Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Are axioms assumed to be true in a formal system? In a logical system, there is assignment of truth values to the sentences in the language, and axioms are assigned the true value.
A logical system is a formal system.
In a formal system, there is no truth value assignment, but there are still axioms. Does that imply t... | For a formal system to be of much interest, it needs to be consistent -- it needs to have at least one model. In that model, the axioms of the system will be true. It doesn't make much sense to talk about the truth of axioms apart from models.
The formal systems of the most interest -- such as Peano arithmetic and Zerm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Separation of variables: when to have exponential solution and when sinusoidal? In separation of variables, one can assume a solution of V(x,y) = X(x)Y(y) and after plugging this into Laplace's equation which is: ${{\partial^2 V} \over {\partial x^2}}$ + ${{\partial^2 V} \over {\partial y^2}}$ = 0 we can get:
${d^2X \... | If you solve the differential equation $\frac{d^2 X}{dx^2} - k^2 X = 0$, then let $X=e^{mx}$ so that you get the auxiliary equation $m^2-k^2=0$. That auxiliary equation has roots $m_1=k,m_2=-k$, which are real. If you plug the roots to the formula
$$y=A e^{m_1 x}+ B e^{m_2 x}$$ you will get a solution with exponents.
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
invertibility of self adjoint operators prove that if $T$ is a self adjoint operator and $a^2$ is less than $4b$ Then $T^2$+$aT$+$bI$ is invertible.
Where $a$ and $b$ are scalars and $I$ is the identity operator
Not: please dont use determinants because the book i am using didn't define determinants yet
| This question originally posited the condition $a < 4b$ sufficient for $T^2 + aT + bI$ to be invertible, provided $T$ is self-adjoint; however, I belive that Byron Schmuland was correct to edit this condition to $a^2 < 4b$, and I hope to make my reasons for this clear in what follows, both by proving the corrected prop... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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tank problem Differential equation A tank is partially filled with 100 gallons of coffee in which 10 lbs of sugar is dissolved. Coffee containing 1/3 lb of sugar per gallon is pumped into the tank at rate 3 gal/min. The yummy well-mixed solution is then pumped out at a slower rate of 1 gal/min.
A- What is the rate at ... | Let $A(t)$ be the number of pounds of coffee in the tank at time $t$. We find an expression for $A'(t)$.
There is a standard pattern for setting up the appropriate differential equatiom. We look separately at the rate sugar is (i) entering the tank and (ii) leaving the tank.
Entering: Liquid is entering at $3$ gallons ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
maximum area of a rectangle inscribed in a semi - circle with radius r.
A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum.
My Try:
Let length of the side be $x$,
Then the length of th... | hint :$x\sqrt{r^2-x^2}=\sqrt{x^2(r^2-x^2)}\le \dfrac{x^2+(r^2-x^2)}{2}=\dfrac{r^2}{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/872223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 2
} |
How to show that $(a+b)^p\le 2^p (a^p+b^p)$ If I may ask, how can we derive that $$(a+b)^p\le 2^p (a^p+b^p)$$ where $a,b,p\ge 0$ is an integer?
| Using Jensen's Inequality, we get for $p\ge1$ or $p\le0$,
$$
\left(\frac{A+B}2\right)^p\le\frac{A^p+B^p}2
$$
Which, upon multiplication by $2^p$, yields
$$
(A+B)^p\le2^{p-1}(A^p+B^p)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/872276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 2
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Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse What are some interesting functions I can use to demonstrate this integration trick:
$$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$
I would like to know of some interesting functions where this trick is not obvious. EDIT: The funct... | I'll throw one in for kicks. This is the first one that came to mind that isn't directly given in the form you stated. It's nothing mind blowing but here we go.
$\int 2xdx = x^2$, obviously. Now let $f(x) = x$. Then we get
\begin{align*}\int_a^b 2xdx & = \int_a^b 2f(x)dx \\
& = \int_a^b (f(x) + f(x))dx \\
& = \int_a^b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 3
} |
Proof of an $\iff$ statement on binary trees
Let $x$ and $y$ be two nodes of a binary tree $B$.
Prove that $x$ is an ancestor of $y$ $\iff$ $x$ stands before $y$ in the pre-order traversal of $B$ and $x$ stands after $y$ in the post-order traversal of $B$.
I can do the $\implies$ part without trouble, but I can't de... | For the converse I would proceed by contrapositive with the following ideas.
(1) Assume $x$ and $y$ are vertices for which neither is an ancestor of the other.
(2) Argue that there is a lowest common ancestor (I am picturing the tree drawn with the root at the top and going downwards) of $x$ and $y$, say $a$.
(3) Argue... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Open and closed equivalence relations I am looking for canonical examples of open and closed equivalence relations, especially ones that are generated by a continuous functions. Intuitively I think that an open /closed continuous function $f:X \rightarrow Y$ should also generate a closed/open equivalence relation by $x... | Every $G_\delta$ equivalence relation $E$ on a standard Borel space $X$ is so called smooth, which means that there is a Borel function $f : X \rightarrow X$ such that $x \ E \ y$ if and only if $f(x) = f(y)$.
All closed and open sets are $G_\delta$ hence if you weaken continuous function to Borel functions, in some s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimum Value of $x_1+x_2+x_3$ For an Acute Triangle $\Delta ABC$
$$\begin{align}x_n=2^{n-3}\left(\cos^nA+\cos^nB+\cos^nC\right)+\cos A\,\cos B\,\cos C\end{align}$$ Then find the least value of $$x_1+x_2+x_3$$
My Approach: I have found $x_1$, $x_2$ and $x_3$
$$\begin{align}x_1=\frac{1}{4}\left(\cos A+\cos B+\cos C\rig... | Use AM-GM inequality,we have
$$\cos^3{x}+\dfrac{\cos{x}}{4}\ge 2\sqrt{\cos^3{x}\cdot\dfrac{\cos{x}}{4}}=\cos^2{x}$$
then we have
$$x_{1}+x_{3}\ge\cos^2{A}+\cos^2{B}+\cos^2{C}+2\cos{A}\cos{B}\cos{C}=2x_{2}$$
so
$$x_{1}+x_{2}+x_{3}\ge 3x_{2}=\dfrac{3}{2}$$
because we have use this follow well know
$$\cos^2{A}+\cos^2{B}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Borel measure supported on $\mathbb{Q}$ Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?
| Yes, because by definition of the support (depending on your definition), you have $\mu ( \Bbb{R} \setminus \Bbb{Q}) \leq \mu(\Bbb{R} \setminus {\rm supp}(mu)) = 0$ and thus for every measurable set $M$:
$$
\mu(M) = \mu(M \cap \Bbb{Q}) = \sum_{q \in \Bbb{Q} \cap M} \mu({q}) = (\sum_{q \in \Bbb{Q}} \mu({q}) \delta_q ) (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Continuity of a function defined by an integral, when the variable is in the region of integration Hi everyone: Suppose that $f$ is locally integrable in $\mathbb{R}^{n}$, $(n\geq2)$; $B(a,r)$ is the ball of center $a$ and radius $r>0$ , and $\lambda$ is the $n$-dimensional Lebesgue measure. It seems clear that the fu... | Assume without loss of generality that $a=0$ and let $B(0,r)=B_r$. We have to prove the following: given $\epsilon>0$ there exists $\delta>0$ such that
$$
0<r<R<\delta\implies\Bigl|\int_{B_R\setminus B_r}f(t)\,d\lambda(t)\Bigr|\le\epsilon.
$$
Since $\lambda(B_R\setminus B_r)\to0$ as $r\to R$, the result is a consequenc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that if the square of a number $m$ is a multiple of 3, then the number $m$ is also a multiple of 3. I'd like to prove that if ${m}^{2}$ is a multiple of $3$, then ${m}$ is also a multiple of $3$. Similarly, I'd like to disprove that if ${n}^{2}$ is a multiple of $4$, then ${n}$ is also a multiple of $4$.
Per the ... | Hint $\ 3\mid (m\!-\!1)m(m\!+\!1)=\color{#c00}{m^3\!-m},\ $ so $\ 3\mid\color{#0a0}{m^3}\,\Rightarrow\, 3\mid \color{#0a0}{m^3}\!-(\color{#c00}{m^3\!-m}) = m$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/872921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 0
} |
Problem regarding Euler's Theorem: $a^{\phi(n)}\equiv 1 \bmod n$ Here's the problem:
If $n \geq 2$, and if $p$ is a prime number s.t $p|n$ but $p^2$ is not a factor of $n$, then $$p^{\phi(n)+1}\equiv p \mod n$$
So since we're dealing with Euler's Phi function, I figured that this was an application of Euler's Theorem... | I think I've got it now:
We know that $p|n$ $\implies$ $\exists \gamma \in \mathbb{Z^{+}}$ s.t $n=p\gamma$. We are given the $p^2$ is not a factor of $n$ $\implies p^2 > n$ and so $p$,$\gamma$ share no common divisors. Thus, $\gcd(p,\gamma)=1$. Then by Euler's Theorem, we have
$$p^{\phi(\gamma)} \equiv 1 \mod p$$
Since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
On a basic tensor product question I am trying to show that
$$
\mathbb{Z} / (10) \otimes \mathbb{Z} / (12) \cong \mathbb{Z}/(2)
$$
by defining a map
$$
h([a]_{10} \otimes [b]_{12}) = [ab]_2
$$
and extend it linearly. I am having trouble trying to prove that this map is
well defined on $\mathbb{Z} / (10) \otimes \mathb... | try to show in general that: let $A$ a commutative ring and $I$ , $J$ two ideals, then $A/I\otimes A/J\simeq A/(I+J)$. And remark that $12\mathbb{Z}+10\mathbb{Z}=2\mathbb{Z}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/873099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Websites for math tests/quizzes Next semester I'm taking calculus at college and I was looking for websites that have quizzes/test for things like trigonometry, trig formulas, pre-calculus, calculus readiness, etc. so I can get ready this summer. I found the "MU Math Tests Homepage"
http://mathonline.missouri.edu/mucgi... | There are a couple of resources that I enjoy using. First, is Brilliant.org:
http://brilliant.org/
There is a subscription fee but it is worth it. It starts with simple questions on the topics you mentioned and progressively gets harder.
Another site that is good is this quiz page written by Terry Tao:
http://scherk.p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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holomorphic function with nonvanishing derivative on unit disk $D$ Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
| Yes: This is true for functions which are smooth enough, and in fact (if $f$ is smooth) we have
$$\int |\cos kx| f(x) dx \to A \int f(x) dx$$
where $A$ is the average of $|\cos x|$ over a single cycle.
Now given $f \in L^1$, choose a sequence of smooth enough $f_n$ converging to $f$ in the $L^1$ norm, and then note
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon? Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice.
Note that removing the "points" of the star yields a similar heptagon. I want to know the side length of this internal... | Refer to the following diagrams:-
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/873412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$ I recently encountered the following equality ($\{\}$ denotes fractional part):
$$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$
and found it very interesting as most of the individual summ... | Given $n$, let $a$ and $b$ be integers coprime with $n$. Then:
$$\sum_{k=1}^n k \left\{ \frac{ak}{n}\right\} = \sum_{k=1}^n k \left\{ \frac{bk}{n}\right\}$$
As Michael Stocker commented:
$$f(a,n) = \sum_{k=1}^n k \left\{ \frac{ak}{n}\right\} = \sum_{k=1}^n k \frac{ak \mod n}{n} = \frac{1}{n}\sum_{k=1}^n k (ak \mod n)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 0
} |
integrate $ \int \frac {x dx}{\sqrt {1+x^{10}} } $ This is a tough one. Thanks.
$$\int \frac {x dx}{\sqrt {1+x^{10}} } $$
This is not a homework problem. I spend 10 hours over the course of 3 days on this. I tried:
1) substituting u for x^5 to get a tangent-like quantity in the denominator
2)subsituting u for 1+x^10
3)... | As mentioned in the comments, you can at least simplify this integral somewhat via the substitution $u=x^2$:
$$\int\frac{x\,\mathrm{d}x}{\sqrt{1+x^{10}}}=\frac12\int\frac{\mathrm{d}u}{\sqrt{1+u^5}}.$$
Next, substituting $t=-u^5$ puts the integral in a form recognizable as the definition of an incomplete beta function: ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Finding Cauchy principal value for: $ \int_1^\infty \frac{ a x^2 + c }{x^4 - b x^2 - c} \mathrm{d}x $ I need to solve the integral
$ \displaystyle \mathcal{P} \int_1^\infty \frac{ a x^2 + c }{x^4 - b x^2 - c} \mathrm{d}x $,
where $\mathcal{P}$ is the Cauchy principal value, $ - 1 \leq c \leq 1$ and $a, b$ are both real... | I actually ended up using a different solution, I found more direct and intuitively.
It is completely equivalent with @Yves method, but I just state it for completeness.
$$ \mathcal{P} \int_{1}^{\infty} \mathrm{d} x
\frac{a x^{2} + c }
{
x^{4} - b x^{2} - c
}
=
\mathcal{P} \int_{1}^{\infty} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Reverse Fatou's lemma on probability space Let $(\Omega, \mathcal{F},\mathbb{P})$ be probability space and $E_{n \in \mathbb{N}}$ be $\mathcal{F}$-measurable sets.
Show example that reverse Fatou's Lemma, $\mathbb{P}(\limsup_n E_n) \geq \limsup_n \mathbb{P}(E_n)$, meets inequality strictly.
I understand this inequality... | The correct example: $\Omega=[0,1]$, $\mathcal F=\mathcal B(\Omega)$, $P=\mathrm{Leb}$, $E_{2n}=[0,1/2]$ and $E_{2n+1}=(1/2,1]$ for every $n$, then $\limsup E_n=\Omega$ and $P(E_n)=1/2$ for every $n$ hence $$P(\limsup E_n)=1\gt1/2=\limsup P(E_n).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/873827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How many of the 9000 four digit integers have four digits that are increasing? How to find the number of distinct four digit numbers that are increasing or decreasing?
The correct answer is $2{9 \choose 4} + {9 \choose 3} = 343$. How to get there?
| The analysis that was used goes as follows:
Not using $0$: We choose $4$ non-zero digits. Once we have done that, we can arrange them in increasing order in $1$ way, and in decreasing order in $1$ way, for a total of $2\binom{9}{4}$.
Using $0$: They can only be decreasing. And we need to choose $3$ non-zero digits to g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Word problem regarding system of linear congruences... Full problem:
A hoard of gold pieces ‘comes into the possession of’ a band of $15$ pirates.
When they come to divide up the coins, they find that three are left over.
Their discussion of what to do with these extra coins becomes animated,
and by the time som... | Have
$$x \equiv 3 \mod 15$$
$$x \equiv 2 \mod 7$$
$$x \equiv 0 \mod 4$$
Using Chinese Remainder Theorem, we first solve
$$x \equiv 3 \mod 15$$
$$x \equiv 2 \mod 7$$
Since $15,7$ are relatively prime, we have
$$15(1)+7(-2)=1$$
$$\implies 15(1)(2)+7(-2)(3)=30-42=-12$$
$$\implies x \equiv 93 \mod 105$$
Now we need to sol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find derivative of an integral of this type $$f(x) = \int _x^{e^x}\:\left(\sin t^2\right)\,dt$$
How to find the derivative $f'(x)$
Attempt: $\sin (e^{x^2}) e^x$
| Use Fundamental theorem of calculus, let $F$ be antiderivative of $\sin t^2$, then you have:
$$f(x)=F(e^x)-F(x)$$
So:
$$f'(x)=e^xF'(e^x)-F'(x)=e^x \cdot \sin ((e^{x})^2)-\sin (x^2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/874046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
A transcendental number from the diophantine equation $x+2y+3z=n$ Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation
$$x+2y+3z=n$$
Prove that
$$
\sum_{n=0}^{\infty} \frac{1}{D_{2n+1}}
$$
is a transcendental number.
| If I have not mistaken something,
$$ [x^{2n+1}]\frac{1}{(1-x)(1-x^2)(1-x^3)} = \frac{(n+1)(n+3)}{3}, $$
when $n\equiv 0,2\pmod{3}$, and
$$ [x^{2n+1}]\frac{1}{(1-x)(1-x^2)(1-x^3)} = \frac{(n+1)(n+3)+1}{3} $$
when $n\equiv 1\pmod{3}$, hence:
$$\begin{eqnarray*}\sum_{n=0}^{+\infty}\frac{1}{D_{2n+1}}&=&\sum_{n=0}^{+\infty}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/874158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Quotient Gaussian Integers Following Quotient ring of Gaussian integers, their extended conclusion is $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$. However it does not convince me, at least, one example below:
Let $a=2,b=0$, I cannot find explicit isomorphism between $\mathbb{Z}[i]/2\mathbb{Z}[i]$... | You'll never find such an isomorphism, because $x+x=0$ for all $x\in\mathbb Z[i]/2\mathbb Z[i]$, but not for all $x\in\mathbb Z/4\mathbb Z$.
The theorem you state is only true if $\gcd(a,b)=1$.
See this answer from the same question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/874231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Differentiability at a point Let $f:\mathbb{R}^{2}\mapsto\mathbb{R}\mathbb{}^{2}$ be given
by
$$f(x,y) = \left(\begin{array}{c}
x^{2}y+2y-x\\
3xy+4y
\end{array}\right)$$
Find a open set containing (0,0) where f has a differentiable inverse?.
I know the inverse function theorum guarentees there exists a neigbourhood
(o... | Let $U$ be an open neighborhood of $(0,0)\in{\mathbb R}^2$ for which $d{\bf f}(x,y)$ is regular at all points $(x,y)\in U$. When ${\bf f}$ is injective on $U$ then ${\bf f}$ maps $U$ bijectively onto an open neighborhood $V$ of $(0,0)$, and the inverse map ${\bf f}^{-1}:\ V\to U$ is $C^1$, by the inverse function theo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving a master equation with linear coefficients I have the following PDE:
$$
\partial_t P(x,y,t)=x\partial_xP(x,y,t)+(y-1)\partial_yP(x,y,t)+2P(x,y,t).
$$
Mathematica suggests that the solution is
$$
\dfrac{f((y-1)/x,t+\log x)}{x^2},
$$
where $f$ is an arbitrary function.
My question is: How to solve this equation? ... | $\partial_tP(x,y,t)=x\partial_xP(x,y,t)+(y-1)\partial_yP(x,y,t)+2P(x,y,t)$
$\partial_tP(x,y,t)-x\partial_xP(x,y,t)+(1-y)\partial_yP(x,y,t)=2P(x,y,t)$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$
$\dfrac{dx}{ds}=-x$ , letting $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/874426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's wrong with my aproach to solving this equation with multiple logarithms? A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?"
My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$
However, this leads to an incorrect answer. Instead... | Taking $\log_{10}$ of both sides and using the rule $\log_{10}(a^b)=b\log_{10}a$ gives
$$(\log_{10}(\log_{10}(x))(\log_{10}(\log_{10}(x))=\log_{10}10000=4\ ,$$
that is,
$$(\log_{10}(\log_{10}(x)))^2=4\ .$$
Hence
$$\log_{10}(\log_{10}(x))=2\quad\Rightarrow\quad
\log_{10}(x)=100\quad\Rightarrow\quad x=10^{100}$$
or
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/874477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find all integer solutions of $1+x+x^2+x^3=y^2$ I need some help on solving this problem:
Find all integer solutions for this following equation:
$1+x+x^2+x^3=y^2$
My attempt:
Clearly $y^2 = (1+x)(1+x^2)$, assuming the GCD[$(1+x), (1+x^2)] = d$, then if $d>1$, $d$ has to be power of 2. This implies that I can assume: $... | The solution is given in Ribenboim's book on Catalan's conjecture, where all Diophantine equations $$y^2=1+x+x^2+\cdots +x^k$$ are studied.
For $k=3$, only $x=1$ and $x=7$ are possible.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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The probability of getting a certain image by random pixelation Well, seeing that I'm terribly bad at math I don't know how to solve this, I'll try to explain, excuse me if I sound dumb.
Just suppose that I've got a photo/image with 320x240 resolution and 24 bit color depth (16,777,216 colors) and suppose that I made a... | The chance of 1 pixel would be 1 in 16777216.
The chance of 2 pixels would be (1/16777216) * (1/16777216).
So the chance of all of them would be (1/16777216) ^ (320*240).
Not very likely at all :) I'd stick with the lottery...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/874700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Trace of power of stochastic matrix I would like to know if this statement is true.
Having a stochastic matrix (rows sum up to 1), with a positive (non-negative) diagonal, then it holds that
$$\text{trace}({W^2})\leq \text{trace}({W}),$$
(or more generally, if $p\geq q$, then $\text{trace}({W^p})\leq \text{trace}({W}^q... | This is (still) trivially false if you just take a small epsilon
$p_{12}=p_{21}=1-\epsilon$ and $p_{11}=p_{22}=\epsilon$
The trace of $P^2$ is $2((1-\epsilon)^2+\epsilon^2)>2\epsilon$, if take $\epsilon$ to be, say, smaller than 0.2
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/874759",
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Why are duals in a rigid/autonomous category unique up to unique isomorphism? I'm having trouble understanding the following statement:
"In a rigid category, duals are unique up to unique isomorphism."
It seems to me that this isomorphism is not unique.
Let me try to give a counterexample: Let $(X,Y,\epsilon: X \otimes... | Thanks to Zhen Lin to bring me on the right track.
The misunderstanding is the following:
The dual object ($Y$ in my example) is unique up to isomorphism, but this isomorphism is not unique. However, a dual is more than merely the object, it's the triple $(Y,\epsilon,\eta)$. This one is indeed unique up to unique isomo... | {
"language": "en",
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What is the remainder when the number below is divided by $100$? What is the remainder when the below number is divided by $100$?
$$
1^{1} + 111^{111}+11111^{11111}+1111111^{1111111}+111111111^{111111111}\\+5^{1}+555^{111}+55555^{11111}+5555555^{1111111}+55555555^{111111111}
$$
How to approach this type of question? I ... | HINT:
$$(1+10n)^{1+10n}=1+\binom{1+10n}1(10n)\pmod{100}\equiv1+10n$$
and $$(5+50n)^{1+10n}=5^{1+10n}+\binom{1+10n}1(50n)5^{10n}\pmod{100}$$
Now, $$5^{m+2}-5^2=5^2(5^m-1)\equiv0\pmod{100}\implies5^{m+2}\equiv25\pmod{100}$$ for integer $m\ge0$
$$\implies5^{1+10n}+\binom{1+10n}1(50n)5^{10n}\equiv25+(1+10n)(50n)25\pmod{100... | {
"language": "en",
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"source": "stackexchange",
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Question on Green's Theorem Consider the vector field $\textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j}$.
Use Green's Theorem to evaluate $\int_C\textbf{f} \dot d\textbf{r}$, where $C$ is the ellipse given by $(x-1)^2+\frac{y^2}{9}=1$, oriented counterclockwise.
$\int_C\t... | After fixing the partial derivative typo, we have
$$\int \int_R 2y\sqrt{x}-2\sqrt{x}y \ dx \ dy$$
$$\int \int_R 0 \ dx \ dy$$
It does not matter what the bounds are, the answer is zero.
| {
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Distinguishability problem / How many ways are there to put 6 balls in 3 boxes if the balls are distinguishable but the boxes are not?
I'm not quite sure how to approach it, $\frac{3^6}{3!}$ is not an integer. Thanks.
| I'm not the best at combinatorics but here's a go. I think I remember problems like this in statistical mechanics. $B$ for box.
$B\mid\quad B\mid\quad B\mid \qquad ways$
$6\mid\quad 0\mid\quad 0\mid \qquad 1 \qquad$ all in one box.
$5\mid\quad 1\mid\quad 0\mid \qquad6\qquad$ one of the six on its own
$4\mid\quad 2\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/875136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of ways distribute 12 identical action figures to 5 children Need a little help with this problem.
Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most three action figures.
So far I have that we are look... | You have $\displaystyle G(x)=(1+x+x^2+x^3)^5=\big(\frac{1-x^4}{1-x}\big)^5=(1-x^4)^5(1-x)^{-5}$
$\displaystyle=\big(1-5x^4+\binom{5}{2}x^8-\binom{5}{3}x^{12}+\cdots\big)\big(\sum_{k=0}^{\infty}\binom{k+4}{4}x^4\big)$,
so now you just have to find the coefficient of $x^{12}$ in this expression.
| {
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Existence of a function with certain integral properties Does there exist a non-negative Borel-measurable function $g:\mathbb [1,\infty)\to[0,\infty)$ such that
\begin{align*}
\int_1^{\infty}g(y)^2\,\mathrm dy<&\,\infty,\\
\int_1^{\infty}\frac{g(y)}{\sqrt{y}}\,\mathrm dy=&\,\infty?
\end{align*}
$g(y)=1/\sqrt{y}$ “almos... | Got it. $$g(y)=\frac{\chi_{[2,\infty)}(y)}{\sqrt{y}\log y}.$$
| {
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Linear Maps from a finite space to an infinite space Suppose V is finite dimensional with dim V > 0. Prove that if W is infinite dimensional then $L(V, W)$ is infinite dimensional.
Help? I really have no idea how to go about this one?
I'm assuming I need to use the fact that if a space is infinite dimensional then ther... | Maybe it's easier to prove the contrapositive: if $L(V,W)$ is finite-dimensional, then $W$ is finite-dimensional.
Let $f_1,...,f_n : V \rightarrow W$ be linear functions that span $L(V,W).$ Fix any nonzero $v \in V$. Then $f_1(v),...,f_n(v)$ span $W$, so $W$ is finite-dimensional.
The reason for this is that, for any $... | {
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Is it possible to create a bigger square using distinct smaller ones? Another user just inquired about possible solutions to the famous $70$x$70$ square puzzle. When I encountered that many years ago and the first idea that came to my mind as to why I wouldn't think it was possible to solve had to do with the $1$x$1$ s... | I commented on the previous post as well!
There is a good reason you couldn't find an example by hand: the smallest example of what's called a perfect squared square is a $112\times 112$ (link).
There is much more research here.
| {
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"source": "stackexchange",
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Joint density distribution and Variance I was wondering if there is a way to calculate the joint distribution of two fully correlated variables, both with known distributions, expected value and variance, without knowing the conditional distribution?
If this is not possible, is there a way of finding Var$(X,Y)$ = E$[(X... | With slightly more general details than in the
original version of André Nicolas's answer, it must be that $Y = aX+b$ where
$$a = \sqrt{\frac{\operatorname{var}(Y)}{\operatorname{var}(X)}}
\quad \text{and}\quad b = E[Y] - aE[X].\tag{1}$$
There is no joint density of $X$ and $Y$ in the sense that $X$ and $Y$ are not joi... | {
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if $\frac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$,find $a_{n}$ Let
$$\dfrac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$$
Find the closed form $$a_{n}$$
since
$$(1-x^4)(1-x^3)(1-x^2)=(1-x)^3(1+x+x^2+x^3)(1+x+x^2)(1+x)$$
so
$$\dfrac{1}{(1-x^4)(1-x^3)(1-x^2)}=\dfrac{1}{(1-x)^3(1+x+x^2+x^3)(1+x+... | Hints :
*
*First, prove that
$$\frac{1}{(1-x^2)(1-x^3)(1-x^4)} = \frac{7}{32(x+1)}-\frac{59}{288(x-1)}+\frac{1}{8(x-1)^2}+\frac{1}{16(x+1)^2}-\frac{1}{24(x-1)^3}+\frac{x+2}{9(x^2+x+1)}+\frac{1-x}{8(x^2+1)}.$$
*Then, use that
$$ \frac{1-x}{x^2+1} = -\frac{1+i}{2(x-i)}+\frac{-1+i}{2(x+i)}.$$
and
$$ \frac{x+2}{x^2+x+... | {
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Inequality involving Jensen (Rudin's exercise)
Exercise (Rudin, R&CA, no. 3.25). Suppose $\mu$ is a positive measure on the space $X$ and let $f \colon X \to (0,+\infty)$ be such that $\int_X f \, d\mu=1$.
Then for every $E \subset X$ with $0<\mu(E)<\infty$ we have
$$
\int_E \log f \, d\mu \le \mu(E) \log \frac{1}... | Write the inequality as
$$\frac{1}{\mu(E)} \int_E \log f\,d\mu \leqslant \log \frac{1}{\mu(E)}.$$
Jensen's inequality gives you
$$\exp \left(\frac{1}{\mu(E)}\int_E \log f\,d\mu\right) \leqslant \frac{1}{\mu(E)}\int_E f\,d\mu.$$
The remaining part should be clear.
| {
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What are some conceptualizations that work in mathematics but are not strictly true? I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to iteratively learn and unlearn along the ... | What about all the basic rules of weight & motion--aren't they just simplifications of terribly complex rules that generally work as long as you don't deal with anything too small or going too fast?
It seems that EVERY problem in early physics/calculus is simplified to eliminate most of the variables because the proble... | {
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Need help with this Geometric sequence problem First, sorry if Im not using the right syntax, im translating the problem and im not sure if im supposed to say "Sequence" or "series", and also thanks to who ever tries to help.
The sum of a geometric sequence is 20, and the sum of its squared terms is 205.
find how many... | You have found that $q^n+1=20.5(q+1)$.
But since the sum of the terms is $20$, we have $q^n-1=40(q-1)$. (In essence you had written down this equation also.)
Subtract, and find $q$, and then $n$. The numbers are disappointingly small, $3$ and $4$ respectively.
| {
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How to evaluate $\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$ using L'Hôpital's rule? I'm stuck on how to evaluate the following using L'Hôpital's rule:
$$\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$$
This is a problem that I encountered on Khan Academy and I attempted to understand it using the resou... | $(1+{2 \over n})^{3n} = ((1+{2 \over n})^{n})^3$.
We have $\lim_{n \to \infty} (1+ {\alpha \over n})^n = e^\alpha$.
To see the latter using l'Hôpital, let $a_n = (1+ {\alpha \over n})^n$.
Then $\log a_n = n \log(1+ {\alpha \over n})= { \log(1+ {\alpha \over n}) \over {1 \over n}}$.
Note that $\lim_{n \to \infty} { \log... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is a densely defined symmetric operator $T$ extended by its adjoint $T^*$? This is a result I've seen stated a few times, but I can't seem to come up with a proof!
Suppose $T$ is a densely defined linear operator with domain $D(T)\subset H$, where $H$ is a Hilbert space then if $T$ is symmetric i.e. $\langle Tx,y\... | For $y \in D(T)$, we have, due to the symmetry of $T$,
$$x \mapsto \langle Tx,y\rangle = x \mapsto \langle x, Ty\rangle,$$
and the latter is easily seen to be continuous, hence $D(T) \subset D(T^\ast)$.
| {
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Frobenius method, why is it an issue when the roots of the indicial equation differ by an integer When solving second-order differential equations by the Frobenius method at a regular singular point, you are supposed to use the two roots of the indicial equation to give you two independent solutions.
If there is only ... | If you look at http://math.creighton.edu/nielsen/DE_Fall_2010/Series%20Solutions/Series_Solutions_Beamer.pdf, they write the resulting recurrence for one of the solutions as $a_n F(n+r) = E$, where $F(r)$ is the indicial equation, and I'm writing $E$ to abbreviate a complicated expression which depends on a variety of ... | {
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How can I calculate the total number of possible anagrams for a set of letters? How can I calculate the total number of possible anagrams for a set of letters?
For example:
"Math" : 24 possible combinations.
math
maht
mtah
mtha
mhat
mhta
amth
amht
atmh
athm
ahmt
ahtm
tmah
tmha
tamh
tahm
thma
tham
hmat
hmta
hamt
hatm
ht... | I worked out a formula for calculating the number of anagrams for an a-letter word where b letters occur c times, d letters occur e times, f letters occur g times, etc.
a!/((c!^b) * (e!^d) * (g!^f)...)
| {
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A question on the proof of 14 distinct sets can be formed by complementation and closure In Munkres, problem 20 of Section 2-6, it says that 14 distinct sets can be formed by complementation and closure. I see only five so far.
Let f be the function of closure mapping and g be the function of complementation mapping.
I... | The following page lists the rest, and since it lets you experiment in real time, may also accelerate your intuitive and pictorial understanding:
http://www.maa.org/sites/default/files/images/upload_library/60/bowron/k14.html
Though the 14-set theorem is more algebraic than topological, constructing a Kuratowski 14-set... | {
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Prove $\,17\mid 2a+3b \,\Rightarrow\, 17\mid 9a + 5b$ So, according to the book, for all $a, b, c$ that are elements of integers, it holds that $a|b$ implies $a|bx$ for all $x$ that is an element of integers. In other words it works for all ARBITRARY $x$ in the universe $Z$.
However, please consider this question:
Whe... | HINT:
Eliminate one unknown
$$9(2a+3b)-2(9a+5b)=?$$
| {
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Trigonometric functions of the acute angle Find the other five trigonometric functions of the acute angle A, given that:
\begin{align}
&\text{a)}\ \ \sec A = 2 \\[15pt]
&\text{b)}\ \ \cos A = \frac{m^2 - n^2}{m^2 + n^2}
\end{align}
Help me. I don't know how to solve this one. Thanks.
| It is useful to know at least some basic trigonometric identities. See List of trigonometric identities at Wikipedia for a very complete list.
a) Since $\cos A=\frac1{\sec A}$, you get $\cos A=\frac12$. Can you get possible values of $A$ from there?
b) If $$\cos A=\frac{m^2-n^2}{m^2+n^2}=\left(\frac m{\sqrt{m^2+n^2}}... | {
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If $a_{n+1}=\cos(a_n)$ for $n\ge0$ and $a_0 \in [-\pi/2,\pi/2]$, find $\lim_{n \to \infty}a_n$ if it exists Let $a_{n+1}=\cos(a_n)$ for $n\ge0$ and $a_0 \in [0,\pi/2]$
Find $\lim_{n \to \infty}a_n$ if it exists.
I drew some sketches and it does seem like the limit exists, it's probably $x$ such that $\cos(x)=x$
I have ... | The "Fixed Point" method in numerical analysis says that if a function is continuously differentiable in a neighbourhood of a fixed point $x_0$ and
$$
|f'(x)| \lt 1
$$
it will converge to a fixed point
source: Fixed point
| {
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Intuitive ways to get formula of cubic sum Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$
I think the cubic sum is squaring the arithmetic sum $$1^3 + 2^3 + \dots + n^3 = (1 ... | Here are some 'Proof without Words' for this identity;
As Hakim showed (but this one might be slightly clearer);
Here's another very clean illustration by Anders Kaseorg;
Here, the total top volume is undoubtedly $1^2+2^2+3^3+\cdots +n^2,$ while on the volume of the bottom part is;
$$1\left(1+2+\cdots+n\right)+2\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/876922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Why is $\frac{d}{d \mu} \big|_{\mu=0} \, F^*_\mu F^*_\lambda t = F^*_\lambda \frac{d}{d \mu} \big|_{\mu=0} \, F^*_\mu t $? In the book "Manifolds, Tensor Analysis, and Applications" by Marsden, Ratiu, Abraham the following relation (see the proof of 6.4.1, third edition) is used:
$$\frac{d}{d \mu} \bigg|_{\mu=0} \, F^*... | Result : If $\phi$ and $\psi$ are $1$-parameter groups s.t. $$ \psi'= X$$
then $$\phi^\ast (L_X\alpha) = \phi^\ast \lim_t \frac{\psi(t)^\ast
\alpha - \alpha }{t} = \lim \frac{(\phi^{-1} \circ \psi \circ \phi
)^\ast
(\phi^\ast \alpha )-\phi^\ast \alpha }{t} $$
Here recall $$ \frac{d}{ds}\bigg|_{s=0}
\phi^{-1}\circ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Derivative of a matrix function with respect to a matrix I am trying to differentiate the following function, with respect to a matrix $X$:
$$
\operatorname{tr}(AX(X^TX)^{-1}B)
$$
where tr corresponds to the trace. Is there an easy way to see what the derivative will be? I've come across rules for $\operatorname{tr}(AX... | Define $W \equiv (X'X)^{-1}$, then $f = BAX : W$ and its derivative is
$$
\frac {\partial f} {\partial X} = A'B'W - XW(BAX+X'A'B')W
$$
The algebra to arive at this result is tedious but straight-forward. The only tricky part is knowing that
$$ \eqalign {
dW &= - W\,\,dY\,\,W \cr
} $$
where $Y \!\equiv W^{-1}\!= X'X$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show if A is denumerable and $x\in A$ then $A-\{x\}$ is denumerable My thoughts:
If $A$ is denumerable then it has a bijection with $\mathbb{N}$
So therefore $A\rightarrow \mathbb{N}$. Then x is a single object in A and A is infinite.
So if a single object is removed from then $A$ is still infinite.
| Your reasoning is informally correct. If you want to give a careful answer to this question you should demonstrate a bijection from $A \setminus \{x\}$ to $\mathbb{N}$.
You have a given bijection between $A$ and $\mathbb{N}$. Can you use this as a stepping stone to get from $A \setminus \{x\}$ to $\mathbb{N}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/877256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof $e^n*n!$ is an asymptote of $(n+1)^n$ I would like to prove $\lim_{n\to \infty}e^nn!-(n+1)^n=0$.
All I have really done is show $(n+1)^n=\sum_{i=0}^n\frac{n!}{(n+1)^i(i!)(n-i)!}$
| (This is too long for a comment, but neither is it yet a full answer.)
Consider the Taylor series expansion of $e^x$.
\begin{align*}
e^x &= \sum_{k = 0}^\infty \frac{x^k}{k!}\\
&= \lim_{n \rightarrow \infty} \sum_{k = 0}^n \frac{x^k}{k!}
\end{align*}
It follows
\begin{align*}
\lim_{n \rightarrow \infty} n! e^n &= \lim_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Polygons with equal area and perimeter but different number of sides? Let's say we have two polygons with different numbers of sides. They can be any sort of shape, but they have to have the same area, and perimeter.
There could be such possibilities, but can someone show me with pictures? I just need visualize it.
So... | ┌─┐ ┌┐
│ │ │└─┐
│ └─┐ │ └┐
└───┘ └───┘
Edit: I like the above figures because they're easy to generalize to many sides. But if it's unclear that they have the same area, here's another pair: the L and T tetrominoes.
You can imagine sliding the square on the right side up and down relative to the 1x3 bar on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Null-homotopic covering space map I'm stuck with the following question, which looks quite innocent.
I'd like to show that if a covering space map $f:\tilde{X}\to X$ between cell complexes is null-homotopic, then the covering space $\tilde{X}$ must be contractible.
Since $f$ is null-homotopic there exists a homotopy $H... | Since $f$ is nullhomotopic, $f_*:\pi_n(\tilde X)\to \pi_n(X)$ are trivial for all $n$. Consequently $\pi_n(\tilde X)$ are all trivial. Whitehead theorem implies $\tilde X$ is contractible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/877477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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In group theory, is it true that $f(X \vee Y) = f(X) \vee f(Y)$? ($\vee$ denotes join).
Let $G$ and $H$ denote Abelian groups, $X$ and $Y$ denote subalgebras of $G$, and let $f : G \rightarrow H$ denote a homomorphism. Then:
$$f(X \vee Y) = f(X+Y) = f(X)+f(Y) = f(X) \vee f(Y)$$
So $f(X \vee Y) = f(X) \vee f(Y)$.
Now su... | That we have $f(X\vee Y)\subseteq f(X)\vee f(Y)$ should be clear (the image of a product of elements in $X$ or $Y$ is a product of elements in $f(X)$ or $f(Y)$).
For the other direction, let $g\in f(X)\vee f(Y)$. Then $g=g_1\cdot\cdots\cdot g_n$ where $g_i\in f(X)$ or $g_i\in f(Y)$. Thus each $g_i$ is of the form $f(x_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluating $\int_{-\infty}^\infty \frac{\sin x}{x-i} dx$ I would like to evaluate the integral
$$\int_{-\infty}^\infty \frac{\sin x}{x-i} dx,$$
which I believe should be equal to $\frac{\pi}{e}$. However, I cannot reproduce this result by hand. My work is as follows: first, we evaluate the indefinite integral.
\begin{a... | The mistake is simple--$\mathrm{Ci}$ has a branch cut across the negative real axis, so $\mathrm{Ci}(-\infty - i)$ should indeed evaluate to $-i \pi$ rather than $i \pi$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$ I need to evaluate the following integral:
$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$
I thought of evaluating the iterated integral $\displaystyle\int_{-\infty}^{\infty}dx\int_{-\infty... | Recall:
$$x^2-xy+y^2=(x-\frac{1}{2}y)^2+\frac{3}{4}y^2$$
With that you get:
$$\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\frac{1}{2}(x^2-xy+y^2)}dxdy=\\\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\frac{1}{2}\left((x-\frac{1}{2}y)^2+\frac{3}{4}y^2\right)}dxdy=\\
\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$\int_\mathbb{R} \bigg( \frac{1}{h} \int_x^{x+h} |f(t)| dt\bigg) dx= ||f||_{L^1}$? $$\int_\mathbb{R} \bigg( \frac{1}{h} \int_x^{x+h} |f(t)| dt\bigg) dx= ||f||_{L^1} \;\;?$$
I worked out that the equality holds for each $\chi_{[a,b]}$, therefore it holds for each piecewise constant function. By a density argument, it ... | Your argument is correct, if you note that the left hand side of the equality is a continuous function of $f\in L^1$. Then the denseness perpetuates the equality from the simple functions to all of $L^1$.
But there is an easier argument, change the order of integration:
$$\begin{align}
\int_\mathbb{R} \left(\int_x^{x+h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Exercise 1.13 of chapter 1 of Revuz and Yor's This is the exercise 1.13 of chapter 1 of Revuz and Yor's.
*
*Let $B$ be the standard linear BM. Prove that $\varlimsup_{t\to\infty}(B_t/\sqrt{t})$ is a.s. $>0$ (it is in fact equal to $+\infty$ as will be seen in Chap. 2)
*Prove that $B$ is recurrent, namely: for... | Re 1., note that the random variable $X=\limsup\limits_{t\to\infty}B_t/\sqrt{t}$ is asymptotic hence, by Kolmogorov zero-one law, if $P(X\gt0)\lt1$, then $P(X\gt0)=0$, that is, $X\leqslant0$ almost surely. If this holds, then, by symmetry, $\liminf\limits_{t\to\infty}B_t/\sqrt{t}\geqslant0$ almost surely, thus, $B_t/\s... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Hide my invoice number I'm not a mathematician, so please forgive any ignorance.
I have a small business - I'm generating invoices incrementally. I'm currently on about invoice number 4000.
I guess I don't want my customers knowing how much business I'm doing (i.e. if they get an invoice for 4500, they know I've been d... | Why make such a simple problem so complex?
Most e-commerce and invoice software supports starting numbers and increments, so just pick a large number and increment by another odd number. So something like:
1034578 = first invoice number
32876 = increment amount
So your invoices would follow 1034578 + 32876X like so:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/877947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 5
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What's the name of this algebraic property? [Complementary Subgroup Test] I'm looking for a name of a property of which I have a few examples:
$(1) \quad\color{green}{\text{even number}}+\color{red}{\text{odd number}}=\color{red}{\text{odd number}}$
$(2) \quad \color{green}{\text{rational number}}+\color{red}{\text{irr... | Don't prime numbers offer a counter-example to the general truth of this property?
Prime $+$ not-prime $=$ not-prime $==> 17 + 4 = 21$
Prime $+$ not-prime $=$ prime $==> 7+4=11$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/878038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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Polynomial $f(x)$ degree problem. Suppose the polynomial $f(x)$ is of degree $3$ and satisfies $f(3)=2$, $f(4)=4$, $f(5)=-3$, and $f(6)=8$. Determine the value of $f(0)$.
How would I solve this problem? It seems quite complicated...
| If you just want to use subtraction, you could take first differences
2 4 -3 8
2 -7 11
-9 18
27
then expand this to the left
-328 -137 -36 2 4 -3 8
191 101 38 2 -7 11
-90 -63 -36 -9 18
27 27 27 27
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Generalising integration by parts for the product of more than two functions Just as the product rule can be generalised to the product of more than two functions, i.e. $$\frac{d}{dx} \left [ \prod_{i=1}^k f_i(x) \right ]
= \sum_{i=1}^k \left(\frac{d}{dx} f_i(x) \prod_{j\ne i} f_j(x) \right)
= \left( \prod_{i=1}^k f_... | The Wikipedia article on integration by parts gives you the generalization you're looking for:
$$\Bigl[ \prod_{i=1}^n u_i(x) \Bigr]_a^b = \sum_{j=1}^n \int_a^b \prod_{i\neq j}^n u_i(x) \, du_j(x),$$
where $u_i(x)$ are your $n$ functions of $x$ that are terms of the product that comprise your integrand.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/878213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do we define the mirror image of a knot in general 3-manifolds How do we define the mirror image of a knot in general oriented 3-manifolds ? For instance for a knot in an irreducible integer homology sphere.
| One method would be to express the manifold as an open book decomposition (see http://en.wikipedia.org/wiki/Open_book_decomposition). Then project the knot onto one of the pages (fibers) of the decomposition recording over an under crossings. Switch over crossings to undercrossings and vice verse. You might want to s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Applications of Algebra in Physics Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics.
I have a good preparation in Algebra and Representation Theory (in particular about Representations of Lie Algebras), and I'm fascinated wit... | Peter Woit, the author of the book "Not Even Wrong" and a blog by the same name, has been working on a book on quantum mechanics as described by representation theory. The latest draft may be found at the following link:
Quantum Theory, Groups and Representations:
An Introduction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/878373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Is there a way to calculate the area of this intersection of four disks without using an integral? Is there anyway to calculate this area without using integral ?
| Let $R$ be its radius and $D$ its diameter: $R = 5$, $D = 10$.
$$\begin{align}
\text{Area of big square} &= D^2 = 100 \\
\text{Area of circle} &= \frac{\pi D^2}{4} \approx 78.54 \\
\text{Area outside circle} &= 100 - 78.54 = 21.46 \\
\text{Area of 4 petals} &= 78.54 - 21.46 = 57.08 \\
\text{Area of single petal} &= \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/878457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 5,
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Solutions to functional equation $f(f(x))=x$ Is there any more solutions to this functional equation $f(f(x))=x$?
I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.
| If you don't make any niceness assumptions about $f$, there are lots. Partition $\Bbb{R}$ (or whatever you want $f$'s domain to be) into $1$- and $2$-element subsets, in any way you like. Then define $f(x)=y$, if $\{x, y\}$ is in your partition, or $f(x)=x$, if $\{x\}$ is in your partition.
Moreover, any such $f$ yield... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Difference isometrically isomorphism and homeomophism. What is the difference between isometrically isomorphism and homeomorphism?is an isometric mapping is continuous?
| An isometry is a map $f:X\to Y$ between metric spaces that preserves distances: $d_Y(fx, fy) = d_X(x, y)$. Such maps are automatically continuous (just use the $\delta$-$\epsilon$ definition of continuity) and injective, but they may not be surjective; an isometric isomorphism is one that's both a bijection and an isom... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Non integer derivative of $1/p(x)$ I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$
It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine.
| Why don't you just use the formula?
For generic $f$ and for 0<$\alpha$<1 you have:
$$D^\alpha(f(x))=\frac{1}{\Gamma(1-k)}\frac{\partial}{\partial x}\int_0^x\frac{f(t)}{(x-t)^\alpha}\,\mathrm{d}t$$
So after deriving [k] (the integer part) times you have the remainder $\alpha\in[0,1]$:
$$D^\alpha(D^{[k]}\left(\frac{1}{p(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/878707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find an angle in range(0, 360) between 2 vectors? I know that the common approach in order to find an angle is to calculate the dot product between 2 vectors and then calculate arcus cos of it. But in this solution I can get an angle only in the range(0, 180) degrees. What would be the proper way to get an angle... | Before reading this answer - Imagine your angle in a 3D space - you can look at it from the "front" and from the "back" (front and back are defined by you). The angle from the front will be the opposite of the angle that you see from the back. So there is no real sense in a value in a range larger than $[0,180]$.
If yo... | {
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"url": "https://math.stackexchange.com/questions/878785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
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Problem involving decomposition of measures Let $\mu$ be a signed measure. We wish to prove that $$\left| \int{f} \> d\mu \right| \leq \int{|f|} \> d|\mu|.$$
(We are given the following defintion: $\int{f} \> d\mu = \int{f} \> d\mu^{+} - \int{f} \> d\mu^{-}$.)
For the proof, I want to just write $$\left| \int{f} \> d\m... | The last equality rests on the fact that $$|\mu|=\mu^++\mu^-,\tag{1} $$ where $\mu= \mu^+-\mu^-$ (Jordan-Hahn decomposition), where $\mu^+$ and $\mu^-$ are positive and muutually singular.
Equality (1) is either the definition of $|\mu|$, or it canbe deduced from the definition of $|\mu|$ with partitions using the sup... | {
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Finite additivity in outer measure Let $\{E_i\}_{i=1}^n$ be finitely many disjoint sets of real numbers (not necessarily Lebesgue measurable) and $E$ be the union of all these sets. Is it always true that
$$
m^\star (E)=\sum_{i=1}^N m^\star(E_n)
$$
where $m^\star$ denotes the Lebesgue outer measure? If not, please giv... | The sentence "$m^{*}$ is not finitely additive" is independent from the theory $ZF+DC$.
Fact 1. (ZF)+(AC). $m^{*}$ is not finitely additive.
Proof
Let $X$ be a subset of the real line $R$. We say that $X$ is a Bernstein set in $R$ if
for every non–empty perfect set $P \subseteq R$ both sets
$P \cap X$, $P \cap (R \setm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/878962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Solving this Exponential Equation This is the equation that I need help with. The fact that there is that extra $1$ is throwing me off. If you move the $4^x$ term over and take the $\log$ of both sides, then you have a $\log$ with a polynomial inside.
$$5^x - 4^x = 1$$
| The function $y(x)=5^x-4^x$ is negative if $x<0$ . So $x<0$ is excluded. For $x>0$ the function is strictly increasing from $0$ to $\infty$. So, there is only one value of $x$ so that $y=1$.
Obviously this value is $x=1$ because $y(1)=5^1-4^1=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/879049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Mathematician who talked about the probability of a "good" graph? In my undergraduate years, one of my professors always talked about this one mathematician who was talking about "good" graphs and wondered about the existence of such a graph. Apparently this mathematician could not find such a graph, and then proceeded... | Not sure without more context, but my best guess is the mathematician was Paul Erdős and technique you are talking about is the Probabilistic Method.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/879117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$F/K$ algebraic and every nonconstant polynomial in $K[X]$ has a root in $F$ implies $F$ is algebraically closed. Let $F/K$ be an algebraic extension of fields in characteristic zero. If $F/K$ is normal, and every nonconstant polynomial $f \in K[X]$ has a root in $F$, then $F$ is algebraically closed. This is obvious... | Let $E/F$ be a finite extension (which is separable as the characteristic is zero), we will prove that $E=F$, hence $F$ is algebraically closed:
Let $a$ be a primitive element of $E/F$. Then $a$ is algebraic over $K$, so there exists a finite Galois extension $N/K$ containing $a$, in particular
$$
E\subseteq NF.
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Proving that $f(x)=2^x$ is $O(x^2)$ Can someone help me with this problem? I don't really know what to do if the x is in exponential form.
| To see that this is way false, we just observe:
$$2^x\le Mx^2\iff x\log 2\le \log M+2\log x$$
$$\iff x\le \log_2(M)+2\log_2(x)$$
But it is clear from L'Hôpital's rule (among other things) that this is false, since
$$\lim_{x\to\infty} {M\over x\log 2}+{2\log x\over x\log 2}=\lim_{x\to\infty} {2/x\over \log 2}=0$$
(in f... | {
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"answer_id": 0
} |
Smooth surfaces that isn't the zero-set of $f(x,y,z)$ The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth function? If not, what do the counterexamples look like?
I was thin... | Georges Elencwajg's "easy proof" is providing a "global solution" to your problem. Locally one can argue as follows: A smooth surface $S\subset{\mathbb R}^3$ is produced by a $C^1$-map
$${\bf g}:\quad(u,v)\mapsto\left\{\eqalign{x&=g_1(u,v)\cr y&=g_2(u,v)\cr z&=g_3(u,v)\cr}\right.$$ with $d{\bf g}(u,v)$ having rank $2$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 0
} |
Lexicographical rank of a string with duplicate characters Given a string, you can find the lexicographic rank of the string using this algorithm:
Let the given string be “STRING”. In the input string, ‘S’ is the first character. There are total 6 characters and 4 of them are smaller than ‘S’. So there can be 4 * 5! s... | There's a similar process, complicated by counting permutations of strings with duplicates. For example, the number of permutations of AAABB is $5!/3!2!$.
With that in mind, here's how we could find the rank of BCBAC. We count the smaller permutations $s$ by considering the first position where $s$ is smaller. For e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Help with simple rotation on an x,y plane I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane.
Something that is hopefully quite easy for you guys.
Anyway, to be concrete,
what I need is, say I have an object (image) on an x... | Take one of the corners, say it has coordinates $(x,y)$. Let $\alpha$ be the angle that $(x,y)$ makes with the positive $x$-axis and let $r=\sqrt{x^2+y^2}$. Then $x=r\cos\alpha$ and $y=r\sin\alpha$. Let's say you are rotating by an angle of $\theta$. Call this linear transformation $T_\theta:\mathbb{R}^2\to\mathbb{R}^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Where can I find the proof of this Ramanujan result? I'm searching for a proof of one impressive Ramanujan result. Not one in particular, the only request I have is to be really impressive.
For example
$$
\sqrt{\phi+2}-\phi=\frac{e^{-2\pi/5}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}}{1+\cdots}}}
$$
where $\phi=\frac{1+\sqr... | The series for $1/\pi$ is proved in J. M. Borwein and P. B. Borwein, Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987.
See also Motivation for Ramanujan's mysterious $\pi$ formula
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/879576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
$f$ is twice differentiable, $f + 2 f^{'} + f^{''} \geq 0$ , prove the following Let $ f : [0,1] \rightarrow R$. $f$ is twice diff.
and $f(0) = f(1) = 0$
If
$f + 2 f^{'} + f^{''} \ge 0$ , prove that $f\le 0$ in the domain.
Please don’t give complete solution, only hints.
| Hint Let $g(x)=f(x)e^x$. Then
$$g''=(f+2f'+f'')e^x \geq 0 \,.$$
That means that $g$ is....
How does this solve the problem?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/879674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Evaluation of Sum of $ \sum_{n=1}^{\infty}\frac{\sin (n)}{n}$. If $\displaystyle S = \sum_{n=1}^{\infty}\frac{\sin (n)}{n}.$ Then value of $2S+1 = $
Using Fourier Series Transformation I am Getting $2S+1=\pi$
But I want to solve it Using Euler Method and Then Use Logarithmic Series.
$\bf{My\; Try::}$ Using $\displayst... | my attempt :
$$\ \ S=\sum_{n=1}^{\infty } \frac{sin(n)}{n}=\sum_{n=1}^{\infty }
= \int_{0}^{\infty } e^{-nw}\sin(n)dw\\ \\ \\$$
$$\therefore S=Im\int_{0}^{\infty }\sum_{n=1}^{\infty
}e^{-(w-i)n}dw=Im\ \int_{0}^{\infty
}\frac{1}{e^{w-i}}dw=Im\int_{0}^{\infty
}\frac{dw}{cos(1)e^{w}-isin(1)e^{w}-1}\\ \\ \\$$
$$\theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Fundamental Theorem of Calculus and limit I've been reading through a paper and my question has essentially came down to this:
Let
$f(\beta) \to M$ as $ \beta \to 0$ and $f(\alpha) \to 0$ as $\alpha \to \infty.$ Prove that $M=-\int_{0}^\infty f'(x)dx$. I wanted to check this before going any further. It is probably tr... | Right , just the signs are here and there.Rest is fine
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/879803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If one number is thrice the other and their sum is $16$, find the numbers If one number is thrice the other and their sum is $16$, find the numbers.
I tried,
Let the first number be $x$ and the second number be $y$
Acc. to question
$$
\begin{align}
x&=3y &\iff x-3y=0 &&(1)\\
x&=16-3y&&&(2)
\end{align}
$$
| Let the first number be $x$.
Let the second number be $y$.
According to question
$$ \tag{1}
x+y=16
$$
$$
\tag{2}
x=3y
$$
So, $x-3y=0 \tag{2}$
Multiply equation $(1)$ by $3$.
Solve both equations:
$$\tag{1} 3x+3y=48$$
$$\tag{2} x-3y=0$$
$$\tag{1) + (2}4x=48$$
$$\tag{3}x=12$$
Putting in equation $(1)$:
$$\tag{1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$? The following integrals are classic, initiated by L. Euler.
\begin{align}
\displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} \zeta(3)+\frac{93}{128} \zeta(5),
\\ \int_{0}^{\pi/2} x^2 \ln\cos x\:\... | I've established some related explicit formulae.
Theorem 1.
Let $n$ be any positive integer.
Set
$$ I_{2n}:=\int_{0}^{\pi/2}\! \! x^{2n} \ln \cos x \: \mathrm{d}x $$
Then
$$
I_{2n} = - \frac{\pi^{2n+1}\ln 2}{2^{2n+1}(2n+1)} - (-1)^{n}\frac{(2n)!}{2^{2n+1}}\sum_{p=1}^{n} \frac{(-1)^p}{(2p-1)!}\pi^{2p-1}\zet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 2
} |
Summation of Infinite Geometric Series Determine the sum of the following series:
$$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} $$
My work:
$$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} = \sum_{n=1}^{\infty } \frac{-1}{7} (\frac{3}{7})^{n-1}$$
$$\sum_{n=1}^{\infty } ar^{n-1} = \frac{a}{1-r} = \frac{\frac{-1}{7}}{1-... | $$\begin{align}
\sum_{n=1}^{\infty } \frac{-3^{n-1}}{7^{n}}
& = - \frac{1}{7} \sum_{n=1}^{\infty } (\frac{3}{7})^{n-1}
\\
& = - \frac{1}{7} \frac{1}{1-\frac 3 7}
\\
& = - \frac 1 4
\\[2ex]
\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}}
& = \frac{1}{7} \sum_{n=1}^{\infty } (-\frac{3}{7})^{n-1}
\\
& = \frac{1}{7} \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Relation between an unsatisfiable set and a tautology In mathematical logic, satisfiability and validity are elementary concepts of semantics. A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true. A formula is valid if all interpretations make the formula true. The op... | It is not true. What is true is that $\lnot \varphi_1 \lor \lnot \varphi_2
\lor \cdots \lor \lnot \varphi_m$ is a tautology.
To see that the sentence with the $\land$ is not necessarily a tautology, let $m=2$, let $\varphi_1=\varphi$ and $\varphi_2=\lnot\varphi$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/880131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Rewriting the matrix equation $AX = YB$ as $Y = CX$? Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have to estimate the two independent polarizations of a signal given the... | In general it is not going to be possible.
Example: Let
$
A = Y =
\begin{pmatrix}
1 & 1 \\
0 & 0
\end{pmatrix}
$
and
$
X = B =
\begin{pmatrix}
1 & 0 \\
1 & 0
\end{pmatrix}.
$
Then $AX=YB$, but no matter what $C$ you choose, $CX$ has $0$'s in its second column and thus cannot equal $Y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/880205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How find this sum $\sum\limits_{i=0}^{2n}\binom{2n}{2i}\binom{2i}{i}y^{2i}$
Find the sum close form
$$f(x)=\sum_{i=0}^{2n}\dfrac{\binom{2n}{2i}\binom{2i}{i}x^{2i}}{2^{2i}}$$
if we let
$$\dfrac{x}{2}=y$$
then
$$f(y)=\sum_{i=0}^{2n}\binom{2n}{2i}\binom{2i}{i}y^{2i}$$
this PDF have this page 5
$$\sum_{k=j}^{n}\binom... | Given any formal Laurent series $\;(???) = \sum \alpha_{k_1 k_2 \ldots k_n} t_1^{k_1} t_2^{k_2} \cdots t_n^{k_n}$, we will use the notation $[ t_1^{k_1} t_2^{k_2} \cdots t_n^{k_n} ](???)$ to denote the coefficient $\alpha_{k_1 k_2 \cdots k_n}$ in front of corresponding monomial.
Instead of $f(y)$, let us denote the po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$ How to evaluate the following integral
$$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$
The solution is supposed to look like this, unfortunately I can't derive it.
$$
\left[\frac{l}{r^2\sqrt{r^2+l^2}}\right]_{l=0}^\infty
$$
| Let $l=r\tan{u}$, then $dl=r\sec^2{u} \ du$. The integral becomes
$$\frac{1}{r^2}\int^{\pi/2}_0\frac{\sec^2{u}}{\sec^3{u}}du=\frac{1}{r^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/880415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$... When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$. What is the denominator of the fraction?
I tried,
Let the numerator of the fraction be $x$ and the denominator be $y$.
Accordingly, $$\frac{x+4}... | Given,
$$\frac{n+4}{d}=\frac{n}{d}+\frac{2}{3}$$
So,
$$\frac{n}{d}+\frac{4}{d}=\frac{n}{d}+\frac{2}{3}$$
Or,
$$\frac{4}{d}=\frac{2}{3}$$
That, gives us $d=6$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/880466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 3
} |
How to prove that $ 1- \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1-\frac{x}{n})^n$ How would I prove this inequality (assuming its true, its from a textbook)
$$1 - \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1+\frac{-x}{n})^n$$
if $n > |x|$, $x\in R$ and $n\in N$
I first rewrote the inequality to
$$1 - \frac{x^2}{n} \... | Deriving both sides on $x$, $$-\frac{2x}{n}\le-n\frac{2x}{n^2}(1-x^2)^{n-1},$$ or
$$-1\le-(1-\frac{x^2}{n^2})^{n-1}.$$
The latter relation is obviously true for $|x|<n$, so that the LHS of the initial relation decreases faster than the RHS, while they are equal for $x=0$.
(If you prefer, $l'(x)\le r'(x)\implies l'(x)-r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Combo Identity: How to prove this using Induction
$$ \sum_{n = 0}^{\infty} \binom{n + k}{k}x^n = \dfrac{1}{(1 - x)^{k + 1}} $$
Could someone suggest how I should get started to prove this using induction?
| HINT:
$$\frac1{(1-x)^{m+1}}=\frac{1-x}{(1-x)^{m+2}}=\frac1{(1-x)^{m+2}}-\frac x{(1-x)^{m+2}}$$
$$\implies(1-x)^{-(m+1)}=(1-x)^{-(m+2)}-x(1-x)^{-(m+2)}$$
Assume that the formula is true for $k=m+2$ and establish the same for $k=m+1$
Alternatively, use $$(1-x)(1-x)^{-(m+2)}=(1-x)^{-(m+1)}$$
Assume that the formula is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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