Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Expected value for the number of tries to draw the black ball from the bag We have a bag with $4$ white balls and $1$ black ball. We are drawing balls without replacement. Find expected value for the number of tries to draw the black ball from the bag.
Progress. The probability to draw a black ball from first trial ... | It is as if you will create a word with $4$ W's and $1$ B. For example $BWWWW$ or $WWWBW$ etc. How many such words can you create? Answer: $5$ and any such word is equally likely.
In other words: the probability that the black ball will be drawn at any place - not only the first - is equal to $1/5$. Not conditional pr... | {
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"timestamp": "2023-03-29T00:00:00",
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What is $\textit{the}$ discriminant of a degree $n$ polynomial? In my high school algebra class the teacher (who is me) says that the discriminant of a quadratic polynomial $ax^2 + bx + c$ is $b^2 - 4ac$.
I have read in the Wikipedia article that the discriminant of a polynomial is the product of the squares of the dif... | Wolfram Mathworld says that the most common definition for the discriminant of a polynomial $p(z) = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z +a_0$ is: $$D(p) \equiv a_n^{2n - 2} \prod_{i, j}^n(r_i - r_j)^2, i<j$$ This definition gives, and I quote,
...a homogenous polynomial of degree $2(n - 1)$ in the coeffici... | {
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"url": "https://math.stackexchange.com/questions/1581283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can you compute $\int_0^1\frac{\log(x)\log(1-x)}{x}dx$ more precisely than $1.20206$ and do a comparision with $\zeta(3)$? I know from Wolfram Alpha that $$\int_0^1\frac{\log(x)\log(1-x)}{x}dx=1.20206$$ and in the other hand, too from this online tool that
$$\int\frac{\log(x)\log(1-x)}{x}dx=\mathrm{Li}_3(x)-\mathrm{Li}... | Let $x = e^{-y}$, we have
$$\int_0^1 \frac{\log x\log(1-x)}{x} dx
= \int_0^1 \frac{(-\log x)}{x} \sum_{n=1}^\infty \frac{x^n}{n} dx
= \sum_{n=1}^\infty \frac{1}{n}\int_0^1 (-\log x) x^{n-1} dx\\
= \sum_{n=1}^\infty \frac{1}{n}\int_0^\infty y e^{-ny} dy
= \sum_{n=1}^\infty \frac{1}{n^3}
= \zeta(3)
$$
Please note that we... | {
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"url": "https://math.stackexchange.com/questions/1581354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
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If $u$ is a unit in $S$, then $u$ is a unit in $R$ I think the proof is fairly simple...but of course one of my big problems in math is overdoing simple problems. So..
Let $R$ be a ring with identity, and let $S<R$ be a subring containing the identity. Prove that if $u$ is a unit in $S$, then $u$ is a unit in $R$. S... | For the first part: That’s not quite correct. You need to argue that $v ∈ R$ rather than $1 = uv ∈ R$.
For the counterexample: I think you should rather disprove that
Whenever $S ⊂ R$ is a subring of a ring $R$, and $u ∈ S$ is a unit in $R$ (that is there is some $v ∈ R$ with $uv = 1$ in $R$) then $u$ is also a unit i... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof of Euclid's Lemma in N that does not use GCD I am looking for a proof of Euclid's Lemma, i.e if a prime number divides a product of two numbers then it must at least divide one of them.
I am coding this proof in Coq, and i'm doing it over natural numbers. I aim to prove the uniqueness of prime factorization (So I... | Claim 2 below should answer the question.
Since the only unit in $\mathbb{N}$ is $1$, we have
$p$ is prime iff $p\mid ab\implies p\mid a\lor p\mid b$
$p$ is irreducible iff $a\mid p\implies a=1\lor a=p$
Claim 1: $p$ is prime $\implies$ $p$ is irreducible
Proof: Assume that $a\mid p$. For some $b$, we have
$$
p=ab\tag... | {
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Why is the number of divisors of an integer $n$ equal to the number of factors in the factorization of the polynomial $x^n - 1$ over the integers? Sloane's OEIS A000005 gives the number of divisors of the integer n. A comment (by a very reputable contributor) in this sequence claims that this is also the number of fac... | The idea is to construct a bijection between divisors $d$ of $n$ and irreducible factors $F$ of $X^n-1$.
Given a divisor $d$, let $F$ be the $d$-th cyclotomic polynomial (the minimal polynomial of a primitive $d$-th root of unity). Then $F$ is irreducible, and $F \mid X^d - 1 \mid X^n - 1$.
Given an irreducible factor ... | {
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Information from a characteristic polynomial
(III) I know that since the highest multiplicity is 4, that the largest possible dimension is 4.
(IV) I believe that there are 3 distinct eigenvalues, but because of the multiplicities, there are exactly 7 eigenvalues?
I don't know how to tell if it's invertible, or the siz... | (I): $A$ can't be invertible because $0$ is an eigenvalue.
(II): $A$ must be $7 \times 7$ because its characteristic polynomial has degree $7$.
(III): Your answer is correct
(IV): I don't think that they mean to count eigenvalues up to multiplicity. Yes, $A$ has exactly $3$ distinct eigenvalues.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dimension of this subspace of $\mathbb R^4$ generated by $\{v_{i1},v_{i2},v_{i3},v_{i4}\}$ with $\sum_{i=1}^4 v_{ij}=0$ for $i,j\in\{1,2,3,4,\}$ Let $$v_i=(v_{i1},v_{i2},v_{i3},v_{i4}),\ \ for\ \ i=1,2,3,4$$ be four vectors in$\mathbb R^4$ such that $$\sum_{i=1}^4v_{ij}=0\ \ for\ \ each\ \ j=1,2,3,4.$$ $W$ be the subs... | Take the four vectors $(1,0,0,0), (0,1,0,0), (0,0,1,0)$ and $(-1,-1,-1,0) .$ These four satisfy the given conditions and the first three are linearly independent . Hence the answer will be $B.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1581832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Measurability for discrete measure Let $(X,\mathcal{B},\mu)$ be a measure space with discrete measure $\mu$ and Borel sigma-algebra. Why can we claim that all real functions on $X$ are measurable?
| Strictly speaking, it is not true. In fact, measurability of a function does not depend on the measure $\mu$, it depends only on its sigma-algebra (i.e. family of measurable sets).
For example, take Borel sigma-algebra on $\mathbb R$, and define $\mu(X) = 1$ when $0 \in X $ and $\mu(X) = 0$ otherwise. The measure $\mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1581941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Kernel and Image I have to find the kernel for this transformation:
$$T(p(x))=xp(1)-xp'(1)$$
My solution so far:
$p'(1)=0$
Hence $T(p(x))=xp(1)$
So $xp(1)=0$
Either $x=1$ or $p(1)=0$?
How do you solve the kernel from this point?
Thank you!
| A polynomial $p(x)$ is in the kernel of $T$ iff it is mapped to the zero polynomial $q(x)=(0,0)^T$.
So, in order to be in the kernel, $(Tp)(x)=xp(1) - xp'(1)= x(p(1) - p'(1))$ needs to be equal to $q$ for all $x \in \mathbb R$. This is only possible if $p(1)=p'(1)$.
For $p(x)=ax^2 + b x + c$, which values of $a,b,c\in... | {
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"timestamp": "2023-03-29T00:00:00",
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Errors in math research papers Have there been cases of errors in math papers, that were undetected for so long, that they caused subsequent errors in research, citing those papers. ie: errors getting propagated along. My impression is that this type of thing is extremely rare.
What was the worst case of such a scenari... | One egregious case recently analyzed in detail by Adrian Mathias is Bourbaki's text Theory of sets and a couple of sequels published by Godement and others. Mathias' paper is:
Mathias, A. R. D. Hilbert, Bourbaki and the scorning of logic. Infinity and truth, 47–156, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Sing... | {
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Solution to the recurrence relation I came across following recurrence relation:
*
*$T(1) = 1, $
*$T(2) = 3,$
*$T(n) = T(n-1) + (2n-2)$ for $n > 2$.
And the solution to this recurrence relation is given as
$$T(n)=n^2-n+1$$
However I am not able to get how this is obtained.
| $C=const$ is the solution of the homogeneous equation $T(n)=T(n-1)$ and $n^2-n$ is a solution of the inhomogeneous equation since $n^2-n=(n-1)^2-(n-1)+2n-2$
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Set questions(discrete mathematics) Let $A$, $B$ and $C$ be sets.
Prove or disprove (with a counter example) each of the following:
(a) If $A / C = B / C$ then $A = B$.
(b) If [($A \cap C = B \cap C)\&( A / C = B/ C)$] then $A = B$.
(c) If [$(A \cup C = B \cup C)\&(A / C = B / C)$] then $A = B$.
I understand why they ... | Hint
For proving equality, apply the definitions:
(i) $A \setminus C = B \setminus C$ iff $x \in A \setminus C \leftrightarrow x \in B \setminus C$, for any $x$
and :
(ii) $x \in A \setminus C$ iff $x \in A$ and $x \notin C$.
Note : for $A \setminus C$, note that it is not necessary that the "subtracted" set (in th... | {
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"url": "https://math.stackexchange.com/questions/1582239",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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do infinite family of lines in $\mathbb{R}^2$ have a common point by knowing that any three of them have common point? Suppose we have given an infinite family of lines; say $\mathfrak{F}$, in the plane $\mathbb{R}^2$ such that any three of the lines in $\mathfrak{F}$ have a common point. How can we prove that all line... | Note that any two distinct lines in the family have at most one point in common. Take any three distinct lines $l_1,l_2,l_3$ from the family, and let $v_0$ be their common point. Take any $l$ in the family distinct from these, and consider $\{l_1,l_2,l\}.$
Added: This result is actually true in all $\Bbb R^n$ (vacuousl... | {
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"timestamp": "2023-03-29T00:00:00",
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find $\lim _{x\to \infty }\left(\frac{n^{k-1}}{n^k-\left(n-1\right)^k}\right)=\frac{1}{2005}$ the value of k $\lim _{x\to \infty }\left(\frac{n^{k-1}}{n^k-\left(n-1\right)^k}\right)=\frac{1}{2005}$
What is the value of k in the given expression?
When I expand the term (n-1) to the k-th power I get n^k-n^k..
| Using Taylor expansions, assuming $k \geq 1$ (so that things do tend to $\infty$ when $n\to\infty$):
$$\begin{align}
n^k - (n-1)^k &= n^k\left(1-\left(1-\frac{1}{n}\right)^k\right) = n^k\left(1-\left(1-\frac{k}{n} + o\left(\frac{1}{n}\right)\right)\right) \\&= kn^{k-1} + o\left(n^{k-1}\right)
\end{align}$$
so
$$
\frac{... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluating the Definite Integral $\int_{0}^{1}\frac{2 \sin \pi x \cos \pi x}{1+x^2}dx$
How can I find this integral
$$I=\int_{0}^{1}\frac{2 \sin \pi x \cos \pi x}{1+x^2}dx$$
Any trick that could compute the definite integral is acceptable. However, it will be more challenging to find a primitive.
Any hint or help i... | Let's try your last idea and start with :
$$J(\alpha):=\int_{0}^{1}\frac{\sin \alpha x}{1+x^2}dx$$
Then $\;\displaystyle J''(\alpha)=-\int_{0}^{1}\frac{x^2\;\sin \alpha x}{1+x^2}dx\;$ and we obtain this ODE :
$$J(\alpha)-J''(\alpha)=\frac {1-\cos(\alpha)}{\alpha}$$
The solution of the homogeneous ODE is simply $\;J(x)=... | {
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"timestamp": "2023-03-29T00:00:00",
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Division in finite fields Let's take $GF(2^3)$ as and the irreducible polynomial $p(x) = x^3+x+1$ as an example. This is the multiplication table of the finite field
I can easily do some multiplication such as $$(x^2+x)\cdot(x+1) = x^3 + x^2 + x +1 = x+1+x^2+x+2 = x^2$$
I am wondering how to divide some random fields s... | If we have a field $K = F[X]/p(X)$, then we can compute the inverse of $\overline{q(X)}$ in $K$ as follows.
Since $p$ is irreducible, either $q$ is zero in $K$ or $(p,q)=1$. By the Euclidean algorithm, we can find $a,b$ such that $a(X)p(X) + b(X)q(X) = 1$. Then $\overline{b(X)} \cdot \overline{q(X)} = \overline{1}$, s... | {
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"timestamp": "2023-03-29T00:00:00",
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Series Solution to $y''+xy=e^x$ I am thoroughly familiar with using power series to solve the differential equation $y''+xy=0$, but how exactly does one go about solving $y''+xy=e^x$?
I would imagine you represent $e^x$ as it's power series, along with everything else first, but then what?
| Just looking at
$y''+xy=e^x$
and ignoring the part
about power series,
I would first let
$y = ze^x$.
Then
$y'
=e^x(z+z')
$
and
$y''
=e^x((z+z')+(z'+z''))
=e^x(z+2z'+z'')
$
so the equation becomes,
dropping the $e^x$,
$1
=z''+2z'+z+xz
=z''+2z'+z(1+x)
$.
From this,
we can readily get
a recurrence for
the coefficients of
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intermediate Value Theorem on $\mathbb{R^n}$ Let $S^2$ denotes the subset of $\mathbb{R^3}$ which includes the points
$(x,y,z)$ s.t $x^2+y^2+z^2=1$
i.e the boundary of a unit sphere. Let $f$ be a continuous function from $S^2$ to $\mathbb{R}$ Prove there exists$ p=(x,y,z) $ s.t $f(p)=f(-p)$
Exam hint: The intermediat... | Think about the function $g(x)=f(x)-f(-x)$. Now the goal of the problem is to show that there is some $P\in S^2$ such that $g(P)=0$. if $g$ is identically $0$ then there is nothing to prove. Otherwise let $Q\in S^2$ be a point such that $g(Q)\geq 0$. then $g(-Q)=f(-Q)-f(Q)=-g(Q)$. Now you can apply the intermediate val... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Method of cylindrical shells Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
$y=32-x^2, \ y=x^2$ about the line $x=4$
My confusion is that what will be the radius 'x' of cylinder shells which we have to put in the integral... | Cylindrical shell: consider the volume element
$$ dV = 2 \pi h\,dr = 2\pi\,2y\,dx = 4\pi\,x^2dx
$$
Then integrate
$$ V = \int_{x=-4}^4 dV = 4\pi \int_{-4}^4 x^2dx
$$
Cylindrical disk: Cut your solid in half so that you only have to consider the bottom part. Then integrate the volume element
$$ dV = \pi x^2 dy = \pi (2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1582946",
"timestamp": "2023-03-29T00:00:00",
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why $ 1 - \cos^2x = \sin^2x $? I'm trying to prove this result $$\lim_{x\to 0} \frac{1 - \cos(x)}{x} = 0$$ In this process I have come across an identity $1-\cos^2x=\sin^2x$. Why should this hold ? Here are a few steps of my working:
\begin{array}\\
\lim_{x\to 0} \dfrac{1 - \cos(x)}{x}\\ = \lim_{x\to 0} \left[\dfrac{1... | Here is another simple way to prove the trig. identity, we know $i^2=-1$ so we have
$$\sin^2 x+\cos^2x$$$$=\cos^2x-i^2\sin^2x$$ $$=(\cos x+i\sin x)(\cos x-i\sin x)$$
using Euler's theorem, $$=(e^{ix})(e^{-ix})=e^{0}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
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Can a point and a compact set in a Tychonoff space be separated by a continuous function into an arbitrary finite dimension Lie group? Given a topological space $X$ which is Tychonoff (i.e., completely regular and Hausdorff), we know that given a compact set $K\subseteq X$ and a point $p \in X$ with $p\not\in K$, we ca... | As far as I remember, each Lie group $M$ is locally Euclidean, so there is a homeomorphic embedding $i:\Bbb R\to M$, which yields the required separating map $f'=i\circ f$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the length of PC Here PE is the tangent of the two circle. PA = 12 ; CD/AB = 2
Find the length of PC [Source: BDMO]
]1
| It holds that $$ PE^2=
\boxed{PA \cdot PD = PB \cdot PC}.$$
Taking advantage of the equality in the box, we have:
$\begin{array}[t]{l}
12\cdot (PC + CD) = (12+AB) \cdot PC\\
12 PC +12 CD = 12 PC + AB \cdot PC\\
PC = \dfrac{12CD}{AB}=24
\end{array}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find base of exponentiation Given the two primes $23$ and $11$, find all integers $\alpha$ such that $\alpha^{11} \equiv 1 \mod 23$.
How to compute this? What to use?
| As $\phi(23)=22$
and if ord$_{23}\alpha=a, a|22\implies a=1,2,11,22$
Now $2^2\equiv4,2^5\equiv9\implies2^{11}=2(9)^2\equiv1\implies$ord$_{23}2=11$
We know, ord$_ma=d,$ ord$_m(a^k)=\frac{d}{(d,k)}$ (Proof @Page#95)
So, ord$_{23}(2^k)=\dfrac{11}{(11,k)}$
$\implies$ord$_{23}(2^k)=\dfrac{11}{(11,k)}=11$ for $1\le k\le10$
a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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summation of determinants of $3\times3$ matrices I have an algebra problem but no idea how to solve it. The problem is: "you can create 9! matrices the elements of which lie in a set $ \{1,2,3,...,9\} \subset \mathbb N$ so that their elements do not repeat, i.e. e.g.
$$
\begin{pmatrix}1&2&9\\3&5&7\\6&4&8 \end{pmatrix}
... | The number of those matrices is even, and note that the sum of the determinants of the pair $\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}$ and $\begin{pmatrix}b&a &c\\e&d&f\\ h&g& i\end{pmatrix}$ is 0. If we pair up these matrices in this way we can see that the required sum is 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Evaluate:$\int_{l}(z^2+\bar{z}z)dz$
Evaluate:
$\displaystyle=\int_{l}(z^2+\bar{z}z)dz\,\,\,\,\,\,\,\,\,\,l:|z|=1,0\leq\arg z\leq\pi$
My try:
$$\int_{l}(z^2+\bar{z}z)dz=\int_{0}^{\pi}(r^2e^{2i\theta}+re^{i\theta}???)dz$$
I'm stuck here, hints?
| Since the integral equals $\int_l (z^2+1)\, dz,$ and this integrand has the antiderivative $z^3/3 + z$ (in all of $\mathbb C$), all you need to do is evaluate $z^3/3+z$ at the end points and subtract. (All we're using is the FTC of ordinary calculus here.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why if $K$ is a finite field, $|K|=p^d$ for a prime $p$? Why if $K$ is a finite field, $|K|=p^d$ for a prime $p$ ?
The solution goes like :
Consider $\varphi:\mathbb Z\longrightarrow K$. Since $K$ is finite $\ker \varphi=p\mathbb Z$ for a prime $p$.
Q1) Why $\ker\varphi=p\mathbb Z$ ? And why $p$ has to be prime ? I try... | *
*For any ring $R$, we can define $$\phi: \Bbb Z \to R$$ to be the homomorphism $$n \mapsto n \cdot 1 = \underbrace{1 + \cdots + 1}_n .$$
(Presumably this is the unspecified map $\varphi$ in the question.) Since $\Bbb Z$ is a principal ideal domain, $\ker \phi = n \Bbb Z$ for some $n$ (in fact, there are two choices ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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First order ODE $(y^2\sqrt{x-x^2y^2}-y)dx - 2xdy=0$ $(y^2\sqrt{x-x^2y^2}-y)dx - 2xdy=0$
Change it into this $$y'=\frac{y^2\sqrt{x-x^2y^2}-y}{2x}$$
Square root disables a lot of methods.
It isn't a total differential, I've tried. Quasi-homogeneous $y=z^a$ is only thing I haven't ruled out but I don't know how to deter... | All first order equation of the first $M(x,y)dx+N(x,y)dy=0$ can be integrated, provided that you can find the right integrating factor. For this equation $(y^2\sqrt{x-x^2y^2}-y)dx - 2xdy=0$, the integrating factor is
$$ \frac{1}{xy^2\sqrt{x-x^2y^2}}.$$
Multiplying both sides with this integrating factor, we get
$$ \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Contour integral. Consider the function $y(x)$ defined by $$y(x)=e^{x^2}\int_{C_1'}\frac{e^{-u^2}}{(u-x)^{n+1}}du$$where $C_1'$ is as shown
The Author makes following claims regarding the behavior of $y(x)$ in the limit of large $x$ (It is assumed that $n>-\frac{1}{2}$, but not integral).
1) As $x\rightarrow+\infty$, t... | In my point of view the better way to understand behavior of this function is to make changes of variables $v=u+x$.
Thus $y(x)$ has the following form
$$
y(x)=e^{x^2}\int_{C''}\frac{{e^{-(v+x)^2}}}{v^{n+1}}dv
$$.
Since $n+1$ is not integer one can represent it in terms of usual (not contour) integral
$$
y(x)=(1-e^{-2i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1583832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What is the probability that, at the end of the game, one card of each color was turned over in each of the three rounds?
Three players each have a red card, blue card and green card. The players will play a game that consists of three rounds. In each of the three rounds each player randomly turns over one of his/her ... | The probability that the three players choose different colors on the first turn is $\frac{3}{3}\cdot\frac{2}{3}\cdot\frac{1}{3}=\frac{2}{9}$. Given that they do, consider the generating function for the colors played on the second turn: $(r+b)(b+g)(r+g)=\color{red}1b^2 2g+\color{red}1b^2 r+\color{red}1b g^2+\color{gre... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does a zero eigenvalue mean that the matrix is not invertible regardless of its diagonalizability? If the matrix $A$ is diagonalizable, then we know that its similar diagonal matrix $D$ has determinant $0$, so the matrix $A$ itself is invertible? However, if $A$ not diagonalizable, how are we sure that the matrix $A$ w... | If $0$ is an eigenvalue, then there is a nonzero vector $v$ with $Mv = 0$. Then the kernel of $M$ is not trivial (it is at least one-dimensional), and so it is not one-to-one viewed as a linear transformation. Then it is not invertible.
The geometric picture: $M$ 'collapses' the subspace spanned by $v$, and so maps its... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1584033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Writing a summation as the ratio of polynomial with integer coefficients
Write the sum
$\sum _{ k=0 }^{ n }{ \frac { { (-1) }^{ k }\left( \begin{matrix} n \\ k \end{matrix} \right) }{ { k }^{ 3 }+9{ k }^{ 2 }+26k+24 } } $ in the form $\frac { p(n) }{ q(n) }$, where $p(n)$ and $q(n)$ are polynomials with integral c... | $\sum _{ k=0 }^{ n }{ \frac { { (-1) }^{ k }\left( \begin{matrix} n \\ k \end{matrix} \right) }{ { k }^{ 3 }+9{ k }^{ 2 }+26k+24 } } = \sum _{k=0}^{n} {\frac { (-1)^k \dbinom{n}{k}}{2(k+2)} }-\sum _{k=0}^{n} {\frac { (-1)^k \dbinom{n}{k}}{(k+3)} }+\sum _{k=0}^{n} {\frac { (-1)^k \dbinom{n}{k}}{2(k+4)} }$
Now consider... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1584105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivative of position [Beginning calculus question.] I saw in a calculus lecture online that for a position vector $\boldsymbol{r}$
$$\left|\frac{d\boldsymbol r}{dt}\right| \neq
\frac{d\left| \boldsymbol r \right|}{dt}$$
but I don't understand exactly how to parse this.
It's my understanding that:
*
*$\frac{d\bo... | We can understand this with one example. Consider angular motion problem where position of a particle is given by $\mathbf{r}=\hat{\imath}\cos(\omega t)+\hat{\jmath}\sin(\omega t)$. Now $\frac{d\mathbf{r}}{dt}=-\hat{\imath}\omega\sin(\omega t)+\hat{\jmath}\omega \cos(\omega t)$, clearly $\lvert \frac{d\mathbf{r}}{dt}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1584181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve the double integral $\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)dxdy\:$ $$\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy.$$
I think you need to be solved by the transition to polar coordinates:
\begin{cases}
x=r\cos(\phi),\\
y=r\sin(\phi)
\end{... | Suppose $x=r\cos(\phi)$ and $y=\frac{1}{2}r\sin(\phi)$. Then the Jacobian is $\frac{1}{2}rdrd\phi$. Then
$$
\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy= \int_{0}^{2\pi}\int_{0}^{1}(...)\frac{1}{2}rdrd\phi.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1584262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that if $\sum_n |a_n|$ converges, then $\sum_n a_n^2$ converges
Show that if $\sum_n |a_n|$ converges, then $\sum_n a_n^2$ converges
whenever all $a_n$ are in $\mathbb{R}$.
Lemma: I have already proved that if $\sum_n a_n$ and $\sum_n b_n$ converge absolutely, then $\sum_{n,m} a_n b_m$ converges absolutely.
Le... | Addressing only question 2: If $\sum|a_n|$ converges, then $|a_n|\to0$ so eventually we must have $|a_n|<1$. Let us assume (Wlog) that $|a_n|<1$ for all $n$. Then $a_n^2=|a_n|^2<|a_n|$ for all $n$ so we can apply the comparison test.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Spatial component of the wave function The spatial component of a wave function is given as
$$\sin\left ( \frac{n \pi x}{L} \right )$$
then for $n=1$,
we get
$$\sin\left ( \frac{ \pi x}{L} \right )$$
and this produces one half cycle of a sine wave over the distance $x=0$ to $x=L$.
It has been a while since I touched w... | First note that for
$$\sin\left(\frac{\pi x}{L}\right)=\sin\left(\frac{2\pi x}{2L}\right)$$
We have
$$ x = \mbox{spatial variable for position}$$
$$ 2L = \mbox{spatial period of the wave}$$
This implies that
$$ L = \mbox{half of the spatial period of the wave}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1584425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Separating differential equatons The initial equation:
$$
y''= g-((C*(y')^2)/m)
$$
and I am trying to separate it into two differential equations.
I also have that the aerodynamic force $F=C*(y')^2$.
The initial equation describes falling motion of a skydiver where I am eventually trying to use python integrate functio... | so I have ended up doing this which I think is write
set z=dy/dx
so z=y'
then
z'=g-((c*z^2)/m)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all $n$ for which $n^8 + n + 1$ is prime Find all $n$ for which $n^8 + n + 1$ is prime. I can do this by writing it as a linear product, but it took me a lot of time. Is there any other way to solve this? The answer is $n = 1$.
| Since $n^2+n+1$ divides $n^8+n+1$ and $1<n^2+n+1<n^8+n+1$ for $n>1$, then $n=1$ is the unique solution (which indeed gives a prime).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Property for comparing two composition series Suppose we have two composition series
$$
M = M_0 \unrhd M_1 \unrhd \cdots \unrhd M_r = 1
$$
and
$$
M = N_0 \unrhd N_1 \unrhd \cdots \unrhd N_s = 1
$$
for a finite group $M$. Then we know that $r = s$. But for given $i$, let $j$ be choosen such that
$$
M_{i-1} \cap N_j \... | To avoid all those suffixes, we have simple subnormal sections $A/B$ and $C/D$ of $G$ with $A \cap D \le B$ and $A \cap C \not\le B$, and we want to prove $B \cap C \le D$.
By the 2nd Isomorphism Theorem $(A \cap C)/(A \cap D) \cong (A \cap C)D/D \le C/D$, and since this quotient is nontrivial, $C/D$ is simple, and $A$... | {
"language": "en",
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How many binary words of length n , that consist an even number of zeros? How many binary words (chars '0' and or '1') of length n that consist an even number of zeros are there?
I know that there are $2^n$ options overall, and that for every $n$, there are $\lceil{\frac{n}{2}}\rceil + 1$ options for even zeros. But no... | Doesn't common sense say that there should be an equal number of strings with an even number of zeros as with an odd number of zeros? Well, not if n is odd I guess. But in that case shouldn't common sense say there are just as many strings with an even number of zeros as there are with an even number of ones (which w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Dual set of the unit ball is part of the unit ball. Define the unit ball centered at the origin as $B=\{x\in\mathbb{R}^d\mid \|x\|\leq 1\}$.
Define the dual set of set $X$ as $X^*=\{y\in\mathbb{R}^d\mid\langle x,y \rangle\leq 1\ \forall x\in X\}$.
I'm attempting to prove that for any set $C\subseteq\mathbb{R}^d$ it hol... | Ok, I think I got it. Thanks to @AhmedHussein for the hint.
Take any $y\in B^*$. If $y=0$, then clearly $y\in B$, so assume otherwise. Notice that $\frac{1}{||y||}y\in B$, and thus by the definition of $B^*$ it holds that $\langle\frac{1}{||y||}y, y\rangle\leq 1$.
Extract the coefficient: $\frac{1}{||y||}\langle y, y\r... | {
"language": "en",
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How do you calculate the exponent of an exponent How do you calculate the exponent of an exponent? In what order do you calculate the exponents?
For example, to calculate
${2^3}^4$
Is it
$({2^3})^4 = 8^4$
or
$2^{3^4} = 2^{81}$
ADDED: Say I'm given $y=x^2$ and then told that $x = m^3$. Can I say that in this case $y... | We usually define the notation $$
x^{y^z}
$$ to mean $$ x^{\left(y^z\right)} $$
Mostly, this is because because this definition is most useful. Note that because of a power rule, if we wanted to write: $$ {\left(x^y\right)}^z$$
Then it's easier to write the equivalent $$ x^{yz}$$
Note that if $x=m^3$ and $y=x^2$, we ca... | {
"language": "en",
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"source": "stackexchange",
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Find the radius of circle $P$
$ABCD$ is a rectangle with $AB = CD = 2$. A circle centered at $O$ is tangent to $BC, CD,$ and $AD$ (and hence has radius $1$). Another circle, centered at $P$, is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$. If line $AT$ is tangent to both circles at $T$, find... | I don't know if you want a picture as an answer, or a solution as an answer, but here's a picture (roughly to scale, but no promises):
| {
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What is the probability of matching exactly 4 numbers? In Canada's national 6-49 lottery, a ticket has 6 numbers each from 1 to 49, with no repeats. Find the probability of matching exactly 4 of the 6 winning numbers if the winning numbers are all randomly chosen.
The answer is $\frac{\binom{6}{4}\cdot\binom{43}{2}}{\b... | If there are 6 winning numbers, then you want to take away the possibility of choosing any of them when you choose the last 2 numbers. So we take away all 6 winning numbers from 49 so we get
$$\frac{{6 \choose 4}{49-6 \choose 2}}{49\choose 6}=\frac{{6 \choose 4}{43 \choose 2}}{49\choose 6}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1585120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exponents and mod (Euler's theorem) I know how to compute $7^{402} \pmod{10}$ using Euler's theorem since $7$ and $10$ are relatively prime.
But is there an easy way without using a calculator to compute $12^{720} \pmod{10}$. I don't think Euler's Theorem can be applied because $12$ and $10$ are not relatively prime...... | First observe $(12)^{720} \equiv (2)^{720} \pmod {10}$ (I think this is obvious).
Anyways we can easily prove it using binomial theorem on $(2+10)^{270}$
Now, try to find $x$ such that $2^{719} \equiv x \pmod 5$. This is easy by Euler's theorem.
$2^{719} \equiv 3 \pmod 5$.
So, $2^{720} \equiv 6 \pmod {10}$.
For your se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1585193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integrating over a tetrahedron Let $S$ be the tetrahedron in $\mathbb{R}^3$ having vertices $(0,0,0), (0,1,2), (1,2,3), (-1,1,1)$. Calculate $\int_S f$ where $f(x,y,z) = x + 2y - z$.
Before I show you guys what I have tried, please no solutions. Just small hints. Now, I have been trying to set up the integral by looki... | Hint to your hint: You can find a linear transformation sending the vectors $(0,0,0)(0,1,2)$ etc to the standard basis (actually, it is easiest to compute the inverse to this first). A linear transformation will change area in a uniform way (think about where arbitrary little cubes are sent), and the bounds after apply... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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It is possible to get a closed-form for $1+2^i+3^i+\cdots (N-1)^i$? Let $i=\sqrt{-1}$ the complex imaginary unit, taking $$arg(2)=0$$ for the definition of the summand $2^i$ in $$1^i+2^i+3^i+\cdots (N-1)^i,$$
as $$2^i=\cos\log 2+ i\sin\log 2,$$
see [1].
Question. It is possible to get a closed-form (or the best approx... | First of all, you make the assumption $$1^i=1$$When this is not true.$$1^i=e^{\pm2\pi n},n=0,1,2,3,\ldots$$
More generally, I will solve$$1^x+2^x+3^x+4^x+5^x\ldots=\sum_{n=1}^{\infty}n^x$$This has solution obtainable via permutation:$$\sum_{n=2}^{m}n^x+n=\sum_{n=1}^{m-1}n^x+n^m$$$$\sum_{n=1}^{m-1}(n+1)^x+n=\sum_{n=1}^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
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What is the numerical range of $A$? Let $A = \left( {\begin{array}{*{20}{c}}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{array}} \right)$
What is the numerical range of $A$?
| Hint: This matrix is symmetric $A^T=A$, so in particular $$A^TA=AA^T$$ which means that the matrix is normal (as real valued). So, its numerical range is the convex hull of its eigenvalues, see here bullet 10. Now, $$\det(A-λΙ)=(λ^2-1)^2$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Divide a line segment in the ratio $\sqrt{2}:\sqrt{3}.$ "Divide a line segment in the ratio $\sqrt{2}:\sqrt{3}.$"
I have got this problem in a book, but I have no idea how to solve it.
Any help will be appreciated.
| let AB be the given line Segment. Draw a line L1 from A which makes an angle 45 degrees, and line L2 from B which makes an angle 60 with AB. Let L1, L2 meet at C.
Now ABC is a triangle with angle A= 45, angle B = 60. Let the angular bisector of C (makes an angle 37.5 with either of AC,BC) meets AB at D. We know that $$... | {
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"source": "stackexchange",
"question_score": "9",
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Two Indefinite Integrals Looking for some hints to evaluate the following integrals (with complex analysis or otherwise):
$$\int_0^\infty\frac{x^{p-1}}{x+1}\,dx,\;\;\;\; 0<p<1,$$
$$\int_{-\infty}^\infty e^{-s^2+isz}\,ds,\;\;\;\;z\in\mathbb{C}.$$
Thanks, I'm pretty stuck on both.
| For the first integral, you can set $t=\dfrac{x}{x+1}$ so that $\mathrm{d}x=\dfrac{\mathrm{d}t}{(1-t)^2}$, and you have
$$\int_0^\infty \frac{x^{p-1}}{x+1}\,\mathrm{d}x=\int_0^1t^{p-1}(1-t)^{-p}\,\mathrm{d}t=\mathrm{B}(p,1-p)$$
where $\mathrm{B}(a,b)$ denotes the Beta function.
For the second integral, consider com... | {
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Digits of $\pi$ using Integer Arithmetic How can I compute the first few decimal digits of $\pi$ using only integer arithmetic? By 'integer arithmetic' I mean the operations of addition, subtraction, and multiplication with both operands as integers, integer division, and exponentiation with a positive integer exponent... | The (or rather a) spigot algorithm for $\pi$ does exactly that: extract digits of $\pi$ one by one based entirely on integer arithmetic. See this paper.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve the equation $8x^3-6x+\sqrt2=0$ The solutions of this equation are given. They are $\frac{\sqrt2}{2}$, $\frac{\sqrt6 -\sqrt2}{4}$ and $-\frac{\sqrt6 +\sqrt2}{4}$. However i'm unable to find them on my own. I believe i must make some form of substitution, but i can't find out what to do. Help would be very appreci... | HINT...Let $x=\cos\theta$
The equation becomes $\cos3\theta=-\frac{1}{\sqrt{2}}$
Can you take it from there?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, proving $WZ_n\rightarrow 0$ Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, I need to prove $WZ_n\rightarrow 0$.
I was given a hint: show that for every $\delta,\epsilon<0$ we have
$$\{|WZ_n|\geq\epsilon\}\subset\{|Z_n|\geq\delta\}\c... | I believe is easier to use the subsequence definition of convergence in probability.
$\{Z_n W\}$ is clearly a sequence of random variables (since it is product of r.v's). To show that $\{Z_n W\} \xrightarrow{ P} 0$, we will show that for any subsequence $\{Z_{n_k} W\}$ there is a further subsequence $\{n_\ell\}$ of $\... | {
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non-negative almost surely I have a probability measure P and a non-negative sequence of random variables $(X_n)$ and the limit $X=\lim X_n$ exists P-almost surely. I would like to show that $X\ge0$ P-almost surely.
| If $X_n$'s are nonnegative, then $\limsup X_n$ and $\liminf X_n$ are nonnegative. If we are given that $\limsup X_n = \liminf X_n$, then $\limsup X_n = \liminf X_n \ge 0$
Also, remember that for a given $\omega$, $X_n(\omega)$ is a sequence of nonnegative numbers and a convergent sequence of nonnegative numbers converg... | {
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"url": "https://math.stackexchange.com/questions/1586054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
efficient way to express large numbers I recently watched the walkthrough of Graham's Number on YouTube (Numberphile). Mind-blowing of course.
I then puttered around in other large number topics like Ackerman and Tree(3) and fast growing hierarchies. It's all fantastic.
I also came across a challenge that was to writ... | Well, to try to make your function easier to write, we could use Knuth's arrow-notation, as mentioned by Deedlit.
$$V(n)=V(n-1)\uparrow\uparrow V(n-1)=V(n-1)\uparrow\uparrow\uparrow 2$$$$V_1(n)=n\uparrow\uparrow n$$
Why we would write it like this? Because it is simply easier to understand.
And since you haven't defin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Bayes Theorem with multiple random variables I came across this expression in the Intro to Probability book I am studying:
$P(A,B|C)=\frac{P(C)P(B|C)P(A|B,C)}{P(C)}$
Could anyone please explain how is this obtained. From a simple application of Bayes Rule, shouldn't it be:
$P(A,B|C)=\frac{P(C|A,B)P(A,B)}{P(C)}$ where $... | The extension of Baye's rule is such that $$P(A,B|C)=\frac{\color{blue}{P(A)}\color{blue}{P(B|A)}P(C|A,B)}{\color{blue}{P(B)}P(C|B)}\tag{1}$$
The formula can be seen as an extension of $$P(A|B)=\frac{\color{blue}{P(A)P(B|A)}}{\color{blue}{P(B)}}\tag{2}$$ This is not a derivation of course; I'm just trying to show you t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Finding an angle in a figure involving tangent circles The circle $A$ touches the circle $B$ internally at $P$. The centre $O$ of $B$ is outside $A$. Let $XY$ be a diameter of $B$ which is also tangent to $A$. Assume $PY > PX$. Let $PY$ intersect $A$ at $Z$. If $Y Z = 2PZ$, what is the magnitude of $\angle PYX$ in degr... | Triangle O'SO fits the description of a 30-60-90 special angled triangle. Therefore, $\angle O'OS = 30^0$
Then, $\angle PYX = 15^0$ [angles at center = 2 times angles at circumference]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Is there a notation for saying that $x_n\geq c$ from some $n$ and on? I am looking for something like $\lim\inf x_n \geq c$, but I need that from some point and on it is $\geq c$, not just that the limit is $\geq c$.
| As @user notes, the usual term for this notion is "eventually" (introduced or anyway popularized by Kelley in his book General Topology). It's a quantifier, sometimes written $\forall^*$. For any predicate $P(n)$ of integers, $P(n)$ holds eventually (with respect to $n$) iff $(\forall^* n)\, P(n) := (\exists N)(\forall... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
Does a polynomial minus the tangent at a certain point have a double root? Given: $P(x)$ is a polynomial of at least the second degree and $L(x)$ is the tangent to $P(x)$ at $x=a$.
Questions: Can one then say that $P(x)-L(x)$ has a double root at $x=a$? If so, why? If not, why not?
| You can always rewrite your polynomial as $P(x) = P(a) + P'(a) (x-a) + \frac{P''(a)}{2}(x-a)^2 + \dots $. Using this form, you can write $L(x)$ as $L(x) = P(a) + P'(a)(x-a)$. Combining these two formulas we get that
$P(x) - L(x) = (x-a)^2 \cdot (\frac{P''(a)}{2} + \dots )$, which suggests that at point $a$ function $P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Proving a sequence is unbounded The sequence is $(u_{n})_{n}$ for which $u_{n}=e^{n}$
The answer I have says that for any given positive real $U$, the term of index/position $n=\left \lfloor \ln(n+1) \right \rfloor + 1$ will be such that $\left | u_{n} \right |> U$.
I don't understand this answer. Can someone please... | Proving a sequence is unbounded means that you must show that for any large number $N$ there is no $M$ such that $u_M < N$ and that $u_k$ is also smaller than $N$ for every $k>M$.
For $u_k = e^k$ this can be done by showing that
$$e^k > N$$ if we pick $k = \lfloor \log (N+1)\rfloor + 1$
Because then you have
$$e^{\lflo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $A $ in $\sqrt{28}+\sqrt{7}=\sqrt{A}$ I got this problem from an old test paper of mine. I tried working on it today but I couldn't solve it. I tried solving it using logarithms but I didn't get an answer. The problem asks to solve for $A$.Can you guys help me solve this problem?
Here's the problem:
$\sqrt{28}+\sq... | Here is how it works
$$\sqrt{28}=\sqrt{4 \cdot 7}=\sqrt{4}\sqrt{7}=2\sqrt{7}$$
and hence
$$\sqrt{A} =\sqrt{28}+\sqrt{7}=2\sqrt{7}+\sqrt{7}=3\sqrt{7}$$
which leads to
$$(\sqrt{A})^2=(3\sqrt{7})^2 \\
A=9 \cdot 7 =63$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Does exist a triangle with sides a integer length where one of height is equal to the side which is the base? Does exist a triangle with sides a integer length where one of height is equal to the side which is the base?
$a,k,l$ -natural,
$a$ is lenght of hight
| In formulas, we want to find positive integers $a,b,c,d$ solving the Diophantine equations
$$
a^2+(a+b)^2=c^2
$$
$$
b^2+(a+b)^2=d^2.
$$
In particular, we obtain the Diophantine equation
$$
a^2+d^2=b^2+c^2,
$$
which has been studied [here](
Diophantine equation $a^2+b^2=c^2+d^2$). hence there are integers $p,q,r,s$ such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Behaviour a function when input is small If I have the function:
$P(N)=\frac{P_0N^2}{A^2+N^2}$, with $P_0, A$ positive constants
For small $N$, am I right in thinking that because $A$ dominates $N$ we have that
$P(N) \approx \frac{P_0N^2}{A^2}$
| More precisely, we can write for $A^2>N^2$
$$\begin{align}
P(N)&=P_0\left(\frac{N^2}{A^2+N^2}\right)\\\\
&=P_0\left(\frac{(N/A)^2}{1+(N/A)^2}\right)\\\\
&=P_0\frac{N^2}{A^2}\sum_{k=0}^\infty (-1)^k\left(\frac{N^2}{A^2}\right)^{k}\\\\
&=\frac{P_0N^2}{A^2}-P_0\frac{N^4}{A^4}+P_0\frac{N^6}{A^6}+O\left(\frac{N^8}{A^8}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
An inverse function and a differential equation What is the inverse function of $f(x)=x+\exp(x)$?
I doubt it's the solution of the differential equation: $$y'+y'\exp(y)-1=0.$$
| The derivative of $f(x)$ is $1+\exp(x)$. So by the inverse function theorem
$$(f^{-1})'(x+\exp(x))=\frac{1}{1+\exp(x)}$$
or
$$(f^{-1})'(y)=\frac{1}{1+\exp(f^{-1}(y))}.$$
Thus $g=f^{-1}$ solves the differential equation
$$g'(x)=\frac{1}{1+\exp(g(x))}$$
with the initial condition $g(1)=0$. Straightforward algebra shows t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
What is the general name for a surface of form $z = f(x, y)$? Is there a generic, geometrical name for a "sheet" of data, where the data fits the form:
$$z = f(x, y)$$
Note that $z$ might be an actual third dimension (such as elevation in terrain data) or any scalar measurement (such as surface temperature).
I know yo... | Non-parametric or graph form are two commonly employed words.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1586982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Past open problems with sudden and easy-to-understand solutions What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius, and the ... | Our complex analysis professor told us that huge amounts of literature was written studying analytic functions that are bounded. Then came https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis) like a hammer. Super easy proof, (according to my professor) very unexpected result at the time.
I haven't fou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "127",
"answer_count": 19,
"answer_id": 9
} |
How to adjust a ratio to create desired proportion Let's say I have a bunch of apples and oranges. Specifically 500 apples and 760 oranges.
I need to create a big basket of fruit. For every 7 apples I include, I need to include 3 oranges.
I want to make the biggest basket possible using as much of my fruit as I can w... | Suppose you put $n$ groups of 7-apples-and-3-oranges into the big basket. Then you will have $7n$ apples and you will have $3n$ oranges in the basket.
If all of the apples are used up, you would have $7n=500$, which means you would have $n=500/7 = 71\frac37$ groups. That means you can have at most $71$ complete groups,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Is there any way to universally define the notion of $\text{Isomorphism}$? Suppose we want to give a very general definition of the term Isomorphism, first of all, we'll want an isomorphism to be a bijective function.
Informally, we want our function to preserve whatever 'structure' we're talking about, may it be: mult... | I think category theory answers the question. An isomorphism in a category is just a morphism which has a two-sided inverse.
Now, what is a morphism? This depends on the category you're working on. For example we have:
*
*the category of topological spaces, where continuous functions are the morphisms;
*the catego... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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$I=3\sqrt2\int_{0}^{x}\frac{\sqrt{1+\cos t}}{17-8\cos t}dt$.If $0Let $I=3\sqrt2\int_{0}^{x}\frac{\sqrt{1+\cos t}}{17-8\cos t}dt$.If $0<x<\pi$ and $\tan I=\frac{2}{\sqrt3}$.Find $x$.
$\int\frac{\sqrt{1+\cos t}}{17-8\cos t}dt=\int\frac{\sqrt2\cos\frac{t}{2}}{17-8\times\frac{1-\tan^2\frac{t}{2}}{1+\tan^2\frac{t}{2}}}dt=\... | HINT:
$$\frac{\sqrt{1+\cos t}}{17-8\cos t}=\dfrac{\sqrt2\left|\cos\dfrac t2\right|}{17-8\left(1-2\sin^2\dfrac t2\right)}$$
Set $\sin\dfrac t2=u$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differentiable function such that $f(x+y),f(x)f(y),f(x-y)$ are an arithmetic progression for all $x,y$ If $f$ is a differentiable function on $\mathbb{R}$ such that $f(x+y),f(x)f(y),f(x-y)$(taken in that order) are in arithmetic progression for all $x,y\in \mathbb{R}$ and $f(0)\neq0,$then
$(A)f'(0)=-1\hspace{1cm}(B)f'(... | For $x=y$, you have $$f(x-x)+f(x+x)=2f(x)f(x)$$ or $$f(0)+f(2x)=2f^2(x)$$ or, $$2f^2(x)-f(2x)=1$$
Differentiating both sides with respect to $x$, we get
$$4f(x)f'(x)-2f'(2x) = 0 $$
Putting $x=0$, you have $$f'(0)=0$$
So options (a) and (b) are wrong.
As for (c) and (d), they are trying to check if $f'(x)$ is an even fu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
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Calculate the range of $f$. Calculate the range of the function $f$ with $f(x) = x^2 - 2x$, $x\in\Bbb{R}$. My book has solved solutions but I don't get what is done:
$$f(x) = x^2 - 2x + (1^2) - (1^2)= (x-1)^2 -1$$
edit: sorry for wasting all of the people who've answered time, I didnt realize that I had to complete the... | From the form $f(x)=(x-1)^2-1$ you can see immediately two things:
*
*Since the term $(x-1)^2$ is a square it is always non-negative, or in simpler words $\ge 0$. So, its lowest possible values is $0$ which can be indeed attained when $x=1$. So, $f(x)=\text{ positive or zero } -1 \ge 0-1=-1$. So $f(x)$ takes values ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Finding the exact values of $\sin 4x - \sin 2x = 0$ So I've used the double angle formula to turn
$$\sin 4x - \sin 2x = 0$$
$$2\sin2x\cos2x - \sin2x = 0$$
$$\sin2x(2\cos2x - 1) = 0$$
$$\sin2x = 0$$
$$2x = 0$$
$$x = 0$$
$$2\cos2x - 1 = 0$$
$$2\cos2x = 1$$
$$\cos2x = \frac{1}{2}$$
$$2x = 60$$
$$x = 30$$
With this info... | First you should use the natural angle unit, which is the radian. Second, you should write the general solution, which is always a congruence.
Your equation factors as
\begin{align*}
2\sin x\cos x(2\cos 2x -1)=0 &\iff\begin{cases}\sin x=0\\ \cos x=0 \\\cos x=\frac12
\end{cases}\\
& \iff
\begin{cases}
x\equiv 0\mod \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Why does $A \in$ SO(2) and $A \ne I \implies $ker$(A-I) = \{0\}$ I've come across this in proofs for Euclidean geometry a few times. Can someone tell me why this is true?
$$A \in \text{SO}(2) \text{ and } A \ne I \implies \text{ker}(A-I) = \{0\}$$
| Let $A\in\textrm{SO}(2,\mathbb{R})\setminus\{I_2\}$, there exists $\theta\in\mathbb{R}\setminus 2\pi\mathbb{Z}$ such that: $$A=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}.$$
Therefore, one gets: $$\det(A-I_2)=(\cos(\theta)-1)^2+\sin^2(\theta)=2(1-\cos(\theta))\neq 0.$$
Finally, $A-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Morera's Theorem and independence of path Determine whether
$$
G(z)=\int_{1}^{z}\frac{1}{t} dt
$$
is independent of the path joining $1$ and $z$ , and discuss the relationship of your answer to Morera's Theorem ..... I believe that as long as the paths goes through a region that excludes the origin ,the integral is i... | Have you thought about traversing the unit circle for $z=1$?
$$
\int_C\frac 1zdz=\int_0^{2\pi}\frac{1}{e^{it}}(ie^{it})dt=\int_0^{2\pi}idt=2\pi i
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1587971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Solve for Rationals $p,q,r$ Satisfying $\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1$. Find all rational solutions $(p,q,r)$ to the Diophantine equation
$$\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1\,.$$
At least, determine an infinite family of $(p,q,r)\in\mathbb{Q}^3$ satisfying thi... | If three rationals $a,b,c$ can be found, these lead to rational $p,q,r$. For simplicity, we'll start with the $a,b,c$. Given the system,
$$a^2+(b+c)^2 = x_1^2\\b^2+(a+c)^2 = x_2^2\\c^2+(a+b)^2 = x_3^2$$
Let $a = m^2-n^2,\;b=2mn-c,$ and it becomes,
$$(m^2+n^2)^2 = x_1^2\\2c^2+2c(m^2-2mn-n^2)+(m^2+n^2)^2 = x_2^2\\2c^2-2c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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For any coprime integers $(x, y)$, $m\geq 2$, $\delta\geq 1$, is it true that $\mid x^m - y^{m+\delta} \mid \geq \delta$? Is it true that if $x,y,m,\delta$ are integers, $\gcd(x,y)=1$, $m\ge2$, $\delta\ge1$, then $$|x^m-y^{m+\delta}|\ge\delta?$$
Any proofs or references will be most welcome.
| $|5^3-2^{3+4}|<4{}{}{}{}{}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove this equality: $\int_0^x(x-t)e^{\cos t}dt=\int_0^x\bigg(\int_0^te^{\cos s}ds\bigg)dt$ I'm trying to show this equality is true:
$$\int_0^x(x-t)e^{\cos t}dt=\int_0^x\bigg(\int_0^te^{\cos s}ds\bigg)dt$$
I used the fundamental theorem of calculus without success. I need a hint how to solve this question.
| Integrate by parts:
$$ \int_0^x (x-t) e^{\cos{t}} \, dt = \left[ (x-t)\int_0^t e^{\cos{s}} \, ds \right]_{t=0}^x - \int_0^x (-1) \int_0^t e^{\cos{s}} \, ds \, dt. $$
The first term vanishes at the endpoints, the second is what you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Let $f(x)=x\cos(x)$, Which of following is true?
*
*Let $f(x) = x\cos x$ for $x\in\mathbb R$. Then
$\quad$(A) There is a sequence $x_n\to -\infty$ such that $f(x_n)\to 0$.
$\quad$(B) There is a sequence $x_n \to \infty$ such that $f(x_n)\to\infty.$
$\quad$(C) There is a sequence $x_n \to\infty$ such that ... | A) is true , if you pick $x_n = \pi/2 - n\pi$ you get $f(X)=0$ for every $x$ real. B) is true, if you pick $x_n = 2n\pi$ $f$ goes to $+\infty$ since $cos=1$. $C)$ is true, if you pick $x_n = (2n + 1)\pi$ $f$ goes to $-\infty$ since $cos = -1$. D) is false cause $f'(X) = cosX - XsinX \simeq -X$ for the right choise of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integration of a square root inside another square root function. Integrate the following $$\int_0^5 \sqrt{5-\sqrt{x}} \ \mathrm{d}x$$ I have done the following:
I got the expression under the square root to equal $$5-x^{1/2}$$ so the integral then becomes $$\int_0^5 \left(5-x^{1/2}\right)^{1/2}\ \mathrm{d}x$$ Here is... | Hint: substitute $u=5-\sqrt{x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Measure theory for self study. I have good knowledge of Elementary Real analysis. Now I'd like to study measure theory by myself (self-study). So please give me direction for where to start? Which book is good for starting? I have Principles of Mathematical Analysis by W. Rudin and Measure Theory and Integration by G. ... | I have discovered Yeh's Real Analysis: Theory Of Measure And Integration recently, and I recommend it warmheartedly! Easy to read I think, especially for self study. It has a problem & proof supplement by the way.
Rudin (not PMA but RCA) is very good on the long term, but hard for a first encounter with measure theory.... | {
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"url": "https://math.stackexchange.com/questions/1588450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove This Inequality ${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$ $${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$$
I see I should use Riemann sum, and that
$$\int_0^{\infty} {dx \over {1+x^2}} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}}$$
But how do I e... | Hint:
$$ \sum_{n=0}^{\infty} {1 \over {1+n^2}}=\frac{1}{2}+\frac{\pi}{2}\coth(\pi) $$
$$\coth(\pi)> 1\simeq 1$$
so
$$\sum_{n=0}^{\infty} {1 \over {1+n^2}}\simeq\frac{1}{2}+\frac{\pi}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Is Lipschitz condition necessary for existence and uniqueness of a solution for a first order differential equation? Is the Lipschitz condition only a sufficient condition for existence and uniqueness of $\frac{dy}{dx}=f(x,y), y(0)=y_0$? Is it also necessary?
| First, it's worth pointing out that for a Cauchy problem
$$
\begin{cases}
y'=f(x,y) \\
y(x_0)=y_0
\end{cases}
$$
we require $f$ to be a Lipschitz function only in the second variable, while it is enough for it to be continuous in the first one (this result is known as Picard-Lindelöf theorem).
A typical example for wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Why do we do mathematical induction only for positive whole numbers? After reading a question made here, I wanted to ask "Why do we do mathematical induction only for positive whole numbers?"
I know we usually start our mathematical induction by proving it works for $0,1$ because it is usually easiest, but why do we on... | All of the contexts in which you propose to apply induction are implicitly using the positive integers anyway.
For instance, suppose we have a property $P$, and we prove that $P(x)\implies P(x+\frac 1 2)$. Then if we know $P(0)$, we can prove $P(0.5)$, $P(1)$, $P(1.5)$, etc. But you're implicitly using the natural numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 2
} |
Prove $1^2-2^2+3^2-4^2+......+(-1)^{k-1}k^2 = (-1)^{k-1}\cdot \frac{k(k+1)}{2}$ I'm trying to solve this problem from Skiena book, "Algorithm design manual".
I don't know the answer but it seems like the entity on the R.H.S is the summation for series $1+2+3+..$. However the sequence on left hand side is squared series... | It can be easily shown that the series on the left reduces to the sum of integers, multiplied by $-1$ if $k$ is even.
For even $k$:
$$\begin{align}
&1^2-2^2+3^3-4^2+\cdots+(k-1)^2-k^2\\
&=(1-2)(1+2)+(3-4)(3+4)+\cdots +(\overline{k-1}-k)(\overline{k-1}+k)\\
&=-(1+2+3+4+\cdots+\overline{k-1}+k)\\
&=-\frac {k(k+1)}2\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 5
} |
Poisson distribution problem about car accidents I have this Poisson distribution question, which I find slightly tricky, and I'll explain why.
The number of car accidents in a city has a Poisson distribution. In March the number was $150$, in April $120$, in May $110$ and in June $120$. Eight days are being chosen by ... | I would say there are $500$ accidents in $31+30+31+30=122$ days, so $\lambda=500/122$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Calculate the variance of random variable $X$ A random variable has the cumulative distribution function $$F(x)=\begin{cases} 0 & x<1\\\frac{x^2-2x+2}{2}&x\in[1,2)\\1&x\geq2\end{cases}.$$ Calculate the variance of $X$.
First I differentiated the distribution function to get the density function, $f_X(x)=x-1$, for $x\in... | Note that $X$ does not have continuous distribution. For $F(x)$ approaches $0$ as $x$ approaches $1$ from the left, while $F(1)=1/2$.
So there is a point mass of $1/2$ at $x=1$. This has to be taken into consideration both for the calculation of $E(X)$ and of $E(X^2)$. (Your expression for the variance in terms of $E(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1588951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Determine the ratio $k$ so that the sum of the series $\sum_{i=0}^{\infty} l_{i}$ is equal to the circumference of the outermost circle. Question:
Consider an infinite series of concentric circles, were the radii $r_{0}, r_{1}, r_{2}, ...$ form a geometric series with the ratio $k$, $0 < k < 1$.
From a point on the out... | Without loss of generality we may assume that the outer circle has radius $1$.
If $l_0$ is as in the OP, then $l_0^2=1-k^2$. So by scaling we have $l_1=k\sqrt{1-k^2}$, $l_2=k^2\sqrt{1-k^2}$, and so on.
The sum of the geometric series is $\frac{\sqrt{1-k^2}}{1-k}$, that is, $\frac{\sqrt{1+k}}{\sqrt{1-k}}$. Set this equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove a function is in $L^2[0,1]$
If $f\in L^2[0,1]$, and $$g(x)=\int_0^1\frac{f(t)\mathrm dt}{|x-t|^{1/2}},\quad x\in[0,1],$$
show that $\|g\|_2\le2\sqrt2\|f\|_2$.
I tried Minkowski's integral inequality (with $p=1/2$, so the inequality reverses), but cannot reach the inequality I need. I also used Holder's ineq... | Here is another answer only uses basic calculus. From the expression of $g$,
\begin{align}
\|g\|_2^2 &=\int_0^1 g(x)^2 dx \cr
& =\int_0^1\int_0^1 \int_0^1 \frac{f(t)}{|x-t|^{1/2}}\frac{f(s)}{|x-t|^{1/2}}dsdtdx \cr
&\leq \int_0^1 \int_0^1 \int_0^2 \frac{f(t)^2+f(s)^2}{2}|x-t|^{-1/2}|x-s|^{-1/2}dsdtdx \cr
&=\int_0^1 f(t)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
If $A^2=A$ then prove that $\textrm{tr}(A)=\textrm{rank}(A)$. Let $A\not=I_n$ be an $n\times n$ matrix such that $A^2=A$ , where $I_n$ is the identity matrix of order $n$. Then prove that ,
(A) $\textrm{tr}(A)=\textrm{rank}(A)$.
(B) $\textrm{rank}(A)+\textrm{rank}(I_n-A)=n$
I found by example that these hold, but I am... | The given condition $A^{2}=A$ says that the matrix is idempotent and so diagonalizable and hence rank is same as the number of eigen values $1$(with repetition ) which is the same as trace of $A.$ For second part use it $$|rank(A)-rank(B)|\leq rank(A+B)\leq n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 4
} |
Find the number of distinct throws which can be thrown with $n$ six faced normal dice which are indistinguishable among themselves. Find the number of distinct throws which can be thrown with $n$ six faced normal dice which are indistinguishable among themselves.
The total outcomes will be $6^n$. But this this has many... | Consider six faces as six beggars and n identical dice to be identical coins.
Now,number of distributions is C(n+6-1,6-1)=C(n+5,5).
If a beggar(say face 6)gets no coin,then it is equivalent to anything which does not appear on the dice.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Markov Chain with two components I am trying to understand a question with the following Markov Chain:
As can be seen, the chain consists of two components. If I start at state 1, I understand that the steady-state probability of being in state 3 for example is zero, because all states 1,2,3,4 are transient. But what ... | Yes, you can. Actually this Markov chain is reducible, with two communicating classes (as you have correctly observed),
*
*$C_1=\{1,2,3,4\}$, which is not closed and therefore any stationary distribution assigns zero probability to it and
*$C_2=\{5,6,7\}$ which is closed.
As stated for example in this answer,
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$p$ is a positive integer and $(p)$ is a maximal ideal in the ring $(\mathbb Z, +,\cdot)$, then $p$ is a prime number I need to prove: $p$ is a positive integer and $(p)$ is a maximal ideal in the ring $(\mathbb Z, +,\cdot)$, then $p$ is a prime number.
My attempt:
1) $(p)$ is a maximal ideal, so it is a prime ideal.... | Let $p$ be a nonprime number, thus $p$ can be factored in to two non-unit coprime numbers, say $x$ and $y$ ($p = xy$, $\gcd(x, y) = 1$). Since $x$ and $y$ are non-units, so $\langle p \rangle \subset \langle x \rangle \neq \textbf{Z}$ and $\langle p \rangle \subset \langle y \rangle \neq \textbf{Z}$ (if $\langle p \ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Theorem that links limits of successions and limits of functions I've a doubt on the theorem that links the limits of functions with the limits of successions.
$$\lim_{x\to c}f(x)=l \iff \forall \big\{x_n\big\}_{x\in \mathbb{N}} \mid x_n\in dom(f) \wedge x\neq c \wedge \lim_{n\to \infty} x_n = c$$ holds that $\lim_{n\... | Yes, the first one: "The condition must be true for every $x_n$". Of course this excludes in particular constant sequences ($x_n=c$). For example let $$f(x)=\begin{cases}0, & x=3 \\ x, & \text{ else} \end{cases}$$ Then if you take the sequence $x_n=3$ for all $n$ "converges" (trivially) to $x_0=3$, so $$\lim_{n\to \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Remainder of the numerator of a harmonic sum modulo 13 Let $a$ be the integer determined by
$$\frac{1}{1}+\frac{1}{2}+...+\frac{1}{23}=\frac{a}{23!}.$$
Determine the remainder of $a$ when divided by 13.
Can anyone help me with this, or just give me any hint?
| By Wilson's theorem, $12!\equiv -1\pmod{13}$.
By Wolstenholme's theorem, $H_{12}\equiv 0\pmod{13}$. Since:
$$ a = \sum_{k=1}^{12}\frac{23!}{k} + (12!)(14\cdot 15\cdot\ldots\cdot 23)+\sum_{k=14}^{23}\frac{23!}{k} $$
we have:
$$ a\equiv 0-10!+0 \equiv \frac{-12!}{11\cdot 12}\equiv \frac{1}{(-2)\cdot(-1)}\equiv\frac{1}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
$x'=x^2$ unstable solutions when $x(0)\geq 0$ but asymptotically stable when $x(0)\leq0$ How do I show that the differential equation $x'=x^2$ has unstable solutions when $x(0)\geq 0$ but asymptotically stable solutions when $x(0)\leq0$?
Usually, I look at the eigenvalues of the matrix to determine stability of solutio... | suppose $x(0) = k.$ then the solution is $$x = \frac 1{t + 1/k }, -1/k < t < \infty$$ the graph is monotone increasing on the domain and $\lim_{t \to \infty} x(t) = 0.$
this solution approaches zero but not expentailly; it approaches zero like $\frac1t.$ i don't this the solution is asymptotically stable; just stable... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Compare 2 algorithms by statistics Let's suppose we have two different processes, each generating some amount of money $M$ every second.
$$0 \leq M \leq 1000$$
We run each process for $50\%$ of available time.
The question is how to compare the productivity (in money; per second) of these two processes if there is no ... | You have two samples - one from each method - with equal sample sizes (and big enough I assume) and you want to see which method generates statistically better results. This is a standard methodology with a confidence interval for the difference of means or a hypothesis testing again for the difference of means. Of cou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the derivative $dy/dx$ from the parametric equations for $x$ and $y$
Let
\begin{cases}
y=2t^2-t+1 \\
x=\sin(t)
\end{cases}
find $\frac{dy}{dx}$
Is this all that I need to do? $$\frac{4t-1}{\cos(t)}$$
| As others have said,
$$
\frac{dy}{dt} = 4t-1 \quad\mbox{and}\quad \frac{dx}{dt} = \cos t \implies \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{4t-1}{\cos t}.
$$
But now we must eliminate the parameter $t$, because we'd like to have $y$ as a function of $x$ in writing $dy/dx$.
For the numerator of $dy/dx$, we use
$$
x =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$ $${\color\green{\sum_{x=2}^\infty (x^2 \operatorname{arcoth} (x) \operatorname{arccot} (x) -1)}}$$
This is an impressive sum that has bothered me for a while. Here are the major points behind the sum...
Believing on a clo... | A possible approach may be the following one, exploiting the inverse Laplace transform.
We have:
$$ n^2\text{arccot}(n)\text{arccoth}(n)-1 = \iint_{(0,+\infty)^2}\left(\frac{\sin s}{s}\cdot\frac{\sinh t}{t}-1\right) n^2 e^{-n(s+t)}\,ds\,dt \tag{1}$$
hence:
$$ \sum_{n\geq 2}\left(n^2\text{arccot}(n)\text{arccoth}(n)-1\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1589913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
How to calculate the exact values of c and d Draw the function $f(x)= x^3-4x^2-13x+10$ into a cartesian plane. If the exact length of that curve from point $P(5,-30)$ to point $Q(c,d)$ is $18$ (with $c>5$), I have no idea how to calculate the values of $c$ and $d$ using the formula $(dx)^2+(dy)^2$. How to approximate t... | The formula for arc length from $(a, f(a))$ to $(b, f(b))$ is $L = \int_a^b \sqrt{1 + (\frac{d (f(x))}{dx})^2}dx$
So the arc length from $(5,-30)$ to $(c, d=f(c))$ is $L = 18 = \int_5^c \sqrt{1 + \frac{d(x^3-4x^2-13x+10)}{dx}^2}dx$
Solve for $c$ and $d = f(c)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Convex Set with Empty Interior Lies in an Affine Set In Section 2.5.2 of the book Convex Optimization by Boyd and Vandenberghe, the authors mentioned without proving that "a convex set in $\mathbb{R}^n$ with empty interior must lie in an affine set of dimension less than $n$." While I can intuitively understand this re... | Look at $d+1$, the largest number of affinely independent points from $C$. Let $x_0$, $\ldots$, $x_d$ one such affinely independent subset of largest size. Note that every other point is an affine combination of the points $x_k$, so lies in the affine subspace generated by them, which is of dimension $d$.
If $d < n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1590133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
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