Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
If $\sum a_n$ converges then $\sum a_n^3$ converges Prove or contradict: if $\sum a_n$ converges then $\sum a_n^3$ converges.
I was able to prove that if $a_n \geq 0$ then the statement is true. But I couldn't prove nor contradict the general case.
| False, counterexample: $$a_n = \frac{\epsilon_n}{\sqrt[3]{\lceil n/3 \rceil}}
\quad\text{ where }\quad \epsilon_n = \begin{cases}+2,& n \equiv 1, \pmod 3\\
-1, & n \not\equiv 1, \pmod 3\end{cases}$$
It is easy to see
$\displaystyle\;\left| \sum_{n=1}^N a_n \right| \le \frac{2}{\sqrt[3]{\lceil N/3 \rceil}}
\quad\implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Very very simple matrix multiplication formula, don't go harsh on me please :) So I'm studying from "Linear algebra and it's applications 3rd edition" - Gilbert Strang.
He gave out this formula to find
$$
\sum_{j=1}^n a_{i,j}x_j
$$
matrix multiplication of the matrices below
$$
\left[
\begin{array}{ccc}
1 & ... | Example: Take $i = 2$, i.e. row number $2$. Then
$$\sum\limits_{j=1}^n a_{i,j} x_j = \sum\limits_{j=1}^3 a_{2,j} x_j = a_{2,1}x_1 + a_{2,2}x_2 + a_{2,3}x_3 = 2u + 5v + 8w. $$
This is because $(x_1,x_2,x_3) = (u,v,w)$ and $n = 3$ is the number of columns of the matrix $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find stationary points of the function $f(x,y) = (y^2-x^4)(x^2+y^2-20)$ I have problem in finding some of the stationary points of the function above. I proceeded in this way: the gradient of the function is:
$$ \nabla f = \left( xy^2-3x^5-2x^3y^2+40x^3 ; x^2y+2y^3-x^4y-20y \right) $$
So in order to find the stationary... | WA gets
$$
\DeclareMathOperator{grad}{grad}
\grad((x^2+y^2-20) (y^2-x^4)) = (-6 x^5-4 x^3 (y^2-20)+2 x y^2, 2 y (-x^4+x^2+2 y^2-20))
$$
(link) and nine real critical points (link).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Nested Hypergeometric series Is it possible to express the following series as a hypergeometric function:
$$\sum_{n=0}^\infty (a)_n \sum_{j_1+j_2+\cdots+j_k=n} \frac{1}{(b)_{j_1} (b)_{j_2}\cdots (b)_{j_k}} z^n $$
where $(a)_n, (b)_n$ are Pochhammer symbols.
Intuitively, if the inner sum can be expressed as a Pochhamme... | $$\sum_{j_1+\ldots+j_k=n}\frac{z^{n}}{(b)_{j_1}\cdots (b)_{j_k}}\tag{1} $$
is the coefficient of $x^n$ in the product:
$$\left(\sum_{m\geq 0}\frac{z^m x^m}{(b)_m}\right)^k = \left(\int_{0}^{1}x z e^{txz}(1-t)^{b-1}\,dt\right)^k=x^k z^k\left(\int_{0}^{1}t^{b-1} e^{(1-t)xz}\,dt\right)^k\tag{2}$$
I wrote $(b)_m=\frac{\Gam... | {
"language": "en",
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$f:S^1 \to \mathbb R$ be continuous , is the set $\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ infinite ? Let $f:S^1 \to \mathbb R$ be a continuous function , I know that $\exists y \in S^1 : f(y)=f(-y)$ where $y \ne -y $ (since $||y||=1$) , so that the set $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$... | If $f$ is constant, then the answer is clearly yes.
Suppose that $f$ is not constant, so there are $p,q\in S^1$ with $f(p) < f(q)$.
There are two paths joining $p$ and $q$ in $S^1$; call them $I$ and $J$. Applying the Intermediate Value Theorem to $f|_I$ and $f|_J$, we can see that every value in the open interval $(f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Suppose that $a$ and $b$ satisfy $a^2b|a^3+b^3$. Prove that $a=b$.
Suppose that $a$ and $b \in \mathbb{Z}^+$ satisfy $a^2b|a^3+b^3$. Prove that $a=b$.
I have reduced the above formulation to these two cases. Assuming $b = a + k$. Proving that any of the below two implies that $a=b$ will be enough.
$$a^2b|(a+b)^3 - 3a... | By hypothesis $\ n = \dfrac{a^3\!+b^3}{a^2b} = \dfrac{a}b + \left(\dfrac{b}a\right)^2\! =\, x+x^{-2}\,\overset{\large {\times\, x^2}}\Longrightarrow\,x^3-n\,x^2 + 1 = 0$
By the Rational Root Test $\ a/b\, =\, x\, = \pm 1\ \ $ QED
Generally applying RRT as above yields the degree $\,j+k\,$ homogeneous generalization
$$a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1827236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
A sum of squared binomial coefficients I've been wondering how to work out the compact form of the following.
$$\sum^{50}_{k=1}\binom{101}{2k+1}^{2}$$
| $$\begin{align}\sum_{k=0}^m \binom {2m+1}{2k+1}^2
&=\sum_{k=0}^m \binom {2m+1}{2k+1}\binom {2m+1}{2m-2k}
\color{lightgrey}{=\sum_{j=0}^m\binom {2m+1}{2(m-j)+1}\binom {2m+1}{2j}\quad \scriptsize (j=m-k)}\\
&=\frac 12 \sum_{k=0}^m \binom {2m+1}{2k}\binom {2m+1}{2(m-k)+1}+\binom {2m+1}{2k+1}\binom{2m+1}{2m-2k}\\
&=\frac 1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$S_n$ is an integer for all integers $n$
Let $a$ be a non-zero real number. For each integer $n$, we define $S_n = a^n + a^{-n}$. Prove that if for some integer $k$, the sums $S_k$ and $S_{k+1}$ are integers, then the sums $S_n$ are integers for all integers $n$.
We have $S_{k} = a^k+\frac{1}{a^k} = m_1$ and $S_{k+1}... | Partial stuff:
Lemma: If $b+b^{-1}$ is an integer then $b^n+b^{-n}$ is an integer for all $n$.
The proof is via induction.
Notice that by Newton's theorem (and symmetry of binomial coefficients) $(b+b^{-1})^n=\sum\limits_{i=0}^{(n-1)/2}\binom{n}{i}(b^ib^{-(n-i)}+b^{n-i}{b^{-i}})+A$ (where $A=0$ if $n$ is odd and $A=\bi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Pi Appoximation: Simpler Solution to Limit? May be a ridiculous question, but I wanted to see if MSE had "simpler" proofs for Viete's approximation (specifically, using an equation derived from Viete's formula) of $\pi$: $$\lim_{x \to \infty}2^x \left(\sqrt{2-\sqrt{2+\sqrt{2 + \sqrt{2 +\sqrt(2)+...} }}}\right) = \pi$$ ... | You can use the fact that
$$\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}=2\cos\frac{\pi}{2^{n+1}},$$
where $n$ is the number of radicals in LHS. Thus your sequence is
$$2^n\sqrt{2-2\cos\frac{\pi}{2^n}}=2^{n+1}\sin\frac{\pi}{2^{n+1}}\to \pi$$
as $n\to\infty$, since $$\lim_{x\to 0}\frac{\sin\pi x}{x}=\pi.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Describing the action of T (linear transformation) on a general matrix I am not familiar with linear transformations in general, and as such, I do not know how to approach this type of question as the examples I'm given/looked up online usually deal with finding the transformation matrix itself.
Suppose $T:M_{2,2}\rig... | As it turns out, I have been overthinking this problem.
Taking $T=\begin{bmatrix}
1 & 1 & 1 & 3\\
-3 & -3 & -1 & -5\\
1 & 2 & 2 & 0\\
-2 & -2 & 0 & -1
\end{bmatrix}$ and multiplying it with the vector of constants $\begin{bmatrix} a\\b\\c\\d\end{bmatrix}$ gives me $$\begin{bmatrix} a+b+c+3d\\-3a-3b-c-5d\\a+2b+2c\\-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove $\sum \frac{\cos nz}{n!}$ converges on compact sets Prove that
$$\sum_{n=1}^{\infty} \displaystyle\frac{\cos nz}{n!}$$
Is an entire function, which means that it uniformly convergent on compact sets
| Sketch:
You can use that $\cos(nx + n iy) = \cos nx \cosh ny - i\sin nx \sinh ny$ to obtain that $|\cos nz| \leq e^{|ny|}$. Since factorials grow faster than exponentials, the complex version of the Weierstrass $M$-test will imply the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1827796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Any unit has an irreducible decomposition The proposition is the following:
Let $R$ be a principal ideal domain. Then every $a \in R$ with $a \neq 0$ has an irreducible decomposition, that is, there is a unit $u$ and irreducible elements $p_1, \dots, p_n$ such that $a = up_1\dots p_n$.
We learned that an irreducible ... | In the definition of an "irreducible decomposition" $a = up_1\dots p_n$, it is possible to have $n=0$. Then you have no irreducible factors $p_i$ at all and just have a unit $u$, so you are saying $a=u$. So, for instance, the irreducible decomposition of $1$ is just $1=1$ (with $n=0$ and $u=1$).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $3 + 3 \times 5 + 3 \times 5^2 + \cdots+ 3 \times 5^n = [3(5^{n+1} - 1)] / 4$ whenever $n \geq 0$
Use induction to show that $$3 + 3 \times 5 + 3 \times 5^2 + \cdots+ 3 \times 5^n= \frac{3(5^{n+1} - 1)}{4} $$whenever $n$ is a non-negative integer.
I know I need a base-case where $n = 0$:
$$3 \times 5^0 =... | I imagine the post on how to write a clear induction proof could be of great service to you. Bob's answer highlights the key points, but I thought I would provide another answer to possibly increase clarity.
You have completed the base case and that's the first part. Great. Now, fix some integer $k\geq 0$ and assume t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1828029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Algebraic geometry look on space-filling curves Can space-filling curves be somehow described in terms of algebraic geometry? It appears to me that they shouldn't, but I'm not sure. Does anyone know of interesting papers on space-filling curves?
| In fact there is an interesting discussion about this in Eisenbuds tome on Commutative Algebra (with a view towards AG). In particular, you may want to read his short and soft introduction to dimension theory chapter. He explains how the discovery of space filling curves helped algebraists understand the need for more ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the value of $\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}$ if $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$? If $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$$ then find the values of $$\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}.$$ How can I solve it? Please help me. Thank you in ad... | HINT:
$$\dfrac{a^2}{b+c}+a=\dfrac{a(a+b+c)}{b+c}$$
$$\sum_{\text{cyc}}\left(\dfrac{a^2}{b+c}+a\right)=(a+b+c)\sum_{\text{cyc}}\dfrac a{b+c}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1828288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Sum of elements of the inverse matrix (without deriving the inverse matrix) using elementary methods.
I have the matrix
$$\begin{pmatrix}
3&2&2&\\
2&3&2\\
2&2&3
\end{pmatrix}.$$ Find the sum of elements of the inverse matrix without computing the inverse.
I have seen this post, but I need much more elementary met... | Notice that $u^T=(1,1,1)$ is an eigenvector with eigenvalue $7$.
So we have $Au=7u$ and hence $A^{-1}u=\frac{1}{7}u$.
But the sum of the elements of $\frac{1}{7}u$ is just the sum of the elements of $A^{-1}$. So the answer is $\frac{3}{7}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1828345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Intuitive reason why the Euler characteristic is an alternating sum? The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups. Since homeomorphic spaces have isomorphic homology groups, however, even the non-alternating sum of the ranks of the homology groups i... | The Euler characteristic can be computed from the number of cells of each dimension. The non-alternating sum (or other functions of the ranks) cannot.
The Euler characteristic has nice formulas when $X=X_1\cup X_2$, or when $X=X_1\times X_2$, or when $X$ is a bundle etc. The non-alternating sum (or other functions of t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Why does $A$ times its inverse equal to the identity matrix? I was trying to come up with a proof of why: $AA^{-1} = I$.
If we know that: $A^{-1}A = I$, then $A(A^{-1}A) = A \implies (AA^{-1})A = A$.
However I don't like setting $AA^{-1} = I$ for fear that it might be something else at this point, even though we know t... | We say a matrix $B$ is an inverse for $A$ if $AB = BA = I$, and the notation for $B$ is $A^{-1}$.
So it's by definition $AA^{-1}=I$, you cannot really prove it.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Lonely theorems What are some instances of theorems which are especially unique in mathematics, i.e. for which there are not many other theorems of a similar character? An example I have in mind is Pick's theorem, since it is the only theorem I have ever seen concerning geometry of polygons with vertices on a lattice.... | The Bieberbach conjecture is an example of a lonely result in the sense that, while it generated much interest and almost competition, its ultimate solution by de Branges pretty much closed the field. It turned out that the result does not have many applications, and is a kind of a very high-level olympiad problem.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of the minimum value of an integral
Let $$f(a)=\frac{1}{2}\int_{0}^{1}|ax^n-1|dx+\frac{1}{2}$$ Here $n$ is a natural number. Let $b_n$ be the minimum value of $f(a)$ for $a>1$. Evaluate $$\lim_{m \to \infty}b_mb_{m+1}\ldots b_{2m}$$
Some starters please. Thanks.
| $$\begin{eqnarray*}f(a) = \frac{a}{2}\int_{0}^{1}\left| x^n-\frac{1}{a}\right|\,dx+\frac{1}{2}&=&\frac{1}{2}+\frac{a}{2}\int_{0}^{1}(x^n-1/a)\,dx+a\int_{0}^{\frac{1}{\sqrt[n]{a}}}\left(\frac{1}{a}-x^n\right)\,dx\\&=&\frac{1}{2}+\frac{a}{2n+2}-\frac{1}{2}+\frac{1}{\sqrt[n]{a}}-\frac{1}{(n+1)\sqrt[n]{a}}\\&=&\frac{a}{2n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1828702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Probability of getting a pair of socks from a drawer if three are drawn I'm really struggling with this concept, hoping you guys could help me out.
Question: You have been provided with 20 pairs of socks within a box consisting of 4 red pairs, 4 yellow pairs, 4 green pairs, 4 blue pairs and 4 purple plairs.
The pairs h... | HINT: with 3 socks you must find the probability that all socks have different color, the probability that 2 socks have the same color, and the probability that all socks have the same color. Name these probabilities as P1, P2 and P3... with the number referencing the different amount of different colors.
For any numbe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find number of solutions of an equation? Given $n$, how to count the number of solutions to the equation $$x + 2y + 2z = n$$ where $x, y, z, n$ are non-negative integers?
| $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
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"source": "stackexchange",
"question_score": "6",
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Permutations of {1 .. n} where {1 .. k} are not adjacent The Problem:
So I was thinking up some simple combinatorics problems, and this one stumped me.
Let N be the set of numbers $\{1 .. n\}$, or any set of cardinality $n$
Let K be the set of numbers $\{1 .. k\}$ where $k < n$, or any subset of N of cardinality $k$
H... | Let's call the elements up to $k$ white and the others black, and consider elements of the same colour to be indistinguishable for now. For arrangements beginning with a white element, glue a black element to the left of all other white elements and choose $k-1$ slots among the resulting $n-k$ objects (excluding the in... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Rings of Krull dimension one I have to write a monograph about commutative rings with Krull dimension $1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate a lot to know if there is some result of the kind:
$$ \dim(A)=1 \iff ~?$$
Thanks in advanc... | An integral extension $S$ of a ring $R$ of Krull dimension $1$ has Krull dimension $1$. This is because any 3-chain of prime ideals $P_1 \subsetneq P_2 \subsetneq P_3$ induces an inclusion $p_1 \subset p_2 \subset p_3$ (where we define $p_j := R \cap P_j$ for $j = 1, 2, 3$), of which not all inclusions can be proper be... | {
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Assume $r,s \in\mathbb{Q}$. Prove $\frac{r}{s},r-s \in\mathbb{Q}$ I have attempted this proof by contradiction. Beginning with assuming to the contrary that each a and b are irrational but was not sure if I did it correctly. Any help would be greatly appreciated.
Assume $r,s \in\mathbb{Q}$.
a) Prove $\frac{r}{s}\in\mat... | $x\in\mathbb Q\iff \exists a,b\in\mathbb Z: b\ne0,\ x=\frac ab$
Let $r=\frac pq$,$s=\frac tu$, where $p,q,t,u\in\mathbb Z$ and $u,q\ne0$. Then,
If $s=0$, $\frac rs$ is not defined. Assuming, $s\ne0$,thus, $t\ne0$. $\frac rs=\frac {pu}{qt}$, where $pu,qt\in\mathbb Z$, as $t,q\ne0$; $qt\ne0$. Thus, $\frac rs\in\mathbb Q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1829237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\int_{0}^{\infty}{1\over x^4+x^2+1}dx=\int_{0}^{\infty}{1\over x^8+x^4+1}dx$ Let
$$I=\int_{0}^{\infty}{1\over x^4+x^2+1}dx\tag1$$
$$J=\int_{0}^{\infty}{1\over x^8+x^4+1}dx\tag2$$
Prove that $I=J={\pi \over 2\sqrt3}$
Sub: $x=\tan{u}\rightarrow dx=\sec^2{u}du$
$x=\infty \rightarrow u={\pi\over 2}$, $x=0\... | $$
\begin{aligned}
I & =\int_0^{\infty} \frac{\frac{1}{x^2}}{x^2+\frac{1}{x^2}+1} d x \\
& =\frac{1}{2} \int_0^{\infty} \frac{\left(1+\frac{1}{x^2}\right)-\left(1-\frac{1}{x^2}\right)}{x^2+\frac{1}{x^2}+1} d x \\
& =\frac{1}{2}\left[\int_0^{\infty} \frac{d\left(x-\frac{1}{x}\right)}{\left(x-\frac{1}{x}\right)^2+3}-\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1829298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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A fair coin is flipped 3 times. Probability of all $3$ heads given at least $2$ were heads Here's a problem from Prof. Blitzstein's Stat 110 textbook. Please see my solution below.
A fair coin is flipped 3 times. The toss results are recorded on separate slips of paper (writing “H” if Heads and “T” if Tails), and the ... | (b) involves two processes: First flip the coins, then read two of the results.
So, for instance, if you flip exactly two coins, what is the (conditional )probability that you the two papers read are those of the two heads?
Let $H_f$ be the count of heads flipped and $H_r$ the count of heads read. Use Bayes' Rule and... | {
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"timestamp": "2023-03-29T00:00:00",
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Simplifying this series of Laguerre polynomials I trying to figure out whether a simpler form of this series exists.
$$\sum_{i=0}^{n-2}\frac{L_{i+1}(-x)-L_{i}(-x)}{i+2}\left(\sum_{k=0}^{n-2-i} L_k(x)\right)$$
$L_n(x)$ is the $n$th Laguerre polynomial.
Wikipedia offers several useful recurrence relations in terms of the... | Let we set $N=n-2$. The generating function of Laguerre's polynomials is:
$$ \sum_{n\geq 0}L_n(x)\,t^n = \frac{1}{1-t}\,\exp\left(-\frac{tx}{1-t}\right)\tag{1} $$
hence:
$$ \sum_{n\geq 0}\left(\sum_{k=0}^n L_k(x)\right)\,t^n = \frac{1}{(1-t)^2}\,\exp\left(-\frac{tx}{1-t}\right)\tag{2} $$
$$ \sum_{n\geq 0}L_n(-x)\,t^n =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1829403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed form of infinite product $\prod\limits_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$ I encountered this infinite product while solving another problem:
$$P(x)=\prod_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$$
$$P(x)=P \left( \frac{1}{x} \right)$$
I strongly believe ... | You could more or less infer the results from the infinite product you have posted. Denote
$$f_k(t)=\frac{2}{1+t^{1/2^k}},$$
we have
$$\frac{f_k(x)~f_k(x^{1/2})}{f_k(x^{3/2})}=\frac{2}{1+x^{1/2^k}}\frac{1+\left(x^{1/2^{k+1}}\right)^3}{1+x^{1/2^{k+1}}}\\
=\frac{2}{1+x^{1/2^k}}\left(1+x^{1/2^{k+1}}+x^{1/2^k}\right)=2\le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the determinant defined in terms of permutations? Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of permutations?
$$
\text{det}(A)=\sum_{p}\sigma(p)a_{1p_1}a_{2p_2}...a_{... | I think Paul's answer gets the algebraic nub of the issue. There is a geometric side, which gives some motivation for his answer, because it isn't clear offhand why multilinear alternating functions should be important.
On the real line function of two variables (x,y) given by x-y gives you a notion of length. It reall... | {
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Can I always say that any "custom" operation in $Z_n$ is commutative and associative? I have a lot of exercises that say something similiar:
Given, in the set $Z_{15}$ the following binary operation $*$ $$\forall a,b \in Z_{15}, a*b = \overline6(a+b)\ -\ \overline5ab$$
This is just an example, the operation could be... | Something you can generally do, is looking for a bijection with a set having a binary operation $\oplus$ of which you know that it has the desired properties. Here it is rather evident: Let
$$
f\colon \mathbf Z/15\rightarrow \mathbf Z/3\times\mathbf Z/5
$$
be the map that sends $\overline a$ to the pair $(a\bmod 3,a\bm... | {
"language": "en",
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Are there more quadratics with real roots or more with complex roots? Or the same? Consider all quadratic equations with real coefficients:
$$y=ax^2+bx+c \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, a,b,c \in \mathbb{R}, \, a≠0 $$
I was wondering whether if more of them have real roots, more have complex roots, or a same number o... | There are two ways to write a quadratic equation.
1) $y = ax^2 + bx + c$ and 2) $y = a(x-h)^2 + k$
Both models have equal numbers of quadriatic formulas; i.e. the cardinality of $\mathbb R^3$.
However "random" distribution results in different concentrations when represented as corresponding elements of $\mathbb R^3$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What are linearly dependent vectors like? How are they different from linearly independent vectors?
| While the above answer is correct, I think some intuitive understanding might help you, too.
Consider $\mathbb{R}^2$. Take the vectors $u= (1,1)$ and $v=(-2,-2)$. These vectors are called linearly dependent because they are equal up to a linear transformation, i.e. $v = (-2) \cdot u$.
This generalizes to $\mathbb{R}^n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find three different systems of linear equation whose solutions are.. Find three different systems of linear equation whose solutions are $x_1 = 3, x_2 = 0, x_3 = -1$
I'm confused, how exactly can I do this?
| Note that your system is described by the matrix
$$
\left[\begin{array}{rrr|r}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -1
\end{array}\right]
$$
Performing any row operation on this matrix yields a system with the same solutions. For example, you could add $\DeclareMathOperator{Row}{Row}\Row_1$ to $\Row_2$ to get ... | {
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Determine the closed form for $\int_{0}^{\infty}\sinh(xe^{-x})dx$ Find the closed form for
$$I=\int_{0}^{\infty}\sinh(xe^{-x})dx\tag1$$
Change to
$$I={1\over 2}\int_{0}^{\infty}(e^{xe^{-x}}-e^{-xe^{-x}}dx)\tag2$$
Any hints? I can't go further.
| By expanding the hyperbolic sine as a Taylor series we have:
$$ I = \sum_{n\geq 0}\frac{1}{(2n+1)!}\int_{0}^{+\infty}x^{2n+1} e^{-(2n+1)x}\,dx = \color{red}{\sum_{n\geq 0}\frac{1}{(2n+1)^{2n+2}}} \tag{1}$$
with a series recalling sophomore's dream. I won't bet on simple closed forms but in terms of integrals involvin... | {
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How can I find the sup, inf, min, and max of $\bigcup\left[\frac{1}{n}, 2-\frac{1}{n}\right]$ $$\bigcup\left[\frac{1}{n}, 2-\frac{1}{n}\right]$$
I'm not sure how to get started with this one. When I graph the two functions I see they intersect at the point $(1,1)$, which I take to be the union of the set.
But how do... | The union consists of all real numbers $x$ such that $\frac{1}{n}\leq x\leq 2-\frac{1}{n}$ for some natural number $n$. This is a much larger set than just the single point $\{1\}$. For instance, taking $n=2$ shows that all real numbers $x$ with $\frac{1}{2}\leq x\leq \frac{3}{2}$ are included in the union.
Some leadin... | {
"language": "en",
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Find radius of circle (or sphere) given segment area (or cap volume) and chord length The goal is to design a container (partial sphere) of given volume which attached to a plane via a port of a given radius. I believe this can be done as follows but the calculation is causing me problems:
A circle of unknown radius i... |
With the notations in this figure (borrowed from https://en.wikipedia.org/wiki/Circular_segment), $c$ is your chord length $L$, $$\theta=2\arcsin(\frac{c}{2R}),$$ the green area is $\frac{R^2}{2}(\theta-\sin(\theta))$. The larger circular segment area (whole area of the disk minus the green area) is given by $$A=R^2(\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Help with proving if $s_n$ converges to $0$ and $x_n$ is a bounded sequence, then $\lim(x_ns_n)=0$
Prove: If $s_n$ converges to $0$ and $x_n$ is a bounded sequence, then $\underset{n \to \infty}{\lim}(x_ns_n)=0$
I'm have trouble getting started on this proof.
Since I know $s_n$ converges to $0$ I feel like I should... | HINT: If $x_n$ is bounded, then $\exists M>0: |x_n|<M$ for all $n$. Hence $|x_ns_n|<M\varepsilon$ for large $n$. It should be clear that $x_ns_n$ also tends to 0.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Distribution of ages of 3 children in a family Please consider Blitzstein, Introduction to Probability (2019 2 edn), Chapter 2, Exercise 30, p 88.
*A family has 3 children, creatively named A,B, and C.
(a) Discuss intuitively (but clearly) whether the event “A is older than B” is independent
of the event “A is olde... | The ages of $A,B,$ and $C$ may be independent†, but the events of pairwise orders are not. If you are told that $A$ is one of the two oldest children (because $A$ is older than $C$) it should raise your anticipation that $A$ is also older than $B$.
(† though, actually, they are not independent if they have the same m... | {
"language": "en",
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Linearly independent subset - a simple solution?
Problem: Let $\{ v_1, \ldots, v_n \}$ be a linearly independent subset of $V$,
a vector space. let $$ v= t_1 v_1 + \cdots + t_n v_n $$ where $t_1,
\ldots t_n \in \mathbb{R}$. For which $v$ is the set $\{v_1 + v ,\ldots , v_n + v \}$ linearly independent?
My ideas... | It is easy to explicitly guess a nontrivial solution for the system of equations $$\lambda_i + st_i = 0$$ By setting $\lambda_i = t_i$ for all $i$, we get $\lambda_i + st_i = t_i-t_i = 0$ for all $i$ since $s=-1$ by assumption. Also since $\sum t_i = -1$, atleast one $t_i$ is nonzero, so the solution is nontrivial.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Preimage of a simply closed curve under the two-dimensional antipodal map
Suppose $p:S^2\to P^2$ is the quotient antipodal map, and $J$ is a simply closed curve in $P^2$, then $p^{-1}(J)$ is either a simply closed curve in $S^2$, or two disjoint simply closed curves in $S^2$.
I think I can prove the second case. Clea... | $S^2 \to P^2$ is a 2-sheeted covering map. Now note, that for any space $X\subset P^2$, the map restricts to a 2-sheeted covering map $p^{-1}X\to X$. If we pick $X=J$, then the covering map restricts to
$$ p^{-1}J \stackrel {2:1} \longrightarrow J\cong S^1.$$
So the preimage $p^{-1}J$ might either be homeomorphic to t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof that the function is uniformly continuous
Proof that the function $f: [0, \infty) \ni x \mapsto \frac{x^{2}}{x + 1} \in \mathbb{R}$ is uniformly continuous.
On the internet I found out that a function is uniformly continuous when
$\forall \varepsilon > 0 \ \exists \delta > 0: \left | f(x)-f(x_{0}) \right | < \... | The best way to start these types of problems is to start by messing with the part $|f(x) - f(y)| < \epsilon$ of the definition. Note that, by combining fractions and multiplying everything out we have
$f(x) - f(y) = \frac{x^2}{x+1} - \frac{y^2}{y+1} = \frac{x^2y-y^2x+x^2-y^2}{xy+x+y+1}$.
After playing around with some... | {
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Does $|z_1+z_2|=|z_1|+|z_2|\implies z_1=kz_2$? Let $z_1,z_2\in \mathbb C$. I was wondering if
$$|z_1+z_2|=|z_1|+|z_2|\implies z_1=k z_2\ \ or\ \ z_2=0,\quad k\in\mathbb R^+.$$
Is it true?
I am having problems trying to show it. I'm sure it's easy to show using brute force, but if it's true, is there an elegant way ... | Squaring both sides, we see this is equivalent to:
$$(z_1+z_2)(\overline z_1+\overline z_2)=z_1\overline z_1 + 2|z_1||z_2| + z_2\overline z_2$$
which is equivalent to:
$$\mathrm{Re}(z_1\overline z_2)=\frac{1}{2}\left(z_1\overline z_2+z_2\overline z_1\right)=|z_1 \overline z_2|$$
So $z_1\overline z_2=\alpha$ must be no... | {
"language": "en",
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Characterizing spaces with no nontrivial covers I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
| The converse is also true, under mild assumptions on $X$.
Namely, if you assume that your space $X$ is path connected, locally path connected, and semilocally simply connected, then $X$ has no nontrivial path connected covers if and only if $X$ is simply connected.
This follows from the standard theorem giving a bije... | {
"language": "en",
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Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c? If so, how could you go about constructing it? If not, why not?
I am new to proofs and I was reading a book where they posed this question. I understand that if we are given a... | Yes indeed the converse for Pythagoras is also true. When such a construction is made (SSS) the relation is satisfied. The vertices lie on a semicircle, and by Thale's thm a right angle is enclosed.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 5
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Different ways of evaluating $\int_{0}^{\pi/2}\frac{\cos(x)}{\sin(x)+\cos(x)}dx$ My friend showed me this integral (and a neat way he evaluated it) and I am interested in seeing a few ways of evaluating it, especially if they are "often" used tricks.
I can't quite recall his way, but it had something to do with an ide... | You're integrating on $[0, \pi/2]$ so replacing $x$ by $\pi/2 -x$ we see that $$I=\int_0^{\pi/2} \frac{\sin{x}}{\cos{x}+\sin{x}}dx.$$ Now sum this integral with the initial expression and notice that $2I=\pi/2$ hence...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$\lim_{x\to \infty} \ln x=\infty$ I'm reading the following reasoning:
Since $\underset{n\to \infty}{\lim}\ln 2^n=\underset{n \to \infty}{\lim}n\cdot(\ln 2)=\infty$ then necessarily $\underset{x\to \infty}{\lim}\ln x =\infty$.
I don't understand how the generalisation was done from $\lim_{n\to \infty}\ln 2^n=\infty$... | For $2\leq n\in N$ and $x\geq n$ we have $$\ln x=\int_1^x(1/y)\;dy\geq \int_1^n(1/y)\;dy=\sum_{j=1}^{n-1}\int_j^{j+1}(1/y)\;dy\geq \sum_{j=1}^{n-1}\int_j^{j+1}(1/(j+1))\;dy=$$ $$=\sum_{j=1}^{n-1}1/(j+1).$$ There are many ways to show that this last sum has no upper bound as $n\to \infty.$ The simplest, I think,is $$1/... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to pull out coefficient from radical in an integral I am in an online Calculus 2 class, and before my professor gets back to me, I was wondering if you guys could help. I was reading through an example:
How was 1/27 pulled out from the coefficient next to u^2? I am probably missing something dumb. Thanks.
| There's an error in the problem-the numerator in the u-substitution should be $u^3$, not $u^2$. The numerator here is $(1/3 u)^3$ (compare to u). So a factor of 1/27 drops out into the denominator. The rest is easy to see by careful substitution.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving for possible orientations of 3 objects on a 3x3 grid Say you have a 3x3 grid, and 3 objects to work with. Each occupies one space. How would I go about solving the amount of ways they can lay on the grid.
Example: (pardon my bad ASCII art)
[][][]
[][][]
[x][x][x]
Any help would be much appreciated, thanks!
| To reiterate what was said in the comments, imagine temporarily that your grid has its positions labeled.
$\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}$
A specific arrangement of the objects will then correspond to a selection of three numbers from $1$ to $9$.
$\begin{bmatrix}\circ&\times&\circ\\\circ&\circ&\times\\... | {
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Prove that order is antisymmetric. (for natural numbers) Prove that order is antisymmetric.(for natural numbers)i.e.
If $ a \leq b$ and $b\leq a$ then $a=b$.
I do not want a proof based on set theory.
I am following the book Analysis 1 by Tao. It should be based on peano axioms.
I tried $ b=a+n$ where $n$ is a natural ... | If you're allowed to rely on trichotomy you might as well suppose by way of contradiction that $a\ne b$. Since $a\leq b$ and $b\leq a$ it follows that $a<b$ and $b<a$, a contradiction to the forementioned trichotomy property.
Remarks.
*
*For this proof to be valid you'd sooner have $a\leq b$ defined as $a<b\oplus a=b... | {
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Why $\langle x^2+1\rangle$ is not prime in $\mathbb{Z}_2[x]$? I am reading ring theory (a beginner) and I stumbled upon a problem which I can't understand
The ideal $\langle x^2+1\rangle$ is not prime in $\mathbb{Z}_2[x]$, since it contains $(x+1)^2=x^2+2x+1=x^2+1$ , but does not contain $x+1$ .
$\langle x^2+1\rangl... | You're comparing apples and oranges. The rings $\mathbb{Z}_2$ and $\mathbb{R}$ are very different and the polynomial $x^2+1$ has different behavior in them and there's no contradiction, because
$$
x^2+1\in\mathbb{Z}_2[x]
$$
and
$$
x^2+1\in\mathbb{R}[x]
$$
are different objects that live in distinct sets.
So it can very... | {
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Composition of a unique arrow with the inverse of another Suppose we have the arrows $u:T \rightarrow Q$, $v:T \rightarrow P$ and $f:P \rightarrow Q$. Furthermore, suppose $u$ is unique and $f$ is iso.
I understand that we can say that $v = u;f^{-1}$, but do we have enough information that $v$ is also unique?
I believ... | Yes this is true. Once you have two arrows (the same thing as two uni ue arrows) you can compose them (composition being a function).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Area of a circle on sphere On a (flat) Euclidean plane, the area of a circle with a radius $r$ can be described by the function $A(r) = \pi r^2.$
But how can one describe the area of the same circle on a spherical manifold? Assuming that the radius of the sphere is an Euclidean distance of $d,$ how would $A(x)$ look?
I... | It depends on how you define $r$. If $r$ is the length of the arc on the sphere, then your area is still $\pi r^2$. If $r$ is the radius in the plane, you need to calculate the length of the arc given by a point on the circle, and the intersection between the sphere and the line that goes through the center of the sphe... | {
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"timestamp": "2023-03-29T00:00:00",
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$\text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N)$ Let $(M_i)_{i\in I}$ be a collection of $R$-moduls. Show that for all $N\in \text{Ob}(_R\text{Mod})$ is
$$
\text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N).
$$
My idea is to make use of the universal proper... | The proof is correct. If $(C_i)_{i\in I}$ is any family of cochain complexes, then by writing out the definitions you immediately see $H^n(\prod C_i)=\prod H^n(C_i)$.
For if $D$ denotes the differential on $\prod C_i$, $d_i$ the differential on $C_i$, then
$H^n(\prod C_i)=Ker D/Im D=\prod Ker d_i/\prod Im d_I=\prod Ker... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the minimum, maximum, minimals and maximals of this relation
Tell if the following order relation is total and find the minimum, maximum, minimals and maximals: $$\forall a,b \in\mathbb Z,\ \ a\ \rho\ b \iff a \leq b\ \text{ and }\ \pi(a) \subseteq \pi(b)$$
where $\pi$ is defined as follows: $\forall n \in\math... | I think you have not yet got a hold of the difference between minimum and minimal and maximum/maximal. A minimum of a partially ordered set is an element smaller than all other elements. An element is minimal if there is no element that is strictly smaller than it.
For a totally ordered set, these are the same. But for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I find the Integral of $\sqrt{r^2-x^2}$? How can I find the integral of the following function using polar coordinates ?
$$f(x)=\sqrt{r^2-x^2}$$
Thanks!
| $\displaystyle\int \sqrt{r^2-x^2}dx$
Let be $\;x=r.\sin\alpha$ or $\quad x=r.\cos\alpha$,
Let be $\;x=r.\sin\alpha$,
and $\quad dx=r.\cos \alpha \;d\alpha$
Integral be,
$\displaystyle\int \sqrt{r^2-x^2}dx=\displaystyle\int r.\sqrt{1-\sin^2\alpha}\;.r.\cos \alpha \;d\alpha=\displaystyle\int r^2.\cos^2\alpha\; d\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Find the probability that a word with 15 letters (selected from P,T,I,N) does not contain TINT If a word with 15 letters is formed at random using the letters P, T, I, N,
find the probability that it does not contain the sequence TINT.
(I just made up this problem.)
| The following Python script confirms the answer given by @McFry:
LENGTH = 15
LETTERS = 'PTIN'
SEQUENCE = 'TINT'
def func(word):
if len(word) < LENGTH:
return sum(func(word+letter) for letter in LETTERS)
else:
return SEQUENCE not in word
count = func('')
total = len(LETTERS)**LENGTH
print '{}/{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Is convexity the most general dividing line between "easy" and "hard" optimization problems? Just got started with Boyd's Convex Optimization. It's great stuff and I see how it directly subsumes the all-important linear programming class of models. However, it seems that if a problem is non-convex, then the only recour... | In my opinion, it is sufficient that the objective is quasi-convex. Indeed, this ensures that all local minimizers are global minimizers and the set of global minimizers is convex. Thus, you do not have to fight against local minimizers (which are not global minimizers).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help with proving a 2 by 2 determinant is the area of parallelogram I have proved a large part of this by the following but get stuck at the last step.
To say $A=ad-bc$, we still need $ad>bc$. I have puzzling over this for hours. Thank you!
| In general, $A \geq 0$ is area and $A^2= (ad-bc)^2$, then $A=\vert ad-bc \vert$, so in fact area of a parallelogram is absolute value of the determinant.
In your case, where $(c,d)$ is between 0 and 180 degrees ccw of $(a,b)$, call this angle $\theta \in $[0, 180]. Note $(d,-c)$ is a rotation of $(c,d)$ 90 degrees cw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$5^{th}$ degree polynomial expression
$p(x)$ is a $5$ degree polynomial such that
$p(1)=1,p(2)=1,p(3)=2,p(4)=3,p(5)=5,p(6)=8,$ then $p(7)$
$\bf{My\; Try::}$ Here We can not write the given polynomial as $p(x)=x$
and $p(x)=ax^5+bx^4+cx^3+dx^2+ex+f$ for a very complex system of equation,
plz hel me how can i solve that... | hint : write the polynomial in this form $$f(x)= a(x-1)(x-2)(x-3)(x-4)(x-5)+b(x-1)(x-2)(x-3)(x-4)(x-6) +c(x-1)(x-2)(x-3)(x-5)(x-6)+d(x-1)(x-2)(x-4)(x-5)(x-6)+e(x-1)(x-3)(x-4)(x-5)(x-6)+f(x-2)(x-3)(x-4)(x-5)(x-6)$$ now finding constants are easy
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 3
} |
How to solve asymptotic expansion: $\sqrt{1-2x+x^2+o(x^3)}$ Determinate the best asymptotic expansion for $x \to 0$ for:
$$\sqrt{1-2x+x^2+o(x^3)}$$
How should I procede?
In other exercise I never had the $o(x^3)$ in the equation but was the maximum order to consider.
| You have the following asymptotic expansion :
$$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}+o(x^3)$$
So :
$$\sqrt{1-2x+x^2+o(x^3)}=1+\frac{-2x+x^2+o(x^3)}{2}-\frac{(-2x+x^2+o(x^3))^2}{8}+\frac{(-2x+x^2+o(x^3))^3}{16}+o((-2x+x^2+o(x^3))^3)\\=1+\frac{-2x+x^2}{2}-\frac{4x^2-4x^3+x^4}{8}+\frac{-8x^3+12x^4-6x^5+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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An erroneous application of the Counting Theorem to a regular hexagon? I'm trying to count the unique orbits of a regular hexagon such that each vertex is either Black or White and each edge is either Red, Gree, or Blue. The group I've chosen to act on the hexagon is the dihedral group $D_7$, $$\{e,r,r^2,r^3,r^4,r^5,r^... | For future reference I would like to document how we can do this
calculation using a cycle index. The key observation here is the
following: the cycle structure of a rotation (but not a reflection)
acting on the vertices and edges is the same for edges and
vertices. So we may compute the cyc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Matrix with orthonormal base I have the two following given vectors:
$\vec{v_{1} }=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$
$\vec{v_{2} }=\begin{pmatrix} 3 \\ 0 \\ -3 \end{pmatrix} $
I have to calculate matrix $B$ so that these vectors in $\mathbb{R}^{3}$ construct an orthonormal basis.
The solution is:
$$B=\begin... | Maybe these calculations would help you.
We need to find vertor $\vec{v}_3$ such that $\vec{v}_3\perp\vec{v}_1$ and $\vec{v}_3\perp \vec{v}_2$, i.e.
$$
\begin{cases}
(\vec{v}_1, \vec{v}_3) = 0, \\
(\vec{v}_2, \vec{v}_3) = 0.
\end{cases}
$$
Here $(\vec{x},\vec{y})$ is a scalar product of vectors $\vec{x}$ and $\vec{y}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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What does it mean to have a "different topology"? On a space, I understand the notion of having different metrics on the same space. It is, in layman's terms, different ways of defining a distance but on the same space.
But I often see the term "different topology" being used, for example in this excellent answer. But ... | The topology of a set is essentially a notion about the shape of the set. If you have a topology on a set, you can talk about neighbours, convergence, etc. There are many ways to give a topological structure to a set, as you know, for example taking different metrics may lead to different topologies. For example, the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
which natural english interpretation of this symbolic statement is correct? Part of Keith Devlin's Coursera MOOC on mathematical thinking requires the translation of this symbolic statement into natural language:
$$ 5 < x < 7$$
Interpretation 1: $x$ is a single unknown number located somewhere between 5 and 7 on the n... | The first interpretation is correct.
For the second one, you would write
$$\{x\in \mathbb R ,\ 5<x<7\},$$
or simply
$$\{5<x<7\},$$
or (thanks to a comment) :
$$x\in(5,7).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$ Revised Question:
Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$ into a bona fide Hilbert space?
In particular, does the di... | Of course as Hilbert spaces $L^2(S^1)$ and $L^2\bigl([0,1]\bigr)$ are isomorphic, and you could also say that $L^2\bigl([0,1]\bigr)$ is the prime example of a Hilbert space arising from Lebesgue theory.
But note that $L^2(S^1)$ is one of the most important Hilbert spaces in the world, and there definitively is an essen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to find the Summation of series of Factorials? $$1\cdot1!+2\cdot2!+\cdots+x\cdot x! = (x+1)!−1$$
I don't understand what's happening here. The given sum of factorials is generalized into a single term. Could somebody please help me finding the logic behind this series.
Thanks in advance!!
| Michael Hardy’s computational proof is the simplest, but there is also a reasonably straightforward combinatorial argument.
There are $(n+1)!-1$ permutations of the set $[n+1]=\{1,2,\ldots,n+1\}$ other than the increasing permutation $\langle 1,2,\ldots,n+1\rangle$. Let $P$ be the set of all such permutations; we’ll no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integrating factor for a non exact differential form I can't find an integrating factor for the differential form
$$
-b(x,y)\mathrm{d}x + a(x,y)\mathrm{d}y
$$
where
$$
a(x,y) = 5y^2 - 3x
$$
and
$$
b(x,y) = xy - y^3 + y
$$
The problem has origin form the following differential equation
\begin{cases}
x' = a(x,y) \\
y'... | The integrating factor $\quad \mu=y^2e^x \quad$ can be found thanks to the method below :
The differential relationship leads to a first order linear PDE. We don't need to fully solve it with the method of characteristics. Only a part of the solving is sufficient to find an integrating factor.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Non constant analytic function from $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}.$ Does there is non constant analytic function from $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}?$ According to me there is no such non constant analytic function because if there is any such function say $f,$ t... | If so then $1/f$ is bounded. Hence $1/f$ has a removable singularity at the origin, giving a bounded entire function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to determine which of the following transformations are linear transformations?
Determine which of the following transformations are linear transformations
A. The transformation $T_1$ defined by $T_1(x_1,x_2,x_3)=(x_1,0,x_3)$
B. The transformation $T_2$ defined by $T_2(x_1,x_2)=(2x_1−3x_2,x_1+4,5x_2)$.
C. The tran... | Is T in A a linear transformation?
*
*Check linearity for addition.
Suppose $T:V \rightarrow W$ .Where $V$ and $W$ are vector spaces over $F$. Let $x_1,x_2,x_3 \in F$ and also let $x_4,x_5,x_6 \in F$. So that $(x_1,x_2,x_3) \in V$ and $(x_4,x_5,x_6) \in V$. Now need to check that $T((x_1,x_2,x_3)) + T((x_4,x_5,x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
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Probability of $\max_i \{X_i\} = X_0$ where $X_i$ are iid binomial We have $M$ Binomial random variables, where $X_0 \sim $ Bin$(n,p)$ and $X_i \sim $ Bin$(n,1/2)$.
Suppose $p > 1/2$. I'm interested in the probability that $\mathbb{P}(\max \{X_1,\dots,X_M\} \geq X_0)$. Is this tractable?
If not, is it tightly boundab... | $P( \max \{X_1,...,X_M \} \geq X_0)=P(X_i \geq X_0$ for some i) = $1-P( \text{each }X_i < X_0)$.
Now $P( \text{each }X_i < X_0)=\sum_{k=0}^n P(\text{each }X_i<X_0 \vert X_0=k) \cdot P(X_0 =k)$ =$\sum_{k=0}^n P( \text{each } X_i<k \text{ and } X_0=k)$ = $\sum_{k=1}^n \big[ P(X_1<k) \big]^M \cdot P(X_0=k)$. Note the k=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find the value of $ [1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$ Assume that [x] is the floor function. I am not able to find any patterns in the numbers obtained. Any suggestions?
$$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$
| Just to better asses the solution given by Kitter Catter, premised that
$$
\eqalign{
& x = \left\lfloor x \right\rfloor + \left\{ x \right\} \cr
& \left\lfloor { - x} \right\rfloor = - \left\lceil x \right\rceil \cr
& \left\lceil x \right\rceil = \left\lfloor x \right\rfloor + \left\lceil {\left\{ x \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
} |
Show that if A is self-adjoint and $A^{2}x=0$, show that $Ax=0$. I feel like i'm overcomplicating this a bit.
Let $X$ be a finite-dimensional inner product space and $A$ be a linear transformation from $X$ to $X$. If A is self-adjoint and if $A^{2}x=0$, show that $Ax=0$.
Here's my thought:
$0=\left<A^{2}x,y\right>=\lef... | It's useful to know that if $X$ and $Y$ are finite dimensional inner product spaces over $F$ (where $F$ is $\mathbb R$ or $\mathbb C$) and $A:X \to Y$ is a linear transformation, then $A$ and $A^* A$ have the same null space. Here's a proof: Clearly $Ax = 0 \implies A^* Ax = 0$. Conversely,
\begin{align*}
& A^* A x =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Alternate proof for Vieta's formula (formula for the summing the roots of a polynomial) I just saw Vieta's formula for the first time, where it was stated that given some polynomial
$$p(x)=a_nx^n+\cdots+a_0,$$
let $x_1,\ldots,x_n$ denote its roots. Then $$\sum_{i=1}^n x_i=-\frac{a_{n-1}}{a_{n}}.$$
I initially tried to... | If we let $r_1, .. r_n$ be the roots of $a_nx^n + .. + a_0$.
$(a_nx^n + .. + a_0)/(x - r_1) = a_nx^{n - 1} + (r_1a_n + a_{n - 1})x^{n - 2} + (r_1^2a_n + r_1a_{n - 1} + a_{n - 2})x^{n - 3} + .. + (r^{n - 2}_1a_n + .. + a_2)x + (r_1^{n - 1}a_n + .. + a_1)$
By inductive hypothesis, we have $\sum_{k = 2}^{n} r_k = -\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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If $\sin(\pi \cos\theta) = \cos(\pi\sin\theta)$, then show ........ If $\sin(\pi\cos\theta) = \cos(\pi\sin\theta)$,
then show that $\sin2\theta = \pm 3/4$.
I can do it simply by equating $\pi - \pi\cos\theta$ to $\pi\sin\theta$,
but that would be technically wrong as those angles could be in different quadrants. So how... | We first rewrite
$$
\cos(\pi/2 - \pi \cos \theta) = \cos(\pi \sin \theta)
$$
(cofunction identity).
We notice that $\cos(x)$ is a periodic function with period $2\pi$, so we need a period offset term to be sure that we find all solutions. We also need to account for the fact that $\cos(x)$ is symmetric, so:
$$
\cos(\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $f(x) = \frac{\cos x + 5\cos 3x + \cos 5x}{\cos 6x + 6\cos4x + 15\cos2x + 10}$then.. If $f(x) = \frac{\cos x + 5\cos 3x + \cos 5x}{\cos 6x + 6\cos4x + 15\cos2x + 10}$ then find the value of $f(0) + f'(0) + f''(0)$.
I tried differentiating the given. But it is getting too long and complicated. So there must be a way... | we can simplify the fraction as $$\frac{2\cos3x\cos2x+5\cos3x}{2\cos^23x-1+6[2\cos3x\cos x]+9\cos2x+10}$$
$$=\frac{(2\cos2x+5)\cos3x}{2\cos^23x+12\cos3x\cos x+18\cos^2x}$$
$$=\frac{(2\cos2x+5)\cos3x}{2(\cos3x+3\cos x)^2}$$
$$=\frac{[2(2c^2-1)+5](4c^3-3c)}{2(4c^3)^2}$$
$$=\frac{(4c^2+3)(4c^2-3)}{32c^5}$$
$$=\frac 12\sec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 0
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Formula for proportion of entropy Let's say we have a probability distribution having 20 distinct outcomes. Then for that distribution the entropy is calculated is $2.5$ while the maximal possible entropy here is then of course $-\ln(\frac{1}{20}) \approx 3$.
How can I describe that $2.5$ is quite a high entropy given ... | First of all, I would use log base 2 instead of natural log because it's easier to talk about its meaning as the number of yes/no questions on average to guess the value.
Given 20 choices, the maximum entropy distribution has entropy of 4.322 bits. While your distribution has 3.607 bits, which is 83% of the maximum pos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find curve parametrization I am asked to find the work of $f(x, y, z) = (x, z, 2y)$ through the curve given by the intersection of two surfaces. I have been doing a series of exercises on this and my question has simply to do with the parametrization of the curve.
The two surfaces are:
$\{(x, y, z) \in R^3 : x = y^2 + ... | The intersection of the two surfaces is given by the equation : $$y^2+z^2-x=3-2y-x$$
or, $$z^2+y^2+2y-3=z^2+(y+1)^2=4$$ which is the equation of a circle in the yz-plane.
We have the parametrization $$z=2cost$$, $$y=-1+2sint$$ and $$x=5-4sint$$
Finaly : $\gamma(t)=(5-4sint,-1+2sint,2cost)$ for $t\in[0,2\pi]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Trigonometric and exp limit
Evaluation of $$\lim_{x\rightarrow \frac{\pi}{2}}\frac{\sin x-(\sin x)^{\sin x}}{1-\sin x+\ln (\sin x)}$$
Without Using L hopital Rule and series expansion.
$\bf{My\; Try::}$ I have solved it using L hopital Rule and series expansion.
But I did not undersand How can i solve it Without Usi... | It is not a direct answer.
By substition $x-\frac { \pi }{ 2 } =t$ $$f\left( x \right) =\frac { \cos { x } -{ \cos { x } }^{ \cos { x } } }{ 1-\cos { x } +\ln { \cos { x } } } $$
$$\lim _{ t\rightarrow 0 }{ \frac { \sin { \left( \frac { \pi }{ 2 } +t \right) - } { \sin { \left( \frac { \pi }{ 2 } +t \right) } ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Determining a basis for a space of polynomials. Let $V = \mathbb R[x]_{\le 3}$
I have the space of polynomials $U_2 = \{ p = a_0 + a_1x + a_2x^2 + a_3x^3 \in V \mid a_1 - a_2 + a_3 = 0, a_0 = a_1 \}$
I am asked to find a basis, so I proceed by noticing that in $U_2$:
$$a_0 + a_1x + a_2x^2 + a_3x^3 = a_0 (1+x-x^3) + a... | These are both bases. Your basis is $\{1+x-x^3, x^2+x^3\}$; the solution gives the basis $\{1+x+x^2, -1-x+x^3\}$. But each of these is expressible in terms of the other:
\begin{gather*}
1+x+x^2 = (1+x-x^3) + (x^2+x^3),\quad -1-x+x^3 = -(1+x-x^3) \\
1+x-x^3 = -(-1-x+x^3),\quad x^2+x^3 = (1+x+x^2) + (-1-x+x^3).
\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is there a trick to drawing cubic equation graphs? This year I started drawing out cubic graphs on graph paper...My teacher have been giving a ton of practices.However,I find that it is very difficult to connect the points of the graph,especially the curves...
So,is there a trick to drawing out on graph paper easily?
O... | It's better to locate the point of inflection first:
Say for a graph of $f(x)=ax^3+bx^2+cx+d$,
$$f''\left( -\frac{b}{3a} \right)=0$$
The graph has a rotation symmetry about the point of inflection namely
$$\left( -\frac{b}{3a}, d-\frac{bc}{3a}+\frac{2b^3}{27a^2} \right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $y(2) $ given $y(x)$ given a separable differential equation Find what $y(2)$ equals if $y$ is a function of $x$ which satisfies:
$x y^5\cdot y'=1$ given $y=6$ when $x=1$
I got $y(2)=\sqrt{6\ln(2)-46656}$
but this answer is wrong can anyone help me figure out the right answer and how I went wrong?
| This equation is again separable (you would benefit from learning how to solve separable equations as they are the most simple class of differential equations to solve).
$$
\frac{dy}{dx}xy^5=1\Rightarrow y^5dy=x^{-1}dx\Rightarrow \frac{y^6}{6}=\ln(x)+c
$$
Using boundary condition $y(1)=6$ yields
$$
\frac{y^6}{6}=\ln(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
A 'bad' definition for the cardinality of a set My set theory notes state that the following is a 'bad' definition for the cardinality of a set $x:$ $|x|=\{y:y\approx x\}$ $(y\approx x\ \text{iff} \ \exists\ \text{a bijection}\ f:x\rightarrow y )$
The reason this is a 'bad' definition is since if $x\neq \emptyset$ t... | It is true that $\bigcup ON=ON$. However, it is not true that $\bigcup On\subseteq\vert x\vert$ - rather, each ordinal is an element of some element of $\vert x\vert$, that is, $ON\subseteq \bigcup \vert x\vert$. This is still enough to get a contradiction, though, by the union axiom.
If $x$ is not finite - no problem!... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
properties of distributions If $$\int_{-\infty}^\infty f dx = 1$$, with $f > 0 \forall x$, then prove or disprove: $$\int_{-\infty}^\infty \frac{1}{1 + f} dx $$ diverges.
The hint I got is to consider the measure of the set$(x:f > 1)$. May be the measure is zero thereby ensuring the divergence of integral?
| Suppose that $f$ is nonnegative (you say it's a distribution?). Notice that $1=\frac{1}{1+f}+\frac{f}{1+f}$. Also $0\leq \frac{f}{1+f}\leq f$ which means $\frac{f}{1+f}$ is integrable. Since the l.h.s. is not integrable, it follows that $\frac{1}{1+f}$ is not integrable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
A very curious rational fraction that converges. What is the value? Is there any closed form for the following limit?
Define the sequence
$$ \begin{cases}
a_{n+1} = b_n+2a_n + 14\\
b_{n+1} = 9b_n+ 2a_n+70
\end{cases}$$
with initial values $a_0 = b_0 = 1$. Then $\lim_{n\to\infty} \frac{a_n}{b_n} = ? $
The lim... | Define
$$ L = \lim_{n \to \infty} \frac{a_n}{b_n} $$
$$ r = \lim_{n \to \infty} \frac{a_{n+1}}{b_n} $$
$$ s = \lim_{n \to \infty} \frac{b_{n+1}}{b_n} $$
then it's not hard to see that $L = r/s$. Also, by substituting in the recursion, since we have $b_n \to \infty$ we can compute
$$ r = \lim_{n \to \infty} \left(1 + 2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 5,
"answer_id": 2
} |
Thinking about $ax + by = ab$ This is embarrassing because I know this is fairly simple, but I'm hitting a mental block and not having much luck with any references I'm aware of.
What properties does $ax + by = ab$ have? (a and b are integers)
I guess I'm sort of thinking...
*
*Is there always a solution? (I think s... | $1.$ There is always a solution: we can write the greatest common denominator as $ax+by$, hence we just need to scale that solution in order to get a solution for $ab$. (If $a, b$ are zero it is trivial to see there is a solution)
$2.$ The number of solutions is infinite. Let $ax+by = \gcd(a, b)$. One of $a$ or $b$ mus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Summation of series involving factorials. I got this question in a maths contest archive and I am completely clueless over how to start.
$$\sum_{m=0}^q(n-m){(p+m)!\over m!}= {(p+q+1)! \over q!}\left(\frac{ n}{ p+1}-\frac {q}{p+2}\right)$$
I thought of transforming $\frac{(p+m)!}{m!}$ into $p+m\choose m$ by multiplying ... | For $q=0$ you have only the term $m=0$,$n\frac{p!}{0!}=n p!$ on the left side of the equation. On the right side you have $\frac{(p+1)!}{0!}\frac{n}{p+1}$. Noting that $(p+1)!=p!(p+1)$ you get the left hand side term.
We now prove by induction. We assume that the given equation is valid for $q$, and we want to prove th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
An example of a power series that has a radius of convergence of 3 The problem states "Give an example of a power series $\sum^{\infty}_{n=0}$a$_{n}$z$^{n}$ that has a radius of convergence of 3 and that represents an analytic function having no zeroes. I'm sorry if this is a little simplistic but I really can't think ... | Take any series you are familiar with and has a finite radius of convergence $r$. Then rescale the argument to be $\dfrac{3z}r$. This multiplies all terms by $\dfrac{3^n}{r^3}$ and yields the desired radius.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Let $f = 2x^4 + 2(a - 1)x^3 + (a^2 + 3)x^2 + bx + c.$ ,Find out $a, b, c ∈ R$ and its roots knowing that all roots are real.
Let $f = 2x^4 + 2(a - 1)x^3 + (a^2 + 3)x^2 + bx + c.$
Find out $a, b, c ∈ R$ and its roots knowing that all roots are real.
The first thing that came into my mind was to use vieta's formula... | As you said, $f''$ has complex roots which shows that double derivative is always positive. So this implies that $f'$ has only one real root because it's always a increasing function. From this you can conclude that $f$ will have either 2 coincident roots ($p,p,q,q$) or all 4 coincident roots ($p,p,p,p$) or no real roo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why $f:R^2\to R$, with $\mbox{dom} f=R^2_+$ and $f(x_1,x_2) =x_1x_2$ is quasiconcave? Why $f:R^2\to R$, with $\mbox{dom} f=R^2_+$ and $f(x_1,x_2) =x_1x_2$ is quasiconcave?
I have tried to use Jensen eniquality to check that superlevel set $\{x\in R^2_+ | x_1x_2 \ge \alpha\}$ is convex.
$$\begin{align*}(\theta x_1 + (1-... | Let $(x_1,y_1),\, (x_2,y_2) \in \{x,y\in\mathbb{R}_+: (x,y)\ge \alpha\}$. Then $y_1 \ge {\alpha \over x_1}$ and $y_2 \ge {\alpha \over x_2}$. Since ${1\over x}$ is convex. $${\alpha \over \theta x_1 + (1-\theta)x_2} \le \theta {\alpha \over x_1} + (1-\theta){\alpha \over x_2}$$ Then $${\alpha \over \theta x_1 + (1-\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $\int\limits_a^b |f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt$
Let $f:[a,b]\to\mathbb{R}$ be continuously differentiable. Suppose $f(a) = 0$. Show that
$$
\int\limits_a^b|f(t)|dt \leq (b-a)\int\limits_a^b|f'(t)|dt
$$
By the mean value theorem, for every $t\in[a,b]$, there exists $c\in[a,t]$ such that
$$
\f... | I think the mean value theorem is a no-go here. If you start with (just the fundamental theorem of calculus)
$$
f(t)=f(a)+\int_a^t f'(s)\,ds=\int_a^t f'(s)\,ds,
$$
you find with the triangle inequality that
$$
|f(t)|\leq \int_a^t |f'(s)|\,ds.
$$
Next, integrate this inequality from $a$ to $b$:
$$
\int_a^b |f(t)|\,dt\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding axis of a cylinder I have to find axis of a cylinder that has the top in the origin and the points $A(-5,6,-4),B(-4,-1,2),C(-1,2,4)$ lie on its lateral area.
Now I know that points A,B,C have the same distance to the axis, but I don't know how I could find this distance ( radius).
I would really appreciate an... | So, the axis is through the origin. Place a unit vector there (two variables to fix) and impose that its cross-product with the position vectors for $A,B,C$ be of the same modulus, which will be distance $r$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving $\sum\limits_{k=0}^n \sum\limits_{j=0}^{n-k} \frac{(k-1)^2}{k!} \frac{(-1)^j}{j!} =1$ without character theory
Let $n \geq 2$ be an integer. I would like to prove the following identity in an easy way:
$$\sum\limits_{k=0}^n \left( \frac{(k-1)^2}{k!} \sum\limits_{j=0}^{n-k} \frac{(-1)^j}{j!} \right)=1$$
You ... | We may start from:
$$ \sum_{k\geq 0} \frac{(k-1)^2}{k!}x^k = (1-x+x^2)\,e^{x}\tag{1} $$
$$ \sum_{k\geq 0}\left(\sum_{j=0}^{k}\frac{(-1)^j}{j!}\right) x^k = \frac{e^{-x}}{1-x}\tag{2} $$
then notice that the original sum is just the coefficient of $x^n$ in the product between the RHSs of $(1)$ and $(2)$, i.e.
$$ [x^n]\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
How can I solve this inequality? Have a nice day, how can I solve this inequality?
$$a<b<-1$$
$$ |ax - b| \le |bx-a|$$
what is the solution set for this inequality
| There may be more efficient ways to do these but for clarity I like to break absolute values into cases:
Case 1: $ax -b \ge 0$ and $bx -a \ge 0$.
[This implies $ax \ge b\implies x \le b/a$ and likewise $x \le a/b$ so $x \le \min(a/b,b/a) = b/a < 1$. Let's keep in mind $b/a < 1 < a/b$]
Then $|ax - b| \le |bx -a| \impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that the number: $z = \det(A+B) \det(\overline A-\overline B)$ is purely imaginary. Problem: Let $A_{n\times n}$ and $B_{n\times n}$ be complex unitary matrices, where n is an odd number. Prove that the number:
$$z=\det(A+B) \det(\overline A-\overline B)$$ is purely imaginary.
My idea:
We have that $AA^*=I$ and $... | As proceeding along the lines given in the hint by @Tsermo , we find that $$ z = det(BA^{*} -AB^{*}) =\Pi_i^{n} e_i $$ where $e_i$ are the eigenvalues of $(BA^{*}-AB^{*}) $ which are purely imaginary(as the matrix$(BA^{*}-AB^{*}) $ is skew symmetric of odd order).Since n is odd therefore the result follows.
By the way ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Differentiability of piecewise functions Check whether the function is differentiable:
$$f:\mathbb{R}^2\rightarrow \mathbb{R}$$
$$f= \begin{cases}
\frac{x^3-y^3}{x^2+y^2} & (x,y)\neq (0,0) \\
0 & (x,y) = (0,0) \\
\end{cases}
$$
So what I did is I calculated the partial derivatives of the function in poi... | This is wrong. Being partially differentiable means that the partial derivatives exist, and you have shown this by showing the limits to exist. The partial derivatives need not coincide! To show that $f$ is differentiable a sufficient conditon is that the partial derivatives exist and are continous. To show that $f$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
I want to show that $\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$ I want to show that
$$\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$$
Expand $(x^4-x+\pi)^2=x^4-2x^3+2x^2-2x\pi+\pi{x^2}+\pi^2$
Let see (substitution of $y=x^2$)
$$\int_{-\infty}... | Elaborating user @Dr. MV's answer, we have
\begin{equation}
\int_0^\infty\frac{1}{a^2x^4+bx^2+c^2}\ dx=\frac{c\pi}{2a\sqrt{b+2ac}}
\end{equation}
Putting $a=1$, $b=a$, and $c^2=b$, then
\begin{equation}
I(a,b)=\int_0^\infty\frac{1}{x^4+ax^2+b}\ dx=\frac{\pi}{2}\sqrt{\frac{b}{a+2\sqrt{b}}}
\end{equation}
Hence
\begin{eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 3
} |
How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$? Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$.
One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$. To show $b^m>1$, you can use induction, and ... | If
$b > 1$,
then
$b = 1+c$
where
$c > 0$,
so
$b^n
=(1+c)^n
\ge 1+nc
\gt 1
$
by
Bernoulli's inequality.
If
$b^{n/m} = a$,
then
$b^n = a^m$.
Since
$b^n > 1$,
$a^m > 1$
so that
$a > 1$.
Then apply the definition
$b^x
=\sup_{r \le x} b^r
$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Additivity of trace Let $A$ be a finitely generated abelian group and $\alpha:A\to A$ be an endomorphism. Since $A=A_{free}\oplus A_{torsion}$, we can induce $\bar \alpha:A_{free}\to A_{free}$, i.e. $\bar\alpha$ is a map from $\oplus\mathbb Z$ to itself. Write $\bar\alpha$ as a matrix form and define $tr(\alpha)=tr(\b... | In the case of free abelian groups, the sequence $$ 0 \to A \to B \to C \to 0 $$
splits and we obtain $ B \cong A \oplus C $.
Choosing bases of $A$ and $C$ yields a basis of $B$ and we see (implicitly using above isomorphism)
$$ \operatorname{tr} (\beta) = \operatorname{tr}(\beta | A) + \operatorname{tr}(\beta | C)$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Finding the limit of a Matrices determinant The problem is as follows:
I've been trying to figure this out with no luck. I'm lost at the $A_k+1$ and $A_0$. I'm not sure what they are implying and how they would apply in finding the limit.
| Note that it can be easily shown that
$$
A^{-1} \;\; =\;\; \left [ \begin{array}{cc}
2 & -1\\
-1 & 1 \\
\end{array} \right ].
$$
This implies that
$$
A_1 \;\; =\;\; \left [ \begin{array}{cc}
3/2 & 0 \\
0 & 3/2 \\
\end{array} \right ]
$$
Implying that
$$
A_2 \;\; =\;\; \left [ \begin{array}{cc}
3/4 & 0 \\
0 & 3/4 \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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