that if a given function $f$ is differentiable at a point $a$, then $f$ is continuous at $a$.
Sketch: If $f$ is differentiable at $a$ then we may write (Theorem) $f(a+h) = f(a) + hf'(a) + \frac{r(h)}{h}h$ where $lim_{h \to 0 } \frac{r(h)}{h} = 0 $, then $lim_{h\to 0} f(a+h) = f(a)$, that is, $f$ is continuo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/937299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Closed-form of integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $ I'm looking for a closed form of this definite iterated integral.
$$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy. $$
From Vladimir Reshe... | I think Math-fun's second approach based on changing the order of integration is a good strategy. Appropriate use of substitutions and trig identities along the way clean up a lot of the resulting "mess":
$$\begin{align}
\mathcal{I}
&=\int_{0}^{1}\mathrm{d}y\int_{0}^{1}\mathrm{d}s\,\frac{\arcsin{\left(\sqrt{1-s}\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/937487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
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I want to know whether the following is periodic or not periodic I have a question about system properties of the following function whether it is periodic or aperiodic.
With an insight, I'd determine the function is aperiodic since the unit-step term looks implying that jump discontinuities occur at odd number times ... | The function is periodic with period $1/2$ since you have
$$
f(t+1/2)=\sum_{n=-\infty}^\infty e^{-(2t-n+1)}u(2t-n+1)=f(t),
$$
as you can easily see by an index shift in the sum.
Hope this helps...
| {
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"url": "https://math.stackexchange.com/questions/937580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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What does $R[[X]]$ and $R(X)$ stands for? I'm reviewing Linear Algebra these days and I saw these two notations in my notes without definition.
Those are, $R[[X]]$ and $R(X)$ where $R$ is a commutative ring with unity.
I remember that one of these denote the field of polynomials, but I don't know which one does..
Moreo... | Typically:
*
*$R[x]$ denotes the set of polynomials over $R$
*When $R$ is a domain, $R(x)$ denotes the set of rational polynomials over $R$
*$R[[x]]$ denotes the formal power series over $R$
*$R((x))$ denotes the Laurent series over $R$
vuur asked an interesting question in the comments which I can speak to he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/937693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that any derivative of a given function is bounded Let the function $f\left( x \right) = \left( {\frac{{1 - \cos x}}
{{{x^2}}}} \right)\cos (3x)$ if $x\ne 0$ and $f(0)=\frac{1}{2}$. Prove that any derivative of $f$ is bounded on $\mathbb{R}$. Thank so much for helping.
| The original function is an entire function hence its derivatives are bounded on $[-1,1]$. Moreover, any derivative of $\frac{1-\cos x}{x^2}$ is bounded on $\mathbb{R}\setminus[-1,1]$ since it is a linear combination of functions of the form $\frac{f(x)}{x^k}$ where $f(x)$ is bounded trigonometric function and $k\geq 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/937804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Solving Diophantine equations involving $x, y, x^2, y^2$ My father-in-law, who is 90 years old and emigrated from Russia, likes to
challenge me with logic and math puzzles. He gave me this one:
Find integers $x$ and $y$ that satisfy both $(1)$ and $(2)$
$$x + y^2 = 8 \tag{1} $$
$$x^2 + y = 18 \tag{2}$$
I found one sol... | You can do without the theory of Diophantine equations.
$$x^2+y=18\implies y=18-x^2.$$
Plugged in the other equation,
$$x+(18-x^2)^2=8$$or
$$x^4-36x^2+x-316=0.$$
With a polynomial solver, you get two positive roots $x=4$ and $x\approx4.458899113761$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/937891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
How find this integral $\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$ let $$D=\{(x,y)|y\ge x^3,y\le 1,x\ge -1\}$$
Find the integral
$$I=\dfrac{1}{2}\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$$
My idea:
$$I=\int_{0}^{1}dx\int_{x^3}^{1}(x^2y+2y^2)dy+\int_{-1}^{0}dx\int_{0}^{-x^3}(xy^2+2x+2y^2)dy$$
so
$$I=\int_{0}^{1}[\dfrac{1}{2}x^2y^2+\dfrac... | $\frac{67}{90}$ doesn't look correct. Here is what wolfram computes
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/937955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Cycles of odd length: $\alpha^2=\beta^2 \implies \alpha=\beta$
Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2=\beta^2$, then $\alpha=\beta$.
I need advice on how to approach this. I recognized that $\alpha,\beta$ are even permutations (because they have odd length). I'm not s... | Suppose that $\alpha$ and $\beta$ both have length $2k + 1$ for some $k \in \mathbb N$. Then observe that:
\begin{align*}
\alpha
&= \alpha\varepsilon \\
&= \alpha^1\alpha^{2k+1} \\
&= \alpha^{2k+2} \\
&= (\alpha^2)^{k+1} \\
&= (\beta^2)^{k+1} \\
&= \beta^{2k+2} \\
&= \beta^1\beta^{2k+1} \\
&= \beta\varepsilon \\
&= \be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/938142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
On multiplicity representations of integer partitions of fixed length This is a follow-up question on the question computing length of integer partitions and it is loosely related with the paper "On a partition identity".
Let $\lambda$ be a partition of $n$, in the multiplicity representation $\lambda=(a_1,a_2,a_3,\dot... | Generating functions to the rescue! We have a sum over all partitions $\lambda$ in the world:
\begin{align*}
\sum_{\lambda} \frac{x^{n(\lambda)} y^{S(\lambda)}}{G[\lambda]} &= \sum_{a_1,a_2,\dots\ge0} x^{\sum_{j\ge0} ja_j} y^{\sum_{j\ge0} a_j} \prod_{j\ge0} \frac1{a_j!} \\
&= \sum_{a_1,a_2,\dots\ge0} \prod_{j\ge0} \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/938280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
optimization of coefficients with constant sum of inverses Does anybody knows if there is an easy solution to the following problem:
Given $A = [a_1, a_2, ... a_n]$ and K, find B = $[b_1, b_2,...b_n]$ that minimizes $AB^T$ such that $\sum_{i}^{n}\frac{1}{b_i} = K$?
| As copper.hat suggests: Lagrange multipliers!
We want to minimize $f(B) = AB^T$ subject to the constraint $g(B) = \sum_{i}^{n}\frac{1}{b_i} = K$. We can find all critical points by solving the system
$$
\nabla f = -\lambda \nabla g\\
g(B) = K
$$
Where
$$
\nabla f = A = [a_1,\dots,a_n]\\
\nabla g = -\left[\frac{1}{b_1^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/938371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Proving that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$ Edit: got it, silly mistakes :)
I need to prove that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$
$$=\frac{\tan x+\tan y-\tan x+\tan^2x\tan y}{1-\tan x\tan y+\tan^2x+\tan x\tan y}$$
$$=\frac{\tan y+\tan^2x\tan y}{1+\tan^2x}$$
$$=\frac{(\tan y)(1+... | Hint: Recall that
$$\tan(u-v)=\frac{\tan u-\tan v}{1+\tan u\tan v}.\tag{1}$$ Let $u=x+y$ and $v=x$.
Remark: If Identity (1) requires proof, use
$$\tan(u-v)=\frac{\sin(u-v)}{\cos(u-v)}=\frac{\sin u\cos v-\cos u\sin v}{\cos u\cos v+\sin u\sin v},$$
and divide top and bottom by $\cos u\cos v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/938432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Characterization of a vector space over an associative division ring Let $M$ be a (left) module over an associative division ring $R$.
Then it has the following properties.
1) For every submodule $N$ of $M$, there exists a submodule $L$ such that $M = N + L$
and $M \cap L = 0$.
2) Every finitely generated submodule has... | Let $R=\mathbb{Z}$ so that $R$-modules are abelian groups and submodules are subgroups. Let $M$ be a cyclic group of prime order. Then $M$ is simple, so 1) is satisfied, and every finite group has a composition series, so 2) is satisfied. However, $R$ is not a division ring.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/938520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Differentiation of $xx^T$ where $x$ is a vector How is differentiation of $xx^T$ with respect to $x$ as $2x^T$, where $x$ is a vector? $x^T $means transpose of $x$ vector.
| Differentiation of $xx^T$ is a 3 dimensional tensor to the best of my knowledge,with each element in the matrix $xx^T$ to be differentiated by each element of the vector $x$,and you can just see the result of a derivative online at site http://www.matrixcalculus.org/.Any further communication is preferred.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/938663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Every non-unit is in some maximal ideal I am trying to prove that every non-unit of a ring is contained in some maximal ideal. I have reasoned as follows: let $a$ be a non-unit and $M$ a maximal ideal. If $a$ is not contained in any maximal ideal, then the ideal $\langle M, a \rangle$ (that is, the ideal generated by $... | Yes, it can happen that $\langle M,a\rangle$ is the whole ring, so your proof doesn't work. Try to reason as follows: let $\Sigma$ be the collection of ideals which contain $a$. Now use Zorn's lemma.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/938777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 1
} |
Kernel of a Linear Map on A Tensor Product Suppose I have the linear maps $ l,k: V \otimes V \rightarrow V \otimes V$ defined by
$ l( e_{i_1} \otimes e_{i_2} ) = e_{i_1} \otimes e_{i_2} + e_{i_2} \otimes e_{i_1}$
and
$ k( e_{i_1} \otimes e_{i_2} ) = e_{i_1} \otimes e_{i_2} - e_{i_2} \otimes e_{i_1}$
where $ (e_{i})_{i... | Here's one way to think about it: If $V$ (over, say, $\mathbb{F}$) is finite-dimensional, say, $\dim V = n$, then given a basis $(e_a)$ of $V$, we may identify $V \otimes V$ with the set of $n \times n$ matrices, so that the simple tensor $e_a \otimes e_b$ is identified with the matrix $E_{ab}$ whose $(a, b)$ entry is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/938882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is this calculus proof I came up with sound? We want to prove there every bounded sequence has a converging subsequence. Let $[l,u]$ be the interval to which we know $a_n$ is bounded.Let $\{a_n\}$ be the sequence and $[l_i,u_i]$ where $i$ is a positivie integer be a sequence of closed intervals such that each of the i... | The proof works except for the last line, where there is some mess with the indexes $n,N,N'$. Moreover, why are you taking into account the midpoint of $[l_n,u_n]$? If I'm guessing correctly, you would like to prove that the sequence $(b_n)_n$ converges to $[l_N,b_N]$, but this is false (indeed, your $N$ depends on $\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/938974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Algebra - proof verification involving permutation matrices
Theorem. Let $\textbf{P}$ be a permutation matrix corresponding to the permutation $\rho:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$. Then $\textbf{P}^t=\textbf{P}^{-1}.$
Proof. First note the following identity for the kronecker delta:
$$\sum_k \delta_{ik}\delta_{kj... | Let $e_1,\dots,e_n$ be a fixed orthonormal basis of a real $n$ dimensional inner product space $V$.
Then $P$ can be regarded as the linear transformation $V\to V$ mapping $e_i\mapsto e_{\rho(i)}$.
Since $\rho$ is a permutation, we will have
$$\langle Pe_i,Pe_j\rangle\ =\ \delta_{i,j}\ =\ \langle e_i,e_j\rangle$$
so, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/939164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Linear transformation in linear algebra Let $e_1= \begin{bmatrix}
1\\
0
\end{bmatrix}
$
Let $e_2= \begin{bmatrix}
0\\
1
\end{bmatrix}
$
Let
$y_1= \begin{bmatrix}
2\\
5
\end{bmatrix}
$
$y_2= \begin{bmatrix}
-1\\
6
\end{bmatrix}
$
Let $\mathbb{R^2}\rightarrow\mathbb{R^2}$ be a linear transformation that maps e1 into y1 ... | If $L$ is a linear transformation that maps $\begin{bmatrix}
1\\
0
\end{bmatrix}
$
to $\begin{bmatrix}
2\\
5
\end{bmatrix}
$, $L$ has a matrix representation $A$, such that $A \begin{bmatrix}
1\\
0
\end{bmatrix}
=\begin{bmatrix}
2\\
5
\end{bmatrix}$. But this means that $\vec{a_1}^{\,}$ is just $\begin{bmatrix}
2\\
5
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/939372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is is true that if $E|X_n - X| \to 0$ then $E[X_n] \to E[X] $? My question is motivated by the following problem:
Show that if $|X_n - X| \le Y_n$ and $E[Y_n] \to 0$ then $E[X_n] \to E[X]$.
I started off by saying that since
$$|X_n - X|\ge 0 $$
then $$E[|X_n - X|]\ge 0 $$ At the same time
$$E[|X_n - X|]\le E[Y_n] $$
a... | $\vert E(X) - E(X_n) \vert =\vert E(X-X_n) \vert \leq E(|X-X_n|) \to 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/939562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Examples of books giving problems that require more than one branch of mathematics I want to know if there are books that give problem sets requiring knowledge of two or more branches of mathematics. For example, there could be a problem requiring geometry, set theory, and number theory, and another problem requiring r... | Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by John and Barbara Hubbard offer many problems that tie together analysis and linear algebra. The problem sets however do not require as diverse of a knowledge as competition level problems (as stated by Alfred).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/939659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Closed Form of Recursion $a_n = \frac{6}{a_{n-1}-1}$ Given that $a_0=2$ and $a_n = \frac{6}{a_{n-1}-1}$, find a closed form for $a_n$.
I tried listing out the first few values of $a_n: 2, 6, 6/5, 30, 6/29$, but no pattern came out.
| Start with $a_n=\frac6{a_{n-1}-1}$, and replace $a_{n-1}$ with $\frac6{a_{n-2}-1}$. We obtain
$$a_n=\frac6{\frac6{a_{n-2}-1}-1}=\frac{6(1-a_{n-2})}{a_{n-2}-7}$$
Doing this again with $a_{n-2}=\frac6{a_{n-3}-1}$ and so forth, we get
$$a_n=\frac{6(7-a_{n-3})}{7a_{n-3}-13}=\frac{6(13-7a_{n-4})}{13a_{n-4}-55}=\frac{6(55-13... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/939725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How can I prove this problem on geometry? I need to prove the following:
If $P$ is an inner point of $\triangle ABC$, then there is a single transverse $EF$ of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, where $E$ is on $\overleftrightarrow{AC}$ and $F$ is on $\overleftrightarrow{AB}$, such that P is the ... | This answer is based on @noneEggs's suggestion.
1: Extend BP to Q such that BP = (0.5) BQ
2: (Can be skipped.) A circle (centered at P and radius = PB) is drawn cutting BP extended at Q.
3: A line through Q is drawn parallel to AB cutting AC (extended if necessary) at R.
4: Join BR.
5: A line through Q is drawn parall... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/939824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$ What tools would you recommend me for evaluating this integral?
$$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$$
My first thought was to use the beta function, but it's hard to get such a form because
of $\cos(2x)$. What o... | Mathematica cannot find an expression of this integral in terms of elementary functions. However, it can be integrated numerically, like this
NIntegrate[Log[Sin[x]] Log[Cos[x]] Log[Cos[2 x]], {x, 0, Pi/4}]
to yield the result
-0.05874864
Edit: I'm told an analytic expression might still exist, even though it cannot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/939937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 2,
"answer_id": 1
} |
Order type relation in poset and well ordered sets I just read the definition:
Two partial ordered sets X and Y are said to be similar iff there a bijective function from X to Y such that for f(x) < f(y) to occur a necessary and sufficient condition is x < y.
As much as I can understand necessary and sufficient is requ... | You are right! If $x<y$ implies $f(x)<f(y)$, then $x\ge y$ implies either $y<x$, in which case $f(y)<f(x)$, or $x=y$, so that $f(x)=f(y)$. Either way, $x\not< y$ implies $f(x)\not<f(y)$, so that sufficiency implies necessity. A symmetric argument works for necessity implying sufficiency. This holds for any totally orde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$|b-a|=|b-c|+|c-a| \implies c\in [a,b]$ We know that if $c\in [a,b]$ we have $|b-a|=|b-c|+|c-a|$. I'm trying to prove that if the norm is induced by an inner product, then the converse holds.
I need a hint or something.
Thanks in advance
| I will prove that generally, in an inner product space $(H,\langle,\rangle)$ the equality
$$\Vert a-b\Vert=\Vert a-c\Vert+\Vert c-b\Vert$$
implies $c\in[a,b]$.
Indeed, let $x=b-c$, $y=c-a$, so that the hypothesis becomes $\Vert x+y\Vert=\Vert x\Vert+\Vert y\Vert$. Squaring and simplifying we get
$$
\langle x,y\rangle=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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