Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Replacing a natural number containing a certain digit with the sum of two without that digit A question in Google Code Jam 2019 qualification round wanted a positive integer n which contains at least one digit 4 to be represented as a sum of two positive integers a and b, neither containing 4.
For my solution, I found ... | For 2 ≤ d ≤ 9, the method in my comment always works as b just contains 1 in the same position as any d and 0 otherwise. E.g. for d = 2, n = 25225222,
25225222
—15115111
—————————
10110111
For d = 0, put 9 in the same position as any 0 and 9 otherwise.
d = 1 is interesting. With this program:
def is_bad(x): return '... | {
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Integrating with different methods leads to different results? Acceleration of a particle is given as $a = 0.1(t-5)^2$. The particle is initially at rest.
Here is the way I initially attempted it. (I didn't use substitution as the question in the textbook appears before substitution is officially taught.)
$$a\:=\:0.1t^... | Note that you ended up with
$$
v(t) = \frac{1}{30} t^3 - \frac12 t^2 + \frac52 t
$$
But the book's answer must have $c = 5^3/30$, so the complete answer is
$$
\begin{split}
v(t) &= \frac{1}{30} (t-5)^3 + \frac{5^3}{30}\\
&= \frac{(t-5)^3-5^3}{30} \\
&= \frac{t^3 - 3\cdot 5 t^2 + 3 \cdot 5^2t - 5^3 + 5^3}{30} ... | {
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Why isn't $e^n$ equal to 1? We know $e^{2\pi i} = 1$, and that $(x^m)^n = x^{mn}$. This way, we can rewrite $e^{n}$ as some version of $(e^{2\pi i})^{\frac{n}{2\pi i}}$ for most n (right?).
But if this is true, then why isn't $e^3 = 1$, for example, if we can rewrite it as $(e^{2\pi i})^{\frac{3}{2\pi i}} = (1)^{\fra... | Good question! The answer is that although the rule ${(x^b)^c} = x^{bc}$ holds when $b$ and $c$ are integers, it does not hold in general when they are not integers.
Consider the following simpler example. As you know, $(-1)^2 = 1$. Raising both sides to the power $\frac12$, we get $${((-1)^2)}^{1/2} = 1^{1/2},$... | {
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Demand for operations research in industry I want to know how much demand there is for OR in the commercial world, and understand the relevance and prevalence of OR.
Is the airline industry the place where this technique is applied the most? What other areas would OR be in demand from your experience?
Also, is it mainl... | Airlines do indeed rely on operations research. So does UPS, which has an in-house operations research group that has generated sizeable cost savings for them.
Rather than listing the various industries that use OR, I'll just recommend that you look at past winners of the INFORMS Franz Edelman Award. You might also wan... | {
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Does every null set have a superset which is an $F_{\sigma}$ null set? Let $A$ be a Lebesgue null set in $\mathbb R$. Can we find a set $B$ with the following properties:
1) $A \subset B$
2) $B$ has measure $0$
3) $B$ is an $F_{\sigma}$ set (i.e. a countable union of closed sets).
I suppose so because that null sets I ... | A closed set of Lebesgue measure zero has empty interior. So a countable union of such sets, an $F_\sigma$ of measure zero, is meager (also called "first Baire category), and so are all its subsets. But there are Lebesgue null sets that are not meager, for example the set of those numbers in $[0,1]$ whose binary expan... | {
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Was there ever an axiom rendered a theorem? In the history of mathematics, are there notable examples of theorems which have been first considered axioms?
Alternatively, was there any statement first considered an axiom that later has been shown to be derived from other axiom(s), therefore rendering the statement a the... | The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms
In the article of Moore linked... | {
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Change of variables in stochastic PDE I have the following stochastic partial differential equation (SPDE):
$d v = -\mu \frac{\partial v}{\partial x} dt + \frac{1}{2} \frac{\partial^2 v}{\partial x^2} dt - \sqrt{\rho} \frac{\partial v}{\partial x} d M_t$,
with $M_t$ standard Brownian motion and $\mu, 0 \leq \rho \leq 1... | Proving these kind of phase shifts can indeed be difficult as the Ito calculus does not provide a nice chain rule as we have in deterministic calculus. There is however a trick to solve this problem. Choose a smooth test function $\zeta$ and define the functional
\begin{align}
\phi(v_t,M_t)=\langle v_t,\zeta\rangle_{H... | {
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Evaluate $\lim_{x\to 1}\frac{(x^n -1)(x^{n-1}-1)\cdots(x^{n-k+1}-1)}{(x-1)(x^2-1)\cdots(x^k-1)}$
Evaluate:$$
\lim_{x\to 1}\frac{(x^n -1)(x^{n-1}-1)\cdots(x^{n-k+1}-1)}{(x-1)(x^2-1)\cdots(x^k-1)}
$$
I'm trying to spot an error in my calculations. It is known that $x^n - 1$ may be factored out as $(x-1)(1+x+x^2+\cdots+... | A quick way to evaluate the limit:
First consider
\begin{align}
\lim_{x \to 1} \frac{x^{n-j+1} -1}{x^j -1} &= \lim_{x \to 1} \frac{(n-j+1) x^{n-j}}{j \, x^{j-1}} = \frac{n-j+1}{j}
\end{align}
now,
\begin{align}
\lim_{x\to 1}\frac{(x^n -1)(x^{n-1}-1)\cdots(x^{n-k+1}-1)}{(x-1)(x^2-1)\cdots(x^k-1)} &=
\lim_{x \to 1} \pro... | {
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Ratio and Proportion quantitative aptitude exam for management entrance in India The following is a question from the Common Admission Test which is an aptitude exam conducted in India for admission into postgraduate management programmes of premier management institutes of the country.
The salaries of Ram and Shyam ar... | HINT #1: Shyam makes $1/6$ more money but only spends $1/7$ more money, so what do you think the ratio of their savings can be?
HINT #2: Put Ram's money on the $x$-axis and Shyam's money on the $y$-axis. The salaries are represented by a vector $S=(6a, 7a)$ for some $a > 0$, and the expenditures are represent by a vec... | {
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Show that there are at least two points on a manifold to which a vector is normal Let $M \subset \mathbb R^3$ be a $2$-dimensional manifold which is also a compact set. And let $v \in \mathbb R^3$ be a vector which satisfies $ ||v|| = 1$.
The task is to prove that there are at least two points $x,y \in M$ such that $v$... | Following the hint you propose, $f(x)=\langle x, v\rangle$ has a maximum and a minimum. If they both coincide, it turns out that your $2$-manifold is a compact subset of a plane in $\mathbb{R}^3$, which would imply that it is a surface with boundary, contradicting your definition of submanifold as something that is loc... | {
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What is the name of this method that can easily find $\lim_{x \to 0^+} \frac{x^x-1}{\ln(x)+x-1}$? I can easily find $\lim_{x \to 0^+} \frac{x^x-1}{\ln(x)+x-1}$ using these steps. However, I don't completely remember the rules my professor told me about it, and I want to know what it's name is so I can look it up.
$$\li... | The rule is if $\lim_{x\to a} f(x) = k$ and $k$ is finite and $\lim_{x\to a} g(x) = m$ and $m$ is finite then $\lim_{x\to a} f(x)g(x) = km$ and $\lim_{x\to a} (f(x)+g(x))=m+k$ and if $m \ne 0$ then $\lim_{x\to a} \frac {f(x)}{g(x}) = \frac km$.
As to we $\lim_{x\to 0}\frac {1}{\ln x +x - 1} = 0$. There is a rule that ... | {
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Is there always a way to generate nontrivial finite extensions of a field? Suppose I have a field $k,$ an algebraically closed field $L,$ and an embedding $k \hookrightarrow L.$ Then we know that for any algebraic extension of $k, E$ we can extend the embedding to $E \hookrightarrow L.$ However, if we take $L$ to be th... | Your idea is almost a complete proof. You start off well:
Let $k$ be a field and $L$ an algebraic closure of $k$. Let $K$ be a finite extension of $L$. Then $K/L$ is algebraic, hence $K/k$ is algebraic, hence there exists an embedding $\iota:\ K\ \longrightarrow L$.
Now let $\alpha\in K$ and let $f\in L[X]$ be its mini... | {
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A property of unique factorization domains I'm asked to prove the following statement:
Let $R$ be a unique factorization domain, let $a,b$ be coprime elements. Show that for any ideal $J$ we have $$(ab)+J=((a)+J )\cap ((b) + J).$$
When $J=0$ it's trivial, but when $J \ne 0$, I can't work it out, although it seems eas... | It's true for a principal ideal domain, since for $J=(x)$ this then becomes $$\mathrm{gcd}(ab,x)=\mathrm{gcd}(a,x)\cdot\mathrm{gcd}(b,x).$$
It's not true in general for unique factorisation domains. For example, let $R=K[x,y]$ be the polynomial ring over some field $K$, and take $a=x$, $b=y$ and $J=(x+y)$. Then
$$(xy)+... | {
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If $\|u+tv\| \ge \|u\|$ for all $t$, prove that $u \cdot v=0$
Let $u, v \in \mathbb R^n$. Prove that if $$\|u+tv\| \ge \|u\|$$ for all $t \in \mathbb R$, then $u\cdot v=0$ (vectors $u$ and $v$ are perpendicular).
I tried writing $v$ as $(n+xu)$, where $u\cdot n=0$, and then try to prove that $x$ must be zero, but was... | Although the original question was interested only in the case where the vector space of interest is $\Bbb R^n$, the existing answer has shown the result is more general than that, applying to any vector space over $\Bbb R$. In fact, we can generalise even further, a point I think is of some interest: if in a space ove... | {
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A question on the implicit function theorem. The question is given below:
Approximate by a second-degree polynomial the solution of $z^3 + 3xyz^2 - 5x^2y^2z + 14 = 0$, for $z$ as a function of $x,y,$ near (1, -1, 2).
Could anyone give me a hint for the solution please?
My ideas:
our desired function will map (x,y) to z... | You don't need the derivative of $F$. The steps I suggest you take are the following:
*
*We know, from the implicit function theorem, that, near the point $(1, -1)$, there exists a function $z:\mathbb R^2\to \mathbb R$ such that $$z(x,y)^3 + 3xyz(x,y)^2 - 5x^2y^2z(x,y) + 14 = 0.$$ That is, in this step, we write $z$... | {
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Complex character of a finite group $G$ We let $G$ be a finite group.
If $\chi$ is a complex character of $G$, we define $\overline{\chi}:G \to \mathbb{C}$ by $\overline{\chi}(g)=\overline{\chi(g)}$ for all $g \in G$, and define $\chi^{(2)}:G \to \mathbb{C}$ by $\chi^{(2)}(g) = \chi(g^2)$. We write $\chi_{S}$ and $\ch... | Recall that a character $\chi$ is irreducible if and only if $\langle \chi,\chi\rangle=1$. Note then that
$$\langle \overline{\chi},\overline{\chi}\rangle =\frac{1}{|G|}\sum_{g\in G}\overline{\chi}(g)\overline{\overline{\chi}(g)}=\frac{1}{|G|}\sum_{g\in G}\overline{\chi(g)}\chi(g)$$
but
$$\langle \chi,\chi\rangle=\fra... | {
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Hadamard product tricks of particular entities Consider the following matrices $\mathbf{Q}_{H}$ order $\left(
T\times n\right) $ and $\mathbf{A}$ of order $\left( T\times T\right) $ and $%
\mathbf{\hat{u}}$ of order $\left( n\times 1\right) $ and denote by $\circ $
the Hadamard product and by $\left\vert \mathbf{\hat{u... | Define the vectors
$$v = Qu, \quad s={\rm sign}(v), \quad b={\rm abs}(v)$$
where the functions are applied elementwise.
Assuming $\,v_k\ne0$, the elementwise division of the vector is
$$v\circ b^{\circ -1} = v\oslash b = s$$
So the main equation reduces to
$$\eqalign{
y = b\circ(As) = BA\,s \cr
}$$
where the Hadamard ... | {
"language": "en",
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Inversion of an almost identity matrix Say we have a square matrix like so
1 c c c ... c
c 1 c c ... c
...
c c c c ... 1
What would be the inverse of this matrix? Calling an inv function is expensive, especially for a very large matrix. I am almost certain there's a simple formula to quickly find this inversion since ... | A few examples with WA seem to indicate that the inverse of that matrix has the same form, except that the diagonal element is $-((n-2)c+1)$ and we have to divide by the determinant, which seems to be $(n-1)c^2-(n-2)c-1=(c-1)((n-1)c+1)$.
Here is the inverse for $n=5$:
This is probably easy to prove directly or using t... | {
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x^2+y^2=2, xy=1, how to find x and y I have problems when doing these equations when I don't know any variable's value.
Can someone please explain how to do this and possibly give some tips when it comes to solving these problems?
Well, I know x = 1 and y = 1, but what about this one:
a-b=3, a:b=3:2, find a^2-b^2
(I am... | Since $xy=1$ it is $x^2+y^2=2xy$, which gives us $x^2-2xy+y^2=0\Leftrightarrow (x-y)^2=0$
This means, it has to be $x=y$. From the condition $xy=1$ we deduce $x^2=1\Leftrightarrow x=y=\pm 1$, which gives the only two solution $(1,1), (-1,-1)$.
| {
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Expectation, Variance and Moment estimator of Beta Distribution I'm given a beta distributed random variable: $X \sim \text{Beta}_{(\theta, 1)} =: \mathbb{P}_\theta$. Where $\theta \geq q$ and
$$\mathbb{f}_\theta(x) = \theta \cdot x^{\theta-1} \cdot \mathbb{1}_{[0,1]}(x)$$
I was asked to compute the expectation and var... | The idea behind the Method of Moments estimator is the following.
Let suppose we have $\underline{X}$ $=$ $(X_{1},...,X_{n})$ iid observations, distributed according to $f(\cdot \mid \boldsymbol{\theta})$, $\boldsymbol{\theta}$ $\in$ $\Theta$ $\subseteq$ $\mathbb{R}^{d}$.
Define:
$$\mu_{k} = E(X_{i}^{k}) = \mu_{k}(\the... | {
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What is correct approach for test of divergence/convergence for $\sum_{n=1}^{\infty} \frac{\sqrt{n^3+n+1}}{n^4}$ What is the correct approach for determining if the following series is convergent or divergent?
$$\sum_{n=1}^{\infty} \frac{\sqrt{n^3+n+1}}{n^4}$$
My thought process was to use the limit comparison test wh... | As this is a series with positive terms, you may use equivalents: a polynomial function is asymptotically equivalent to its leading term, so
$$\frac{\sqrt{n^3+n+1}}{n^4}\sim_\infty\frac{\sqrt{n^3}}{n^4}=\frac 1{n^{5/2}},$$
which is a $p$-series ($p=5/2$) converging to $\zeta(5/2)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to show a function is not invertible locally? I have a silly question:
If a smooth function $f$ has a invertible derivative $Df(x_0)$ at a point $x_0$, then $f$ has a smooth inverse locally. This is what I learned in Inverse function Theorem. If we have a function $g$ such that $Dg(x_0)$ is not invertible, can we ... | A sufficient condition for $f$ not to have a local inverse at $x_0$ is that $x_0$ is a local extremum. For any $\delta$ small enough, $f(x_0+\delta), f(x_0 - \delta) \leq f(x_0)$, and hence we an order them as
$$f(x_0 \pm \delta) \leq f(x_0 \mp \delta) \leq f(x_0)$$
Hence, using the intermediate value theorem, we can ... | {
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Area of a rectangle inside a triangle with given coordinates
Given a triangle with vertices at points $(0, -a), (0, a), (b, 0)$, where $a > 0$, find the maximal area and the dimensions (base and height) of a rectangle that can be contained within the triangle.
I tried to find a function, differentiate it, and find t... | Let $A(0,b),$ $B(b,0),$ $C(0,-a)$ and $KLMN$ be our rectangle, $KL=x$, where $K\in BC$ and $N\in AB$.
Thus, since $\Delta ABC\sim\Delta NBK$, we obtain:
$$\frac{NK}{2a}=\frac{b-x}{b}$$ or
$$NK=\frac{2a(b-x)}{b}.$$
Id est, by AM-GM
$$S_{KLMN}=\frac{2a(b-x)x}{b}\leq\frac{2a\left(\frac{b-x+x}{2}\right)^2}{b}=\frac{ab}{2}.... | {
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Is $\frac{lnk}{\pi}$ irrational or rational?Here $k$ is a positive integer. I ask the question because I want to prove $\{\frac{lnn}{2\pi}\}$ （fractional part) is dense in $[0,1)$.
If we take $n=2^{k}$.It will become $\{\frac{ln2}{2\pi}k\}$.So we need to prove $\frac{ln2}{\pi}$ is irrational.It's difficult for me.But ... | You don't know which of $\frac{\ln 2}{\pi}$ or $\frac{\ln 3}{\pi}$ are irrational. But you know that at least one of them is. That's enough. You are allowed to pick "the irrational one", even though you don't know which one it is, as long as you know it exists.
(Small note: If it happens that they are both irrational, ... | {
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Proving condition for spanning vectors I have highlighted the bit I don't understand, what does it mean by in the linear form? And didn't the proof already state that 4 lines above?
| The linear form is as outlined in the last sentence of the first paragraph, namely the linear function of the entries of b: $l$(b)$ = \sum_{i=1}^n \alpha_i b_i$ corresponding to a zero row.
The statement which is four lines above what you highlighted states that the coefficient of $b_i$ is $1$ at the start i.e., befor... | {
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how $\int_a ^ b |f'(x)|$ gives the length of the arc of the contour $f$ : $(f(x) : x \in [a , b])$ I got to know that $\int_a ^ b |f'(x)|$ gives the length of any contour. Where $f(x)$ is a piece-wise differentiable function from $[a,b]$ to $\mathbb R^2$. I was reading complex integral . Can anyone please enlighten me ... | What is the length of a curve?
Let $a = x_0 < x_1 < x_2 < ... < x_n = b$ a partition of $[a,b]$
An approximation of the length can be
$Len(f) \approx \sum_{k=0}^{n-1}|f(x_i+1)-f(x_i)|$
You can see that when at the limit where the partition is delicate enough you get
$Len(f) = \sum_{k=0}^{\infty}|f(x_i+1)-f(x_i)|=\sum_... | {
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Prove that $ \int_X |f_n|^2d\mu<\infty $ for all $ n\in\mathbb N $
Let $ (X,\mathcal M,\mu) $ be a measure space with $ \mu(X)<\infty $. Let $ (f_n)_{n=1}^\infty $ be a sequence of functions in $ L^1(X,\mathcal M,\mu) $, and let $ f\in L^1(X,\mathcal M,\mu) $. Suppose that
$$ \lim_{n\to\infty}\int_X |f_n-f|d\mu=0 .$... | $$\int |f_n|^2 d\mu\leq\sqrt{ \int d\mu}\cdot\sqrt{\int |f_n|^4 d\mu}$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Number of solutions of this equation *. The number of solutions of the equation $a+2b=k$ is $\left\lfloor\frac{k}{2}\right\rfloor+1,$ where $a,b\in \mathbb{N}\cup\{0\}\mbox{ and fixed }k\in\mathbb{N}.$ More general what is the number of solutions of the equation $a_{1}+2a_{2}+3a_{3}+\ldots+(r-1)a_{r-1}+ra_{r}=k,$ where... | The number of solutions of $\sum_{i=1}^r ia_i=k$ in the nonnegative integers is equal to the number of integer partitions of $k$ whose parts all have size at most $r$. The numbers $a_i$ represent the number of parts of size $i$.
The number of such partitions is sometimes denoted $p_{\le r}(k)$. There is no closed form... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3182509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
What is a good way to prove that the function $\Bbb Z_n \rightarrow \Bbb Z_u \times\Bbb Z_v$ is well defined? Given a function of the type $\Bbb Z_n \rightarrow \Bbb Z_u \times\Bbb Z_v$ ($[x]_n\rightarrow([x]_u, [x]_v)$) where of course $n=u*v$ how can I prove that is a well defined function?
| For $f:\mathbb{Z}_n\to\mathbb{Z}_u\times\mathbb{Z}_v$, $[x]_n\mapsto ([x]_u, [x]_v)$
It might go like this:
To prove that this function is well-defined, we have to check if it really does not matter which representative of the equivalence class we choose.
So let $[x]_n=[y]_n$. Then $x=nk+r$ and $y=nl+r$ with $0\leq r<n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3182643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Show that $X_n \overset{p}{\rightarrow} 0$ as $n \rightarrow \infty$
Let $X_1,X_2 ,\ldots$ be random variables defined by the relations
$P(X_n = 0) = 1−\frac{1}{n}$, $P(X_n = 1) = \frac{1}{2n}$ and $P(X_n = −1) = \frac{1}{2n}$ , $n\ge 1$
Show that:
$X_n \overset{p}{\rightarrow} 0 \quad \text{ as } \quad n \rightarro... | The random variable $\left\lvert X_n\right\rvert$ takes the value one with probability $1/n$ and $0$ with probability $1-1/n$ hence it is also the case for the random variable $\left\lvert X_n\right\rvert^r$. We deduce that $\mathbb E\left\lvert X_n\right\rvert^r=1/n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3182868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Maximum size of a subcritical birth-death process Beginning with a population of $n_0$ individuals, let each individual have a probability $p$ to survive until it replicates into two independent and identical individuals, where $p<\frac12$.
It follows that the population goes extinct in the long-run with probability 1,... | I have written this $\texttt{R}$ code to compute a point estimate for the mean and variance of $N$ given fixed values for $p$ and $n_0$:
rm(list=ls())
N <- 10000
maxpop <- numeric(N)
p <- 0.3
n_0 <- 2
for(n in 1:N) {
pop <- n_0
maxpop[n] <- n_0
while(pop > 0) {
if(runif(1)>p) {
pop <- pop - 1
}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3183006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
CW complexes are the cofibrant objects in the Quillen model structure on Top? If $J$ is a class of maps in a category, the $J$-cellular maps are by definition transfinite compositions of pushouts of coproducts of maps in $J$.
Now if $J$ denotes the family of inclusions $S^{n-1}\hookrightarrow D^n$ for all $n$, then CW... | They are not all the cofibrant objects in the Quillen model structure on spaces, they are "among" the cofibrant objects- as is stated in the nlab article you linked in the question.
They are all of the cofibrant objects in the mixed model structure, see here, at least up to actual homotopy equivalence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3183111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
how to find zero solutions and stability? Show that the zero solution of $$ \large \ddot{x}+bx^2 \dot{x}+kx=0$$
is asymptotically stable if $b>0$ and unstable if $b<0$. Does this depend on the sign of $k$?
I know for 1st order equation but how to find stability of 2nd order equation?
Can I convert it into system of 1st... | Any constant solution obviously has $\dot x=0$, $\ddot x=0$ so that the equation $kx=0$ remains.
For $k<0$ you get a saddle point, thus $k>0$.
For the stability consider
$$
\frac{d}{dt}\frac12(\dot x^2+x^2)=-bx^2\dot x^2
$$
which tells you in which direction the solutions cross the circles around the origin.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3183226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a formalization of the link between geometry and analytical geometry? Geometry and algebra/calculus can be formalized by axioms.
Is there a global theory that combines both and establishes correspondences such as
*
*the equation of a straight line is $ax+by+c=0$,
*the length of a segment is $\sqrt{(x_b-x_a... | The "basics" are developed by Hilbert into The Foundations of Geometry (1899).
The book states the axioms for plane Euclidean geometry.
Hilbert defines the fundamental geometrical object : segment.
Indipendently, he states the laws for real numbers.
Finally, Hilbert develop an "algebra of segments", i.e defines the ope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3183391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$L^2$ and Sobolev space In Raymond's book on Pseudodifferential Operator page 18, he says , where $S'$ is the tempered distributions, we define sobolev space of exponent $s$ as
$u \in S'$ with $\lambda^s \hat{u} \in L^2$. This is equivalent to $\hat{u}$ is a function satisfying
$$ ||u||_s^2 =(2 \pi)^{-n} \int (1+|... | It i,s just the definition of $L^{2}$ norm. Just square $\lambda^{s}\hat {u}$ and integrate. I thing the book defines $L^{2}$ space by taking the reference measure as Lebesgue measure divided by $(2\pi)^{n}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3183536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving $\int_0^{\infty}f(x) $ converges using Lagrange Let $f:[0,∞) \to \Bbb R$ be a differentiable such that $f(x) > 0$ for every $x \in [0, \infty)$ and a positive function. Assume there exits an $0 < L < \infty$ such that $$\lim_{x\to \infty}\,[\ln(f(x))]'=-L.$$
Prove that $\int_0^{\infty}f(x) $ converges.
We were... | Since $\lim\limits_{x\to \infty}\,[\ln(f(x))]' = \lim\limits_{x\to \infty}\, \dfrac{f'(x)}{f(x)}=-L < 0$,
$$\forall \varepsilon \in (0,L), \exists M > 0: \forall x \ge M, \frac{f'(x)}{f(x)} \le -L + \varepsilon < 0.$$
Fix any $\varepsilon \in (0,L)$. Multiply both sides by $f(x)$. (We can do so since $f(x) > 0$ for a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3183698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can a sum of a finite number of exponentially growing numbers be calculated as a function of the growth rate and the number of growths? This is a question based on the exponential growth pattern of a game mechanic in World of Warcraft: The Heart of Azeroth item in the game has a level that can be increased by gathering... | You are looking at a geometric series
$$
x + px + p^2x + \ldots xp^n = x\sum_{k=0}^n p^k = \frac{x\left(1-p^{n+1}\right)}{1-p}
$$
If $|p|<1$, $n$ is large and you are happy with an approximation, $p^{n+1}$ becomes very small, so your sum is approximately $$\frac{x}{1-p}.$$
For your example this does not approximate wel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3183834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $ab+bc+ca \geq 3k^2-1$, prove that: $a^3+b^3+c^3-3abc \geq 9k$.
If $ab+bc+ca \geq 3k^2-1$, prove that: $a^3+b^3+c^3-3abc \geq 9k$.
I recently came across a question in which we had to prove the above inequality using the given condition as mentioned above. Here $a,b,c$ are distinct positive integers and $k$ is als... | Use that $$a^3+b^3+c^3-3abc=\left( a+b+c \right) \left( {a}^{2}-ab-ac+{b}^{2}-bc+{c}^{2}
\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3184009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve $x^{x^x}=(x^x)^x$? How can we solve the equation :
$x^{x^x}=(x^x)^x$
with $x \in {\mathbb{R}+}^*$
Thanks for heping me :)
| We can check when the exponents are equal.
It is $x^{x^x}=(x^x)^x\Leftrightarrow x^{(x^x)}=x^{x^2}\Leftrightarrow x^x=x^2$
Now x^x-x^2=0\Leftrightarrow $x^2(x^{x-2}-1)=0$.
So $x^2=0$ or $x^{x-2}=1$.
Since $x\neq 0$ we have $x^{x-2}=1$ left, which holds if $x-2=0$. So $x=2$
And $2^{2^2}=2^{2\cdot 2}$
Edit: And x=1 is an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3184200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Prove that the rotation of sums is equal to the rotation of products So the question starts off:
Prove $$\ e^{t_1+t_2} = e^{t_1}e^{t_2}$$
E(t) is a unique solution to $\dot{E} = E, E(0) = 1$.
Let $E_1(t) = E(t_1 + t)$, and E_2(t) = E(t_1)E(t)
$$\dot E_1 (t) = \dot E_1 (t_1 + t) = E(t_1+t) = E_1 (t); E_1(0) = E(t_1)$... | If you want to repeat the argument, you can look at the equation $R''=-R$.
Another way is to notice that
$$
R(t)=e^{it}\begin{bmatrix}1/2&i/2\\ -i/2&1/2\end{bmatrix} + e^{-it}\begin{bmatrix} 1/2&-i/2\\ i/2&1/2\end{bmatrix}.
$$
Then
\begin{align}
R(s)R(t)&=e^{it}e^{is}\begin{bmatrix}1/2&i/2\\ -i/2&1/2\end{bmatrix}^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3184350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
arranging 2 blue balls, 2 red balls and 1 green ball How many ways are there to arrange 2 blue balls, 2 red balls and 1 green ball?
My Answer: $$\frac{5!}{2!*2!}$$
If this is incorrect then please help me understand where I am wrong.
But if this is correct then please help me with this:
"A bag holds $4$ red marbles, $5... | Well,
$$
\color{blue}{\binom 52 \binom 32 \binom 11 = \frac{5!}{2! \times 2!} = 30}
$$
So it seems that what you are doing is also correct. However, your thought processes when computing the answer were different, so I will just highlight that.
*
*What was the person who wrote the "correct answer" thinking? He was... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3184604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Immediate consequence of standard Schwarz-Pick lemma! How to interpret? Let $g:D(0,1)\to D(0,1)$ be analytic function and $g(0)=a\in[0,1)$;
suppose $$G(z)=\frac{z+a}{1+az},\text{for}\ z\in D(0,1).$$
we want to prove that:$$g(D(0,r))\subset G(D(0,r))\ \forall \ r\in(0,1).$$
In the origin paper, it says that this is an i... | Let $|z|<r$. Define $H(z)=\frac {z-a} {1-az}$. Verify that $G=H^{-1}$. We have to show that $g(z)=G(\zeta)$ with $|\zeta| <r$. Define $\zeta$ as $H(g(z))$. Then $g(z)=G(\zeta)$. To show that $|\zeta| <r$ apply Schwarz Pick Lemma with $f$ changed to $g$ and $\alpha =0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3184700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing $|x_n -x|$ converges to x I'm given a generic sequence $x_n$, and I know that $\lim_{n ->\infty} \frac{|x_{n+1}-x|}{|x_n-x|} < 1$ and exists. I'm trying to show that $x_n$ converges to $x$.
I tried a bunch of approaches, but the farthest I got was a proof by contradiction where I try show that $\lim_{n->\infty... | Hint:
It is equivalent to show tha $|x_n-x|\to 0$. Now, since $\lim_{n\to\infty} \frac{|x_{n+1}-x|}{|x_n-x|} $ exists and is $<1$, there exists $k<1$ and $N_0$ such that
$$\frac{|x_{n+1}-x|}{|x_n-x|}\le k\quad\forall n \ge N_0.$$
Deduce that, for all $n\ge N_0$, one has $\;|x_n-x|\le k^{n-N_0}\,|x_{N_0}-x|$ (use induct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3184844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Reference request: Laplace-Beltrami eigenfunction bases for Sobolev spaces I'm working on a smooth $(d-1)$-dimensional surface $M\subset \mathbb{R}^d$. Let $(\phi_k)_{k\in\mathbb{N}}$ be an orthonormal basis of $L^2(M)$ consisting of the eigenfunctions of the Laplace-Beltrami operator $-\triangle_{M}$, with correspondi... | One definition of $H^s(M)$ is $$H^s(M)= \{u \in \mathcal{S}(M) : \sum_{j=1}^\infty \lambda_j^{2s} \lvert\langle u, \bar{\phi}_j \rangle \rvert^2\equiv \lVert u \rVert_{H^s(M)}^2<\infty\},$$ where $\mathcal{S}$ is the Schwartz space of distributions on $M$ (i.e. the dual of $C^\infty_c(M)$, which is just $C^\infty(M)$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3184941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Infinite sum of prime-counting function The function, $\pi (n) $ is the number of prime number less than or equal to $n$.
So, my question is the radius of convergence about
$$\sum_{n = 0} ^ {\infty} {\frac {1}{\pi (n)} x^n } $$
What is radius of convergence in this power series?
| Hint: $\pi(n)\sim {n\over \log{n}}$ and $\pi(n+1)\le \pi(n)+1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3185083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Uniqueness of solutions of diffusion equation with initial condition In his PDE, Walter A. Strauss claims that the diffusion equation on the whole real line has a unique solution, given an initial condition. However he only proves uniqueness given an initial-boundary condition for solutions on a finite interval (in sec... | It seems that he is only claiming here that $u$ is a solution of (1), (2). He does not prove uniqueness in the book, and it should probably be understood that a uniqueness result (with some qualifier) must be supplied by external sources (As a commenter said, there is no uniqueness without qualifiers). Keep in mind tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3185198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\lim_{n\rightarrow 0}\frac{1}{n}\int_{0}^{1}\ln(1+e^{nx})dx$ $$\lim_{n\rightarrow 0}\frac{1}{n}\int_{0}^{1}\ln(1+e^{nx})dx$$
My try:
$$\frac{b-a}b\leq \ln b-\ln a\leq \frac{b-a}a \implies \frac{1}{1+e^{nx}}\leq \ln(1+e^{nx})-\ln e^{nx}\leq \frac1{e^{nx}}$$
Then I integrated and multiplied by $\frac{1}{n}$ and I ... | You may squeeze the integral as follows using
*
*$nx = \ln e^{nx} \leq \ln (1+ e^{nx})$ and
*$\ln (1+ e^{nx}) = nx + \ln \left(1 +\frac{1}{e^{nx}}\right)\stackrel{0\leq x \leq 1}{\leq} nx + \ln 2$
So, you get
$$\color{blue}{\frac{1}{2}} = \frac{1}{n}\int_0^1 nx \; dx \leq \frac{1}{n}\int_0^1 \ln (1+ e^{nx}) \; dx \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3185317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Gagliardo-Nirenberg inequality for fractional Sobolev spaces Wikipedia states two versions of the Gagliardo-Nirenberg inequality for nonfractional Sobolev spaces. I'm interested in generalizations to fractional (Slobodeckij) Sobolev spaces.
Such a generalization of the version for functions on $\mathbb{R}^n$ can be fo... | It is straightforward. You have $u$ defined on $\Omega$ and then the extension $\tilde u$ defined on $\mathbb{R}^n$. Now the Gagliardo semi-norm of $u$ can be estimated trivially by the Gagliardo semi-norm of $\tilde u$ (just because it's an extension). Since $\tilde u$ is defined on $\mathbb{R}^n$ you can now use the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3185419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Does the inverse function theorem provide a path to interpretation of more general infinitesimal quotients? Let us first bring up inverse function theorem : If $y=f(x)$ and if $f'(x)$ exists at some point $x=a$, then exists in some neighborhood of $(a,f(a))$ an inverse function $f^{-1}(x)$ which around corresponding $b... | If we stick with $\mathrm d$ rather than $\partial$, then there's little problem with handling single-variable in this sort of way. We define $\mathrm d (f(t))=f'(t)\mathrm dt$ (and similarly for any other variable), and then the algebra works out nicely for first-order derivatives.
For example:
$$\dfrac{\mathrm{d}\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3185544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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A challenging system of coupled recursive sequences So I was runing through my old school drafts, and I've just come upon this challenging problem that years ago, one of my former math teacher had let for enthousiastic students to try.
Consider the real sequences $\left(a_n\right)_{n\in\mathbb{N}^*}$ and $\left(b_n\rig... | Hint.
Make
$$
B_n = \frac{n+2}{(n+1)^2+1}b_n
$$
NOTE
$$
\left(
\begin{array}{c}
a_n\\
B_n
\end{array}
\right) = \left(
\begin{array}{cc}
n & 1\\
-1& -n
\end{array}
\right)\left(
\begin{array}{c}
a_{n-1}\\
B_{n-1}
\end{array}
\right)
$$
Calling now
$$
M_n = \left(
\begin{array}{cc}
n & 1\\
-1& -n
\end{array}
\right)
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3185639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
A (nontrivial) nonlinear Poisson equation Let $U \subset \mathbb R^3$ be bounded, with smooth boundary. Does the equation
$$ -\Delta f = e^{-f} \text{ in } U, \quad f = 0 \text{ on } \partial U$$
has a solution? Is there a name for this equation? Any idea or hint or reference is highly appreciated.
| I don't have a definitive answer, but this makes it a little more tractable. The differential operator is uniquely invertible and has a Green's function $G(x,y)$ that is positive. This allows us to reformulate our equation as $$f(x) = \int_U G(x,y)e^{-f(y)}dy = \Phi(f).$$ If you can show that the integral operator on t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3185784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Lower bound given expectation and standard deviation. A random variable X with integer values only has mean 3 and standard deviation 2. Under those assumptions, which is the best lower bound for $P[0\leq X \leq 6]?$. By my calculations, it is $\frac{5}{9}$ however it is not the right answer. Your thoughts please.
| You can WLOG assume symmetry about $3$ this way: Suppose $X$ is integer-valued and satisfies $E[X]=3, Var(X)=4$. Then $Y=6-X$ also satisfies those constraints, as does $$Z = \left\{ \begin{array}{ll}
X &\mbox{ with prob $1/2$} \\
Y & \mbox{ with prob $1/2$}
\end{array}
\right.$$ where the choice is decided by an in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3186027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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LCM of even and odd integers Is the least common multiple of an even $2k$ and odd number $2l+1$ always the product of both numbers $2k(2l+1)$ ?
And also is the least common multiples of two odd numbers the product of both odd numbers?
Thank you.
| If we take 6 and 15 then their lcm is 30 so your first question has answer "no"
If we take 3 and 15 then their lcm is 15 so your second question has answer "no"
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3186128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Sum $\sum_{k=1}^{\infty}\frac{1}{k!k^k}$ I came across this crazy sum, and I have no idea how to tackle it
$$\sum_{k=1}^{\infty}\frac{1}{k!k^k}.$$
I tried to approximate it using wolfram alpha widget,
and got something like $1.13134$.
However, I'd like to get an exact solution.
EDIT: i was thinking that maybe we coul... | I'm afraid I have to concur with @MariuszIwaniuk. I'll summarise what I found.
In a variant on the famous sophomore's dream calculation, note first that, since in terms of modified Bessel functions $\sum_{n\ge 0}\frac{y^n}{n!(n+1)!}=\frac{I_1(2\sqrt{y})}{\sqrt{y}}$, the substitution $u=-\ln x$ obtains $$\int_0^1\frac{I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3186286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Homotopy between unitary element and identity elements, Operator Theory Let $\mathcal{T}$ be the Toeplitz algebra. I.e. the $C^*$ algebra generated by the shift operator $S\in B(l^2(\Bbb N))$.
In page 6, line 8 of a proof we have a unitary element $u \in \mathcal{T} \otimes \mathcal{T}$, and it is claimed that $u$ is ... | The unitaries you consider there are self-adjoint. In that particular case, you can write down an easy formula for the path. Let $u$ be a self-adjoint unitary in a C*-algebra. Define $h := (1-u)/2$. Then $e^{\pi i h} = u$ and the path $t \mapsto e^{\pi i th}$ connects $1$ to $u$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does one prove $C([0,1)\otimes A \cong C([0,1],A)$? $C([0,1])$ is a $C^*$ algebra of complex functionals. $A$ is a $C^*$ algebra. Hence $C([0,1],A)$, the continuous functions from $[0,1]$ to $A$ is also a $C^*$ algebra.
We construct its tensor product in the category of $C^*$ algebras. This is unique as $C([0,1])$... | The map is $\pi:f\otimes a \longmapsto f a$. It's obvious that it is linear and multiplicative, and preserves adjoints, so it is a $*$-homomorphism. It is injective: if $\pi(\sum_j f_j\otimes a_j)=0$, we may choose the $a_j$ so that they are linearlity independent. Then
$$
0=\pi(\sum_j f_j\otimes a_j)=\sum_j f_j a_j. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3186543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $f:(0,\infty)\to(0,\infty)$ be uniformly continuous function, is the following statement true?
Let $f:(0,\infty)\to(0,\infty)$ be uniformly continuous function. Does it imply $$\lim_{x\to\infty} {f(x+{1\over x})\over f(x)}=1\;?$$
By uniform continuity , for any $\epsilon>0$, $\exists\delta>0$ such that $|f(x)-f(... | The statement is not true. Any bounded, continuous function $f:(0,\infty) \to (0,\infty)$ where $f(x) \to 0$ as $x \to \infty$ is uniformly continuous.
Construct such a continuous function which is piecewise linear and where $f(x) = 1/n$ for $x = n $ and $f(x) = 2/n$ for $x = n + 1/n$ where $n \geqslant 2$ is an intege... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Robust & Linear Control Systems The characteristic polynomial of a control system is the following uncertain polynomial:
$$s^3 + a_2 s^2 + a_1 s + 3.5 $$
Where $a_1 \in [1.5,4.2],a_2 ∈ [1.2,4.25]$ and $ 4.2 ≤ a_1 + a_2 ≤ 6.3$
.
Is this uncertain polynomial stable?
How do I solve this problem, when I have got two value... | The Hurwitz matrix of the polynomial is:
$$ \begin{bmatrix} a_2 & 3.5 & 0 \\ 1 & a_1 & 0 \\ 0 & a_2 & 3.5 \end{bmatrix}$$
To stability this matrix must be definite-positive, this is, all subdeterminants must be positive. The conditions are:
$$a_2 >0$$
$$a_2 a_1 >3.5 $$
If the second subdeterminant is positive, it follo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3186826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $T_n=\{11(k+h)+10(n^k+n^h)\mid1\leq k,h \leq 10\}$ for $n\in\mathbb N$. Find all $n$ such that there is no $a\neq b\in T_n$ with $110 \mid (a-b)$. With every positive integer $n$, we have a set $T_n=\{11(k+h)+10(n^k+n^h) \mid 1\leq k,h \leq 10\}$. Find all $n$ such that there is no $a\neq b\in T_n$ with $110 \mid (... | We can solve this without resorting to brute force.
Firstly, define, for positive integer $n$ and $1\leq k,h\leq10$, $f_n(k,h)=11(k+h)+10(n^k+n^h)$. Then the condition $\exists a,b\in T_n$ such that $a\ne b$ and $a\equiv b\pmod{110}$ is equivalent with the conditions below.
$$\begin{cases}
n^{k_1}+n^{h_1}&\equiv n^{k_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Pigeon Hole explanation I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.
Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure y... | There are different versions of PigeonHole Principle, but they all basically amount to:
if number of containers is exceeded by number of objects, then some container has something happen with certainty (or certain probability in a non-equiprobable case)
The logic of this statement, can be done via repeated use of the... | {
"language": "en",
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"source": "stackexchange",
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Complex impedence between 2 terminals I have this problem to solve.
I have this so far. I am struggling with this so looking for some help please.
$$\frac{1}{5j}+\frac{1}{5+8.66j}+\frac{1}{15}+\frac{1}{-10j}=-\frac{j}{5}+\frac{5-8.66j}{5^2+8.66^2}+\frac{1}{15}+\frac{j}{10}=a-bj$$
$$a:=\frac{5}{5+8.66}+\frac{1}{15},\,b... | Impedances sum in series; admittances (reciprocals of impedances) sum in parallel. Thus the total admittance in $\Omega^{-1}$ is $$\frac{1}{5j}+\frac{1}{5+8.66j}+\frac{1}{15}+\frac{1}{-10j}=-\frac{j}{5}+\frac{5-8.66j}{5^2+8.66^2}+\frac{1}{15}+\frac{j}{10}=a-bj$$with $$a:=\frac{5}{5+8.66}+\frac{1}{15},\,b=\frac{1}{5}+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3187306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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matrix raised to a matrix i wanted to know if it was possible to raise a matrix to a matrix and i wanted to confirm if i have it. i tested it out with numbers and notation and i want to know if i'm right. $$
x =\left [ \begin{matrix}
1 & 1 \\
0 & 1 \\
\end{matrix} \right ]
$$
$$
\left [ \begin{matrix... | Defining the exponential function of a (complex) matrix $A$ is not too difficult, it is just $\exp A:=\sum_{n=0}^{\infty} \frac{A^n}{n!}$, which can be shown to converge. The same is therefore true for $\cos A$ and $\sin A$.
Working out what $\exp A$ is another question. In the "easy" case, when $A$ is diagonalizable w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $HThe question is as follows:
Find an example of a group $G$ with a subgroup $H$ so that
$$\{(x, y) | xx^{−1} y^{−1} \in H\}$$
is not an equivalence relation on $G$.
I've just been working on this problem set for hours now and I'm having a hard time coming up with an example for this question.
| Consider $G=V=\Bbb Z_2\times \Bbb Z_2$ given by the presentation $$\langle a, b\mid a^2, b^2, ab=ba\rangle$$ and the subgroup $H\cong \Bbb Z_2$ given by $\langle b\mid b^2\rangle$. Pick $a\in G\setminus H$. Then $a\not\sim a$.
Another way to see this is that the condition $xx^{-1}y^{-1}\in H$ is equivalent to $ey^{-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3187776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Topological semiconjugacy preserves topological transitivity Let $(X, f )$, where $X$ is a compact metric space and $f : X → X $ is a continuous function and let $(Y, g)$ where $Y$ is a compact metric space and $g : Y → Y$ is a continuous function. Suppose they are topologically semiconjugate, i.e., there is a continu... | Let $U,V$ be non empty open subsets of $Y$. Since $h$ is surjective, $h^{-1}(U)$ and $h^{-1}(V)$ are not empty. Since $f$ is topological transitive, there exists $n$ such that $f^n(h^{-1}(U))\cap h^{-1}(V)$ is not empty. Let $x\in (h^{-1}(U))$ such that $f^{n}(x)\in h^{-1}(V)$, $h(x)=y\in U$ and $g^n(y)=g^n(h(x))=h(f^n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving $\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca}$ for positive $a$, $b$, $c$ I'm at the end of an inequality proof that started out complex and I was able to simplify it to:
$$\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca} \quad\text{where}\quad a, b, c > 0$$
I'm able ... | A straightforward proof can be posed as follows, the inequality can be written as such
$$
(3ab+bc+2ca)\left(\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c\right) \ge 6abc
$$
which is equivalent to
$$
3a^2b + 6ab^2 + 2ca^2 + 6c^2a + 2b^2c + 3b^2c \ge 22 abc.
$$
Now using that fact that $P(2, 1, 0) \ge P(1, 1, 1)$ (known as Mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3188234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing $\mathbb Z_4 \times \mathbb Z_2$ to be a group I've been posed the question:
Let $P$ be the pairs $(a,b)$ where $a \in \Bbb Z_4$, and $b \in \Bbb Z_2$
An operation, $*$, is defined by: $$(a,b)*(c,d)=(a+c \pmod 4, b+d \pmod 2)$$ for all $(a,c),(b,d)\in P$
How do I show that this is a group?
I know how to do t... | Associativity follows from that of $(\Bbb Z_4, +_4)$ and of $(\Bbb Z_2, +_2)$.
The identity is $(0\pmod 4, 0\pmod 2)$. (Why?)
The inverse of $(a,b)$ under $*$ is given by $(-a, -b)$ since $$\begin{align}(a,b)*(-a, -b)&=(a+(-a)\pmod 4, b+(-b)\pmod 2)\\
&=(0\pmod 4, 0\pmod 2).
\end{align}$$
Closure follows from the closu... | {
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Showing equivalence relations a=b/a=-b I have this equivalence relation where $S=\mathbb{R}$ and $a\sim b$
$ \iff a=b$ or $a=-b$
I know this is an equivalence relation and that it is also very simple but I am just confused about how to test for reflectivity, symmetry and transitivity? Like how can I see what $a\sim a$ ... |
Like how can I see what $a\sim a$ actually means?
$a\sim a$ is a statement. A statement can be either true or false. What that statement is is defined by what $\sim$ means, and in your case, for any arbitrary $a,b$, the statement "$a\sim b$" is the same as the statement "$a=b$ or $a=-b$. Therefore, the statement $a\s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Topological Algebraic Independence of power series Let $p$ be a prime number, let $x$ be a variable, and consider two power series over the ring $\mathbb{Z}_p$ of $p$-adic integers:
$a(x):=\underset{n\geq 1}{\sum}{\frac{p^n}{n!}x^n}=px+\frac{p^2}{2}x^2+\frac{p^3}{6}x^3+\cdots$
$b(x):=\underset{n\geq 1}{\sum}{\frac{p^n}... | It’s late at night, and I hope I’m not getting egg all over my face here, in this argument tailored to your particular example.
First, I’m going to define $\log(x)=-\sum_{n\ge1}(-x)^n/n=x-x^2/2+x^3/3-\cdots$ and $\exp(x)=\sum_{n\ge1}x^n/n!$, so that this log and exp are inverse power series of each other, defined over ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving two graphs have the same chromatic number Let $G= (V,E)$ be a graph, and let $G'= (V',E')$ be a copy of $G$. That is, for each $v ∈ V$ there is a corresponding $v' ∈ V'$ and for each edge $(u,v)∈E$ there is a corresponding edge $(u',v')∈E'$. Construct a graph $G\widehat{}$ by drawing an edge from ea... | If I'm understanding your question correctly, the new graph $\hat{G}$ is just two copies of $G$ with the corresponding vertices connected right? If so, just think about permuting the colors on the second copy. Explicitly, say you colored $V$ with the colors $\{1, 2, \ldots k\}$. Then for $v'\in V'$, if the correspondin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Compute matrix norm induced by weight l1 vector norm For a strictly positive collection of weights $\{w_{i}\}$, consider the weighted $l_{1}$ vector norm:
$$
||x||_{W} = \sum_{i}^{N} w_{i}|x_{i}|
$$
What is (or more accurately, how would you compute) the matrix norm induced by this vector norm? That is, what is $||A||_... | Let $W$ be the diagonal matrix of weights. Notice then that,
$$
\|x\|_W
= \sum_{i=1}^{N} w_i |x_i|
= \sum_{i=1}^{N} |w_ix_i|
= \|Wx\|_1
$$
By definition,
$$
\|A\|_W = \sup_{\|x\|_W = 1} \|Ax\|_W
$$
Thus, letting $y=Wx$ (so that $x = W^{-1}y$),
$$
\|A\|_W
= \sup_{\|Wx\|_1 = 1} \|WAx\|
= \sup_{\|y\|_1 = 1} \|WAW^{-1}y\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3189124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using $P(A'|B')$ to find $P(A\cup B)$ We have $P(B) = \frac 3 5$ and $P(A'|B') = \frac 1 3$.
I started with $P(A\cap B)' = P(B)' \cdot P(A'|B') = \frac 2 5 \cdot \frac 1 3 = \frac 2 {15}$.
Then converted $P(A\cap B)'$ to $P(A'\cup B')$.
Is $P(A'\cup B') = P(A\cup B)'$, meaning we get $P(A\cup B) = 1 - \frac 2 {15} = \f... | You have made two mistakes, but got the right answer! $P(A'\cap B')=P(A'|B') P(B')=\frac 1 3 (1-\frac 3 5)= \frac 2 {15}$. $P(A\cup B)=1-P(A' \cap B')=1-\frac 2 {15}=\frac {13} {15}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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preimage of a torsion-free subgroup
Let $\phi: G \to H$ be a surjective group homomorphism such that $\ker(\phi)$ is torsion-free. Let $B$ be a torsion-free subgroup of $H$. Show that $A = \phi^{-1}(B)$ is torsion-free.
I'm confused why we need the condition that $\ker\phi$ is torsion-free.
| We let $a\not=e$ is torsion, so $\varphi(a)^n=\varphi(a^n)=e$.
Now since $B$ is torsion-free, $\varphi(a)=e$, hence $a \in \ker \varphi$. This is a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
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How does fundamental theorem of calculus and chain rule work? I came across a problem of fundamental theorem of calculus while studying Integral calculus.
A problem:
$\frac{d}{dx}\int_{\pi}^{x^2}\cot^2t\ dt$
which was salved as :
Step I : Let, F(x) = $\int_{\pi}^{x^{ }}\cot^2t\ dt$
⇒ $F'(x)= \frac{d}{dx}\int_{\pi}^{x^... | I think you're confusing $F'(x^2)$ and $(F(x^2))'$. The first is the function $F'$ $\bf{evaluated}$ at $x^2$ and the second is the derivative of the function $x \mapsto F(x^2).$ These are two different things !
If you take the function $x \mapsto F(x) = 2x + 1$. Then $F'(x) = 2$ so $F'(x^2) = 2$ but $$(F(x^2))' = (x^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Fourier series of translated square I can't seem to find the correct Fourier series coefficients ($s_n$) of the following periodic function. I know how to get the Fourier series of the same one that is not vertically translated and has no negative values for any $t$ (Result is $A sinc(n\pi)$), but this one I'm having t... | Since it is said that $x$ is $T$-periodic, you have that
$x(t) = -A$ for $t\in(-T/2,-T/4)$, so you will have to re-evaluate the integrals.
Then, note that
$$\text{sinc}(n\pi) = 0$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Greatest Integer Function linear equation Given that $2[x]=x+2(x)$, $[x]$ if the Greatest Integer Function and $(x)$ is the fractional part of $x$, find the value (s) of $x$.
I tried replacing $(x)=x–[x]$ but for an equation in $x$ and $[x]$. How do I proceed???
| Let $x = n + r$ where $n \in \mathbb{Z}$ and $0 \le r \lt 1$. Then $[x] = n$ and $(x) = r$, so your equation becomes $2n = (n + r) + 2r \; \Rightarrow \; n = 3r$. Since $0 \le 3r \lt 3$, this gives $3$ choices for $n$ of $0, 1, 2$. You can then determine the matching values of $r$ and $x$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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prime numbers and expressing non-prime numbers My textbook says if $b$ is a non-prime number then it can be expressed as a product of prime numbers. But if $1$ isn't prime how it can be expressed as a product of prime numbers?
| This is mainly just an extended comment on Peter Foreman's answer. The (relatively difficult) uniqueness aspect of the Fundamental Theorem of Arithmetic is not needed for the OP's question, just the (easier) existence aspect.
What's missing from the OP's textbook is the qualifier in the correct assertion that every n... | {
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"timestamp": "2023-03-29T00:00:00",
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Tensor product notation as a power I chance upon a notation while reading a paper which I do not quite understand.
Suppose that $\hat{J}$ is a operator in a tensor product of two N$ dimensional Hilbert space. Explicitly, it is given by
$\hat{J} = \frac{1}{\sqrt{2}}(\hat{I}^{\otimes N} + i\hat{\sigma}_{x}^{\otimes N}... | In general, when dealing with any product-like operation $\star$, you should understand $a^{\star n}$ as $$\underbrace{a \star a \star \dots \star a}_{n \text { times}}.$$ This might be written as simply $a^n$ in case where it's completely obvious which product is meant, but putting $\star$ in the exponent specifies wh... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why are these two definitions of Martingales equivalent? I am recently reading books on Probability theory, in Durrett's book Probability: Theory and Examples, the definition is following:
If $X_{n}$ is sequence with
(1) $\mathbb{E}\left|X_{n}\right|<\infty$
(2) $X_{n} \text { is adapted to } \mathcal{F}_{n}$
(3) $\ma... | From Cinlar's definition, if $X_t$ is a martingale, it is both a supermartingale and a submartingale.
The submartingale property implies $E_s[X_t - X_s] \ge 0$, and the supermartingale property implies $E_s[-(X_t - X_s)] \ge 0$, and thus $E_s[X_t - X_s] = 0$. Since $X_s \in \mathcal{F}_s$, this can be rewritten as $E_s... | {
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Regarding steady state solution of $u_{t}= c^2 u_{xx}$ with $u_{x}(0,t) = c_{1}$ and $u(L,t) = c_{1}$? Suppose we have the one dimensional diffusion equation $u_{t} = c^2 u_{xx}$ with the boundary condition $u(L,t) = c_{2} $ and $u_{x}(0,t) = c_{1}$. I donot recognize which type of condition is it? badly, it seems to b... | Let $u(x,t) = w(x) + v(x,t)$ where $w(x)$ is the steady-state. Then we have
$$ w''(x) = 0, \quad w'(0) = w(L) = c_1 $$
which gives $w(x) = c_1(x-L) + c_1$
Now you can use separation of variables to find $v(x,t)$, which is homogeneous on the boundary (of "mixed" type).
| {
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"timestamp": "2023-03-29T00:00:00",
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Matrix such that $A^n=aA$ Let $A\in M_n(\mathbb{C})$ be a matrix such that $A^n=aA$,where $a\in \mathbb{R}-\{0,1\}$.
I wanted to find $A$'s eigenvalues and I thought that they are the roots of the polynomial equation $x^n=ax$. Is this correct?
| If $A$ has eigenvalue $b$, and $v$ is a corresponding (non-zero) eigenvector, then $$0 = (A^n-aA)v = A^nv - aAv = b^nv-abv = (b^n-ab)v$$This means $b^n-ab = 0$, which does make $b$ a root to the polynomial equation $x^n = ax$.
Apart from that, there is not much we can say. $A$ could have one, some, or all of those root... | {
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"timestamp": "2023-03-29T00:00:00",
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Multiplicative Chernoff Bound for coins We flip 1000 fair coins and mark the results with $X_1, . . . , X_{1000}$. A pair of adjacent coins are $X_i,X_{i+1}$ coins - i count $X_{1000}$ and $X_1$ as adjacent. We refer to the number of pairs of adjacent coins showing both heads as X.
I want to use the multiplicative Ch... | Let $Y$ be the number of pairs of consecutive heads of the form $(X_{2i},X_{2i+1})$, and let $Z$ be the number of pairs of the form $(X_{2i-1},X_{2i})$. Note $X=Y+Z$, so
$$
P(X\ge 300)\le P(Y\ge 150)+P(Z\ge 150)
$$
You can then apply the Chernoff bound to each of $Y$ and $Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3190974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is wrong with my argument using of Collatz–Wielandt formula? If $A$ is a positive square matrix, then the Collatz–Wielandt implies that
$\min_{=1,…,;_\neq 0}\frac{(Ay)_i}{y_i}≤≤\max_{=1,…,;_\neq 0}\frac{(Ay)_i}{y_i}$,
Where $r$ is the largest eigenvalue of $A$.
By replacing $y=e_j$ in the previous expression wou... | No. In $e_j$, $j$ is not related to $i$. You would have
$$ \min_i a_{ji} \leq r \leq \max_i a_{ji} $$
(... or possibly "$a_{ij}$" in both places, since I don't know whether you are using row first or column first indexing for matrices). You are pulling the minimum and maximum entries from the $j^\text{th}$ column... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3191125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the length x such that the two distances in the triangle are the same I have been working on the following problem
Statement
Assume you have a right angle triangle $\Delta ABC$ with cateti $a$, $b$ and hypotenuse $c = \sqrt{a^2 + b^2}$. Find or construct a point $D$ on the hypothenuse such that the distance $|CD|... | Not sure if this is less barbaric but using simple trig: $DE=(a-x)\tan A$, $DC=\frac{x}{\cos A}$ so the equation to solve is $$(a-x)\frac{b}{a}=\frac{x\sqrt{a^2+b^2}}{a}$$ or $$x=\frac{ab}{\sqrt{a^2+b^2}+b}$$
Just another idea to construct point $E$: since $\triangle{DCE}$ is isosceles, it's easy to find $\angle{ACE}=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3191278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is it possible to calculate $\int_{0}^{\pi}(a+\cos{\theta})^nd\theta$, where $a$ is a nonzero integer? I have tried to answer by taking change the variable $\theta$ to $\theta/2$, so the integration is now over unit circle, then I have taken $z=e^{i\theta}$. Now I tried to use residue formula for integration, but I fai... | $\int_0^{\pi} (a+\cos x)^n \ dx = \frac 12 \int_0^{2\pi} (a+\cos x)^n \ dx\\
\cos x = \frac 12 (e^{ix} + e^{-ix})\\
\frac 1{2^{n+1}} \int_0^{2\pi} (2a+e^{ix} + e^{-ix})^n \ dx
z = e^{ix}\\
dx = \frac {1}{iz}\ dz$
$\frac 1{2^{n+1}i} \oint_{|z| = 1} \frac {1}{z}(2a+z + z^{-1})^n \ dz$
Now the trick. When we expand $(2a+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3191402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove or disapprove that the integral converges using Taylor's series $\int_0^\infty\sin(\frac{\sin x}{\sqrt x})dx$
I have an idea to do smth like that:
$\int_0^\infty\sin(\frac{\sin x}{\sqrt x})dx = \int_0^\infty(\frac{\sin x}{\sqrt x} + O(\frac{1}{\sqrt x}))$ and this integral should converge, but I have a feeling th... | $$
\begin{align}
\int_0^\infty\sin\left(\frac{\sin(x)}{\sqrt{x}}\right)\,\mathrm{d}x
&=\int_0^\infty\left(\frac{\sin(x)}{\sqrt x}+O\!\left(\frac{x^{3/2}}{1+x^3}\right)\right)\,\mathrm{d}x\\
&=\sqrt{\frac\pi2}+O\!\left(\frac{2\pi}3\right)
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3191643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Describe all integral solutions of the equation $x^2 + y^2 = 2z^2$ such that $x,y,z > 0$, gcd$(x,y,z) = 1$, and $x > y$. As the title states, the question tasks me with finding all the integral solutions of the equation under the specified constraints. I have an idea of where to start due to a somewhat similar problem ... | start with a primitive Pythagorean triple $x^2 + y^2 = z^2.$ there is a recipe for these. Then
$$ (x+y)^2 + |x-y|^2 = 2 z^2 $$
And those are all
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Summation formula for this? I have found the following summation formula based on a recurrence. It supposes $n = 2^k$ where k is an integer. I've intuitively discovered that the following closed form may be true (following the constraint on n), but I'm not sure why.
$\sum_{\textstyle i=0}^{\textstyle \lg n} {\frac n{... | Let $m$ be a nonnegative integer. Then $$\sum_{i=0}^m \frac{1}{2^i}$$ is simply a finite geometric series with common ratio $r = 1/2$. In general, $$\sum_{i=0}^m r^i = \begin{cases} \frac{r^{m+1} - 1}{r - 1}, & r \ne 1, \\ m+1, & r = 1, \end{cases} \tag{1}$$ from which your desired result follows immediately.
The pro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving $\frac{dy}{dx} = ae^{-bx} - cy(x)$ How would you solve an equation in the form
$$\frac{dy}{dx} = ae^{-bx} - cy(x) $$
where $a, b, c$ are just constants. My ultimate goal is to find $y(x)$ without the derivative in there.
My confusion comes from the fact that the right hand side has both $y(x)$ and $x$ itself i... | $$y'+cy = ae^{-bx} \tag 1$$
Solving with the variation of parameter method :
First, solve the associated homogeneous ODE
$$\quad y'+cy = 0 \tag 2$$
The solution is :
$$y=\lambda e^{-cx}$$
where $\lambda$ is a constant with respect to $x$.
Second, apply the method of variation of parameter. This means that the constant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3192075",
"timestamp": "2023-03-29T00:00:00",
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Power sets : Do the relations between P(A) and P(B) always mirror the relations between the sets A and B? If I am correct it is true that:
(1) "$P(A)$ is included in $P(B)$" implies "$A$ is included in $B$".
(2) "$P(A) = P(B)$" implies "$A = B$".
Might I conclude from this that the power sets of two sets always have... | Let's take these sets as an example:
$$
\begin{align}
A &= \{1\} \\
B &= \{2\} \\
P(A) &= \{\emptyset, \{1\}\} \\
P(B) &= \{\emptyset, \{2\}\}
\end{align}
$$
A relation is in this context a function that takes two sets and gives true or false.
A simple counterexample relation is:
$$f(X, Y) = X \text{ contains a set}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3192205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Example of a sequence needed that satisfies certain conditions Can you please provide examples of sequence $\{a_1,a_2,\cdots\}$ such that $\sum_{i=1}^{\infty}a_i\to \infty$ while $\sum_{i=1}^{\infty}(a_i)^2<\infty$ with each $a_i\in[0,1)$. Thank you.
| A classical example is the sequence
$$ a = (1, 1/2, 1/3,1/4...) .$$
It is a well known result that
$$ \sum_{k=1}^{\infty} a_k = \sum_{k=1}^{\infty} 1/k = \infty. $$
On the other hand, it holds that
$$ \sum_{k=1}^{\infty} a_k^2 = \sum_{k=1}^{\infty} 1/k² = \pi²/6 \quad (\dagger). $$
Showing the equality in $(\dagger)$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3192349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Special case of Bertrand Paradox or just a mistake? I've been working on a question and it seems I have obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in case anyone is interested.
I want to know what is the average distanc... | Thanks, Erick Wong for your feedback. After your answer, I calculated the distribution of the arc length subject to the uniform distribution of the point on the diameter. In fact: if we want to express the arc length $l$ as a function of $x$, $l = f(x)$ we obtain:
$l = \arccos(1-x), x = 1-\cos{l}, |\frac{d}{dl}(x)| = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3192613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Convergent series by the Root Test Determining the convergence of $$\sum_{n=1}^{\infty}\frac{n^3\left(\sqrt{2}+(-1)^n\right)^n}{3^n}$$
Applying the Root Test: $$r=\lim_{n\to\infty}\sqrt[n]{\frac{n^3\left(\sqrt{2}+(-1)^n\right)}{3^n}}=\lim\sqrt[n]{n^3\cdot\left(\frac{(\sqrt{2}+(-1)^n}{3}\right)^n}=\left(\lim n^{\frac{3}... | It does converge
but your proof is not correct.
What you should write is
$\left|\dfrac{\sqrt{2}+(-1)^n}{3}\right|
\le \left|\dfrac{\sqrt{2}+1}{3}\right|
\lt 1
$
and this shows convergence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3192715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove the following ideal $I$ is not a principal ideal. Prove the ideal $I = \left<X^2,3\right> \space $of$ \space \mathbb{Z}[X]$ is not a principal ideal.
The solution I have been given is the following:
Assume for contradiction that I were a principal ideal, i.e., $I = \left<f(X)\right>$ for some $f(X) \in \mathbb{Z}... | I'll assume you are happy with the first line.
Equation (1) is saying $f$ must be a combination of the generators given. Similarly, equations (2) and (3) say the generators given must be recoverable from $f$. Thus, the equations together give equality.
The degree of a univariate polynomial in $x$ is simply the highest ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3192853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Why do early math courses focus on the cross sections of a cone and not on other 3D objects? Conic sections seem to get special attention in early math classes.
My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some other 3D (or any dimensiona... |
My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some other 3D (or any dimensional) object?
Because there's nothing else. Look at the common 3D solids. The cross sections of a cuboid (and in fact any other polytope) are just a bunch of stra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3193067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 4
} |
Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$ I am really satisfied that $(x \to a) ≠(x=a)$ and if that is not right , Then all the process of $Limits$ is dividing by zero and that is a crime.
Since $(x \to a) + h = (x=a)$ , $h ≠ 0$,So Why does $f(x \to a) = f(x = a)$ ?
NOTE:I am talking about continuous... | Your question seems unclear, but perhaps emphasizing this distinction will help:
A function $f: \mathbb R \longrightarrow \mathbb R$ is continuous at $a \in \mathbb R$ if (and only if) for every $\varepsilon>0$ there exists $\delta>0$ such that $|f(x)-f(a)|<\varepsilon$ whenever $|x-a|<\delta$.
We write $\lim_{x \to a}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3193342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Proof verification: Prove $\sqrt{n}$ is irrational. Problem
Let $n$ be a positive integer and not a perfect square. Prove that $\sqrt{n}$ is irrational.
Proof
Consider proving by contradiction. If $\sqrt{n}$ is rational, then there exist two coprime integers $p,q$ such that $$\sqrt{n}=\frac{p}{q},$$ which implies $$p^2... | Proof by contradiction is not needed. It suffices to "take the contrapositive". This is when you switch the antecedent and the conclusion of an implication and negate them. Formally, it looks like this:
$$ P \implies Q \text{ has the contrapositive } \neg Q \implies \neg P$$
It applies to the proof in the following way... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3193554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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How do I evaluate the following combination of random variables? Is it martingale? I'm about to analyse the following expression
$$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$
where $Y_j$ for all $j\in \mathbb{N}$, are independent and $\Gamma(\beta... | I assume that the family $\left(X_i,Y_j,i,j\geqslant 1\right)$ is independent.
There exists a formula for the conditional expectation of a function of two independent vectors with respect to the first vector. Let $f\colon \mathbb R^{k}\times\mathbb R^\ell\to\mathbb R$ be a measurable function and $U$ and $V$ two indepe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3193698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Semigroups with no morphisms between them Given two monoids we always have a morphism from one to the other thanks to the presence of the identity element.
Are there examples of non-empty semigroups that have no morphisms from one to the other? The destination semigroup can't be finite, because if we have an idempotent... | No morphism in either direction
Choose two distinct primes $p,q\in\Bbb N_+$ and consider the additive semi-groups
$$\begin{align}
P&:=\Big\{\frac n{p^m}\mid n,m\in\Bbb N_+\Big\}\subseteq\Bbb Q_+,\\
Q&:=\Big\{\frac n{q^m}\mid n,m\in\Bbb N_+\Big\}\subseteq\Bbb Q_+.
\end{align}$$
Assume there is a morphism $\phi:P\to Q$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3193852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 0
} |
Does Fourier imply Laplace? Can we find a function $f(t)$ for which $$\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt,$$ converges but $$\int_{-\infty}^{+\infty}f(t)e^{-st}dt,$$ does not ?
Here, $j^2=-1$, $\omega$ is a real number and $s$ is a complex number.
I am thinking that we can find such an $f(t)$, for example, whe... | $f(t)=\frac 1 {1+t^{2}}$ is such a function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3194078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding the floor of $\sqrt{d^2-1}$ If we are given $d$ an integer, then I've seen it written that the floor of $\sqrt{d^2-1}=d-1$, but how is this found?
It's not immediately obvious, at least not to me.It makes me feel there must be an algorithm that can be used .
Would anyone be able to impart their knowledge on ho... | It is because $$(d-1)^2 \le d^2-1 < d^2,$$ so $$d-1 \le \sqrt{d^2-1} < d,$$ so $$\lfloor \sqrt{d^2-1}\rfloor = d-1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3194249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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