Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
A is independent of B, and B depends on A? I learned in class that the following three conditions are equal:
*
*$P(A\cap B)=P(A)\cdot P(B)$
*$P(A|B)=P(A)$
*$P(B|A)=P(B)$
But suppose event $A$ is a subset of $B$ such that the (2) condition applies and $P(B)<1$. In that case the probability $P(B|A)$ must equal $1$, ... | If $A$ is a subset of $B$ and $P(B)< 1$, then all conditions are wrong. For (2), you get $P(A|B) = P(A)/P(B)\not=P(A)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the limit of sequence with respect to a given topology
Find the limit of sequence $x_{n} = (\frac{1}{n},\frac{1}{n})$ with respect to the topology $$\tau = \{(n,\infty)\times (n, \infty): n \in \mathbb{N} \} \cup \{\emptyset,\mathbb{R^2}\}.$$
My work. Let $x$ be a limit.
If $x<0$, every number can be a limit, b... | Let's abstract a bit: the topology is just a decreasing sequence of open sets $U_1 \supset U_2 \supset U_3 \supset \ldots $ with empty intersection, and the compulsory $\emptyset, X$ are open too. It's quite easy to check that this always gives a topology.
Now in our case, $U_n = (n,\infty) \times (n, \infty)$ of cours... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to show that $p(1)\ \text{is real} \iff \ p(-1)\ \text{is real}$ I have been working on this problem and I cannot figure it out! I spent hours of time on it with no use. Can anybody help? The question is:
Suppose $p(x)$ is a polynomial with complex coefficients and even degree($n=2k$). All zeros of $p$ are non-real... | Note that (assuming $p(-1)\ne 0$ to begin with
$$ \frac{p(1)}{p(-1)}=\prod_{j=1}^{2k}\frac{1-w_j}{-1-w_j}$$
where the $w_j$ run over the complex roots (with multiplicity).
For a single factor,
$$\frac{1-w}{-1-w}=-\frac{(1-w)(1+\bar w)}{|1+w|^2}=\frac{|w|^2-1+(w-\bar w)}{|1+w|^2}. $$
As we are given that $|w|=1$ for al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Asymptotic behavior $\sum_{n=1}^x\phi_k(n)$, a variant of Euler's Totient function Let $$\phi_k(x)=\sum_{1\le n \le x \\(n,x)=1} n^k$$
What's the asymptotic behavior of
$$\sum_{n=1}^x\phi_k(n)?$$
According to the wikipedia $\sum^x_{n=1} \phi_0 (n) \approx \frac{3}{\pi^2}x^2 $. It also appears in page $69$ and $70$ wh... | Not an answer.
$$\sum_{n=1}^\infty\frac{\phi_k(n)}{n^s}=\frac{1}{\zeta(s-k)}\sum_{l=0}^{k+1}c(k,l)\zeta(s-l)$$
Where $c(k,k+1)=\frac{1}{k+1}, c(k,k)=\frac{1}{2}$ and $c(k, k-l+1) = \frac{B_lk!}{l!(k-l+1)!}$
where $B_k$ is the $k$th Bernoulli number which we define in terms of Stirling numbers of the second kind.
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the mistake in $\lim_{x\rightarrow 1^-} \frac{\sum_{n=0}^\infty x^n}{\sum_{n=0}^\infty x^n}=1 \Rightarrow 1=\frac{1}{2}$ It is obvious that we have:
$$\lim_{x\rightarrow 1^-} \frac{\sum_{n=0}^\infty x^n}{\sum_{n=0}^\infty x^n}=\lim_{x\rightarrow 1^-}1=1.$$
But let us now write this sum in two ways, let $a_n=x^n$ a... | In order to use the Moore-Osgood Theorem, you must make sure that $(f_n)_{n \geq 0}$ converges uniformly toward $f$.
$i.e. \sup\limits_{[0,1]}|f_n - f| \rightarrow 0$.
This is not the case here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Help understand beta reduction example I am currently reading a text book on distributed computing systems that includes a short introduction to $\lambda$-calculus. There is an example of evaluating the sequence $(((if \space \space true) \space \space 4) \space \space 5)$ which is written below.
*
*$\space\space(... | When going from line 4 to line 5, we substitute $x=4$ into $\lambda y. x$, because the inner argument is bound to the $\lambda x$. The 5 is then passed to the result and bound to the $\lambda y$, so that we substitute $y=5$ into $\lambda y. 4$. Since $y$ does not appear free in $4$, this does not alter the expression... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3012109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Counting question on bit strings - problem with using cases
How many bit strings of length 10 either begin with three 0s or end with two 0s?
I solved this question using cases but I do not seem to be getting the answer of $352$.
My attempt:
Consider two cases:
*
*Case 1: The string begins with three $0$s and does ... | $$\underbrace{2^7}_{\text{begin with three zeros}}+\underbrace{2^8}_{\text{end with two zeros}}-\underbrace{2^5}_{\text{double-count}} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3012277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is "the set of all polynomials in $\pi$"? From Ian Stewart's Galois Theory (2015, 4e, p. 20):
What does, for example, "the set of all polynomials in $\pi$" mean?
| The set of all polynomials in $\pi$ with rational coefficients is the set of real numbers of the form $p(\pi)$, where $p(x)$ is a polynomial with rational coefficients; that is, it is the set $\{ p(\pi) \mid p(x) \in \mathbb{Q}[x] \}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3012397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Tensor product terminology in category theory? Say that I have any homomorphisms of commutative rings, $A \rightarrow B, A \rightarrow C.$ I recently read that $B \otimes_A C$ is the pushout of the morphisms in the category of commutative rings. However, I understood tensor products as defined for modules over a commut... | Take all rings here to be commutative. A ring homomorphism $f:A\to B$
makes $B$ into an $A$-module. In detail, the module action is $a\cdot b=f(a)b$.
With another ring homomorphism $g:A\to C$ then we have two $A$-modules,
and can form the tensor product $B\otimes_A C$.
At first $B\otimes_A C$ is just a module. But it h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3012515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
$(a) = R$ if and only if a is a unit. Let $⟨R;+,−,0,·,1⟩$ be a commutative ring. For $a \in R$, define $(a)$ $:= \{a · r | r \in R \}.$ How can i prove that $(a) = R$ if and only if a is a unit.
So if there exist $a' \in R$ such that $a*a' = 1$ we have $ \forall r \in R $ $ (a*a')*r = r$ hence $(a) = R$ , right ?
If ... | The first part is correct: if $a$ is a unit, then $ab=1$ for some $b\in R$; therefore, for every $r\in R$,
$$
r=r1=r(ab)=a(rb)\in(a)
$$
hence $R\subseteq (a)$ and therefore $R=(a)$.
The second part is simpler: if $(a)=R$, then $1\in (a)$, so there exists $b\in R$ with $ab=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3012674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Singularity of Morphism and Its Extension I really need help with this question.
Assume you have a morphism $\varphi: \mathbb{A}^2 \rightarrow
> \mathbb{A}^1$ such that $\varphi (x,y)= x^2-y^4$.
1) Find all points in $\mathbb{C}$ such that $\varphi^{-1}(a)$ is
singular. And its type of singularity.
2) If $Y_{a}$ is... | 1) As you said, by the Jacobian criterion indeed $\varphi^{-1}(0)$ is the unique singular fiber and the singularity type is two tangent parabolas $(x-y^2)(x+y^2) = 0$.
2) The fiber has homogenous equation $x^2z^2 = y^4 + az^4 $. Its intersection with the line at infinity given by $z=0$ and the previous equation, i.e ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3012815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to show that $2^n > n$ without induction I'm solving exercises about Pascal's triangle and Binomial theorem, and this problem showed up, however I don't have any clue on how to solve it
The sum of ${n\choose p}$ from $p=0$ to $n$ is the same thing as $(1+1)^n=2^n$, how can I use this information?
Maybe comparing wi... | hint
Consider $x\mapsto \frac{\ln(x)}{x}$ for $x\ge 1$.
$$f'(x)=\frac{1-\ln(x)}{x^2}$$
the maximum if $f(e)=\frac{1}{2}<\ln(2)$.
thus
$$\ln(x)<x\ln(2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Birthday problem: why not: combinations without/with replacement?? My first intuition on the birthday problem was:
*
*The number of ways $k$ people have different birthdays is the combinations $\binom{365}{k}$
*The number of ways $k$ people can have birthdays is the combinations with replacement.
So the probabilit... | The calculation $\binom{365}{k}$ only chooses $k$ days; it does not assign particular days to the people. So your first statement is incorrect. I am not sure where your second formula comes from, but you're probably making a similar error.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find a partial derivatives by definition I have to check whether the function is able to be differentiated on M(0, 0), and find partial derivatives $f_x'(0, 0), f'_y(0, 0)$. Is it correct?
Let $z = {x}+{y}+\sqrt{\mid{xy}\mid}$.
By definition of partial derivative, $$\frac{\partial{z}}{\partial{x_k}} = \lim_{\Delta{x}\t... | That part is OK.
Now, to see that $f$ is NOT differentiable at $(0,0)$, remember that if $f$ is differentiable at $(0,0)$, then for any direction $\vec v$ such that $\|\vec v\|=1$ it is true that
$$f'_{\vec v}(0,0)=\langle \nabla f(0,0),\vec v\rangle.$$
But you proved that $\nabla f(0,0)=(1,1)$, and you can use the def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the Legendre polynomial Let us consider the numerical integral $ \ \int_{-1}^{1}w(x) f(x)dx=\sum_{i=0}^{N} f(x_i)w_i$, where $w_i$ are the weights and $w(x)$ is the weight function.
Legendre polynomials, denoted by $ \{p_n \}$ are a list of orthogonal polynomial supported on $[-1,1]$ with weight $w(x)=1$. Then th... | (Up to now I see no correction to the question, here is the computation of the first Legendre polynomials)
This is a simple exercise in integration and using the orthogonality relations.
Let's write $(f,g)=\int_{-1}^1 f(x)g(x) dx$. Then you have to compute the polynomials $p_n$ of degree $n$ with
$(p_n, p_m) = \frac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Divisors of $-1$ are only $1$ and $-1$? I'm working through a discrete math textbook and I've come across this question with answer:
Prove that the only divisors of $−1$ are $−1$ and $1$.
Answer:
We established that $1$ divides any number; hence, it divides $−1$, and any nonzero number divides itself. Thus, $1$ and $−1... | We are assuming that $d$ is a divisor of $-1$ that is
$$dk=-1$$
and multiplying each side by $-1$ we obtain
$$-1\cdot dk=-1\cdot (-1)=1 \iff d(-k)=1 \iff d=1,-1 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Making sure if it is Cauchy In my real analysis exam I had a problem in which I proved that given any positive number $a\lt 1$ if $|x_{n+1} - x_n|\lt {a^n}$ for all natural numbers $n$ then $(x_n)$ is a Cauchy sequence.
This was solved successfully but the question is if $|x_{n+1} - x_n|\lt \frac 1n$ does that mean $(x... | Take $x_n=1+\frac 1 2+\cdots+\frac 1 n$. This is not Cauchy because the harmonic series $1+\frac 1 2+\cdots$ is divergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 1
} |
Smooth a graph as if placing a rope across the data…? I'm not sure how to correctly phrase this question, in fact if I knew exactly what I needed to ask I could probably work it out myself, so please bear with me.
What I need to do is smooth out a line, but weight the smoothing so that the smoothed graph never goes be... | You're possibly looking for the envelope of the data. There is a Wikipedia article on it. https://en.m.wikipedia.org/wiki/Envelope_(waves)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Let $f$ be integrable on $[a,b]$ and suppose for each integrable function $g$ defined on $[a,b]$, $\int^{b}_afg=0$, then $f(x)=0,\forall x\in[a,b]$ I do not think this is true,
but at the same time I am not sure.
I know that if we assume that f is continuous instead of integrable then this statement is true. I just do ... | Another counter example is that consider $f:[0,1]\rightarrow [0,1]$ given by $f(x)=0$ if $x$ is irrational or $x=0$ and $f(\frac{p}{q})=\frac{1}{q}$ where $p\in \Bbb Z-\{0\},q\in \Bbb N,gcd(p,q)=1$ , then $f$ is Riemann integrable and $\int_0^1 f=0$ and for any other Riemann integrable $g$ we have using Cauchy-Schwarz ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Finding $\lim_{x\to\pi/2}\left(\frac{1-\sin x}{(\pi-2x)^4}\right)(\cos x)(8x^3 - \pi^3)$ using algebra of limits Let $$
F(x) = \left(\frac{1-\sin x}{(\pi-2x)^4}\right)(\cos x)(8x^3 - \pi^3)
$$
Then find the limit of $F(x)$ as $x$ tends to $\pi/2$.
How can we find the limit using algebra of limits?
The limit of $\dfrac... | The limit of $\frac{1-\sin x}{(\pi-2x)^4}$ as $x$ tends to $\pi/2$ is NOT some non zero finite number.
Note that $(8x^3 - \pi^3)=(2x-\pi)(4x^2+2x\pi+\pi^2)$, then
$$F(x)=\frac{1-\sin x}{(2x-\pi)^2}\cdot\frac{\cos x}{(2x-\pi)}\cdot (4x^2+2x\pi+\pi^2)$$
Now evaluate the limit of each factor. This time they are all finit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Minimal elements Let $A$ be the set $A = \{1,2,3,...,20\}$.
$R$ is the relation over $A$ such that $xRy$ iff $y/x = 2^i$, $i$ is natural including $0$.
I am supposed to find the minimal and maximal elements in relation to $R$.
Does that mean the elements are members of $A$ or members of $R$?
The elements of $R$ are pai... | You're searching for members of $A$.
$a \in A$ is maximal, whenever $aRa'$ implies $a'=a$.
$a \in A$ is minimal, whenever $a'Ra$ implies $a'=a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to show $2^{ℵ_0} \leq \mathfrak c$ I want to show $2^{ℵ_0}=\mathfrak c$.
I already showed $\mathfrak c \leq 2^{ℵ_0}$ as follows:
Each real number is constructed from an integer part and a decimal fraction. The decimal fraction is countable and has $ℵ_0$ digits. So we have
$\mathfrak c \leq ℵ_0 * 10^{ℵ_0} \leq 2^{ℵ_... | $2^{\aleph_0}$ is the cardinality of all reals (belonging to $(0,1)$ if you prefer) that you can write by using only $0,1$. Those numbers clearly form a subset of $\mathbb R$ which must therefore have cardinality at least $2^{\aleph_0}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Factorial Proof - ${n \choose r-1}+{n \choose r}={n+1 \choose r}$ ${n \choose r-1}+{n \choose r}={n+1 \choose r}.$
So what I tried to do was expand the first and second term.
$\frac{n!}{(r-1)!(n+1-r)!}+\frac{n!}{(r)!(n-r)!}.$
Then what I did was try to get common denominators.
$\frac{n!}{(r-1)!(n+1-r)(n-r)!}+\frac{n!}{... | You start well:
$$
\binom{n}{r-1}+\binom{n}{r}=
\frac{n!}{(r-1)!\,(n-r+1)!}+\frac{n!}{r!\,(n-r)!}
$$
Now it's best to collect all common terms. Not so different from what you did, but with less complications. Note you can collect
*
*$n!$ in the numerator
*$(r-1)!$ in the denominator, using $r!=r\,(r-1)!$
*$(n-r)!$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Truth tables from word problem Sentential Logic I am reading the book "how to prove it" and the answers says that this argument is valid and that I'm to construct a truth table to verify it but I just can't see how this argument is valid and I'm not sure how to construct the truth table to prove it. Here is the argumen... | Hint
1) Either sales or expenses will go up.
2)If sales go up, then the boss will be happy.
3) If expenses go up, then the boss will be unhappy.
4) Therefore, sales and expenses will not both go up.
In symbols :
1) $S \lor E$
2) $S \to H$
3) $E \to \lnot H$
4) $\lnot (S \land E)$
Having said that, you have to b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Rewriting a logical statement Only lakers are irrational people.
I believe it technically should be translated as:
All irrational people are lakers.
Is there is any way at all to rewrite the above statement to mean the following and be logically correct:
All lakers are irrational people.
How would you justify it? (If i... | Indicating with $L$ the set of lakers $l$ and with $\Pi$ the set of irrational people $\pi$, the first statement is equivalent to
$$\forall \pi\in \Pi \quad \pi\in L$$
the second one is
$$\forall l\in L\quad l\in \Pi $$
which is not equivalent to the first one, indeed from this last one we could also have $\pi \not \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Problem on dual basis
Let V be the real vector space of all polynomials, in a single
variable and with real coefficients, of degree at most $3$.
Let $V^*$ be its dual space.
Let $t_1 = 1, t_2 = 2, t_3 = 3, t_4 = 4.$
Which of the following sets of functionals $\{f_i |1 \leq i \leq 4\}$
form a basis for $V^*$?
... | One approach is as follows: just as the linear map
$$
p \mapsto \pmatrix{p(1)\\p(2)\\p(3)\\p(4)}
$$
has a trivial kernel, show that the map
$$
p \mapsto \pmatrix{p(1)\\p(2)\\p'(1)\\p'(2)}
$$
has a trivial kernel.
For c, note that we can rewrite $f_4(p) = p(b) - p(a)$. So, whether or not the $f_i$ are linearly independ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $f(x)=\log\sqrt{\frac{1+x}{1-x}}$ is surjective from $(-1,1)$ to $\mathbb{R}$. I have to prove that the function $\;f:(-1,1)\to \mathbb{R}\;$ defined by $f(x)=\log\sqrt{\frac{1+x}{1-x}}\;$ is bijective.
I have already proved that it is injective:
$$f(x)=f(y)$$
$$\log\sqrt{\frac{1+x}{1-x}}=\log\sqrt{\frac{1+y... | We have that $f(x)$ is defined in $(-1,1)$ and
$$f(x)=\log\sqrt{\frac{1+x}{1-x}}\implies f'(x)=\frac1{1-x^2}>0$$
then $f(x)$ is injective, moreover
$$\lim_{x\to 1^-} f(x)=\infty \quad \lim_{x\to -1^+} f(x)=-\infty$$
and since $f(x)$ is continuous by IVT it is surjective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Evaluate $\lim_{h \rightarrow 0} {(x+h)^{99}-x^{99}\over h}$
$$\lim_{h \rightarrow 0} {(x+h)^{99}-x^{99}\over h}$$
I need to factor this in order to get a limit.
I tried:
$$\lim_{h \rightarrow 0} {[(x+h)^{33}]^3-[(x)^{33}]^3\over h}
\\ \lim_{h \rightarrow 0} {[(x+h)^{33}-(x)^{33}][(x+h)^{66}+(x+h)^{33}(x)^{33}+(x)^{6... | Because both $\lim\limits_{h\to 0}((x-h)^{99}-x^{99})=0$ and $\lim\limits_{h\to 0}h=0$ you may also use L'Hospital's rule:
$\displaystyle\lim\limits_{h\to 0}\frac{(x-h)^{99}-x^{99}}{h}=\displaystyle\lim\limits_{h\to 0}\frac{\frac{d}{dh}((x-h)^{99}-x^{99})}{\frac{d}{dh}h}=\displaystyle\lim\limits_{h\to 0}\frac{99(x-h)^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3014988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
approximation of trigonometric functions in form $a + bx + cx^2 +...$ Is there any way to express the trigonometric functions as infinitely long polinomials? If so, how? If not why?
Obviously doing $x(x-\pi)(x-2\pi)(x-3\pi)...$ does not work as it doesn't match for values in between the zeroes.
| I have heard that there are ways of making your approach work, but I don't know anything about it personally.
The standard approach would be Taylor series. The idea of Taylor series is to make an "infinite degree" polynomial for which the function value and all derivatives agree with those of your function.
Specificall... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3015082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Locality and Hilbert Curve I have a hilbert curve index based on this algorithm. I take two to four values (latitude, longitude, time in unix format and an id code) and create a 1-d hilbert curve.
I'm looking for a way to use this data to create a bounding box query (i.e. "find all ids within this rectangle).
I'm looki... | I just found the answer in this paper:
https://www.researchgate.net/publication/3296936_Analysis_of_the_Clustering_Properties_of_Hilbert_Space-filling_Curve
It is a dense paper, so I don't have the exact answer. But if you're motivated you can find it !
PS: you can also find the paper here
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3015176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
distribution associated with a discontinuous function Let $f\colon\mathbb{R}\to\mathbb{R}$ be such that, for every $n\in\mathbb{Z}$, $f$ is differentiable on $\left(n,n+1\right)$ and $n$ is a discontinuity of first kind of $f$. We define
$$T_f(\phi)=\int_{\mathbb{R}}f(x)\phi(x)dx,\quad\text{where }\phi\text{ is a test ... | If $f$ is absolutely continuous on each segment $[n, n + 1]$ when defined to take the values $f(n + 0)$ and $f((n + 1) - 0)$ at the endpoints, integration by parts shows that the distributional derivative is
$$f'(x) + \sum_n (f(n + 0) - f(n - 0)) \delta(x - n),$$
where $f'(x)$ is the ordinary derivative. If the conditi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3015318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Second derivative of a matrix quartic form I need to compute the second derivative of the following quartic expression: $$x^H A^H x x^H A x$$ where is Hermitian. I have tried to compute the first derivative, and if I am not wrong, it should be: $$(A+A^H) x x^H (A+A^H) x$$
But then, I do not know how to proceed to calcu... | Define the scalar variables
$$\eqalign{
&\phi &= x^HAx = (A^Tx^*)^Tx \cr
&\phi^* &= x^HA^Hx = (A^*x^*)^Tx \cr
&\psi &= \phi^*\phi \cr
}$$
Find the gradient of your function $(\psi)$ with respect to $x$, treating $x^*$ as an independent variable.
$$\eqalign{
d\phi &= (A^Tx^*)^T\,dx \cr
d\phi^* &= (A^*x^*)^T\,dx \cr
d\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3015464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$E(X^2|X-Y) E(X^3|X-2Y)$ for Gaussians? For independent gaussians with following the normal distribution with expectation zero and variance one, how do I compute:
$E(X^2|X-2Y), E(X^3|X-2Y)$
I know that $X-2Y$,$X+2Y$ are independent. However, this does not seem to be enough to deduce the result, without a restriction su... | $X=\frac {(X+2Y)+(X-2Y)} 2$. Compute $X^{2}$ and $X^{3}$ in terms of $X+2Y$ and $X-2Y$ from this. [ $$E(X^{2}|X-2Y)=\frac {E((X+2Y)^{2}|X-2Y)+E((X-2Y)^{2}|X-2Y)+2E((X+2Y)E((X-2Y)|X-2Y)} 4$$ $$=[E((X+2Y)^{2} +(X-2Y)^{2}+2(X-2Y)E(X+2Y)] /4=[E((X+2Y)^{2} +(X-2Y)^{2}] /4$$ $$=\frac 5 4+\frac {(X-2Y)^{2}} 4.$$
This answer w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3015559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Recurrence relation/with limit Let $F_{n+1}=F_{n-1}+F_{n-2}$ for $n \in \mathbb{N}$ with $n \geq 2$
$F_0:=0$ and $F_1:=1$.
How to compute
$\lim\limits_{n\to\infty}\frac{F_{n-1}}{F_{n+1}}$?
I tried to use Binet's formula:
$\lim\limits_{n\to\infty}\frac{F_{n-1}}{F_{n+1}}=\lim\limits_{n\to\infty}\frac{\frac{1}{\sqrt{5}}(\... | After$$\lim_{n\to\infty}\frac{F_{n-1}}{F_{n+1}}=\lim_{n\to\infty}\frac{\frac{1}{\sqrt{5}}\left(\xi^{n-1}-\phi^{n-1}\right)}{\frac{1}{\sqrt{5}}\left(\xi^{n+1}-\phi^{n+1}\right)},$$you should have obtained$$\lim_{n\to\infty}\frac{\xi^{n-1}\left(1-\frac{\phi^{n-1}}{\xi^{n-1}}\right)}{\xi^{n+1}\left(1-\frac{\phi^{n+1}}{\xi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3015694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Picking marbles
We have 15 urns each of them having a different number of marbles, from 1 to 15. We start by picking the same number of marbles from each of the urns we choose. We repeat the process until we have picked all marbles. What is the minimum number of days we can finish picking all marbles? Just to clarify ... | It is possible in 4 days:
First day you reduce the number of balls by 8 in urns with at least 8 balls. So now each urn has at most 7 balls.
Second day you reduce the number of balls by 4 in urns with at least 4 balls. So now each urn has at most 3 balls.
Third day you reduce the number of balls by 2 in urns with at lea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3015813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Let $f:[a,b]\rightarrow \mathbb{R}$ a continuous function, with continuous derivate in $[a,b]$ such that: $0<f'(x)<M \ \ \forall \ x\in[a,b]$
Find $c,d\in R$ such that $c\leq f(x)\leq d$.
| If you know a bit of general topology, there is an easy proof of a stronger statement:
Lemma. Let $(X, \tau), (Y, \sigma)$ be topological spaces and let $f \colon (X, \tau) \to (Y, \sigma)$ be continuous. If $C$ is compact in $(X,\tau)$. Then $f[C]$ is compact in $(Y, \sigma)$.
Prove this and combine it with the fact t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Given a book with $100$ pages and a $100$ lemmas, prove that there is some lemma written on the same page as its index
A book consists of 100 pages and contains 100 lemmas and some images. Each lemma is at most one page long and can't be split into two pages (it has to fit in one page). The lemmas are numbered from 1 ... | We claim more generally that a book of $n$ pages and $n$ lemmas numbered $1$ through $n$ has at least one lemma on a page matching its number.
Proof by induction on $n$: The case $n=1$ is obvious. Now suppose the statement is true for some $n$, and suppose we have a book of $n+1$ lemmas and $n+1$ pages. If lemma $n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 7,
"answer_id": 5
} |
Type/codomain of constant function $f(x)=5$ A function like $f(x) = 2x$ can be defined over the reals so its “type signature” or in set theory domain and codomain is $f: \mathbb{R} \rightarrow \mathbb{R}$.
I want to define a function $f(x) = 5$ (or some other constant number) and restrict the codomain/return type to b... | The answer to this really depends on your type theory. The type theories of proof assistants like Coq or Agda will let you express any of the following classes as types:
$$
\{f : \Bbb{R} \to \Bbb{R} \mid \forall x:\Bbb{R}\cdot f(x) = 5\} \\
\{f : \Bbb{R} \to \Bbb{R} \mid \exists c:\Bbb{R}\cdot \forall x:\Bbb{R}\cdot f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\forall n, \, \exists N,x :\lfloor{x^{N}}\rfloor =n \, \land \,\lfloor{x^{N+1}}\rfloor =n+1$ The question is related to the interesting problem raised by following OP.
Notably I'm trying to prove the following fact
For any $n\in \mathbb{N} \quad \exists x\in \mathbb{R} \quad x>1$ and $\exists N\in \mathbb{... | This is long after you received the excellent answer by Ingix(+1), but I'd like to present an arguably more elementary solution.
We'll use $x = n^{1/N}$ as in your Q. We need to argue that $\exists N \in \mathbb{N}:$ \begin{align} n+1 &\le n^{1 + 1/N} < n+2 \\ \iff \frac{\ln (n+1)}{\ln n} &\le 1 + \frac{1}{N} < \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Outer measure and set with full measure I've seen two different definitions for an outer measure $\mu$ on $\mathcal{P}(X)$, where $X$ is a set, obtained from a given probability measure on $X$.
D1 = a set of full outer measure is a subset $A\subseteq X$ such that $\mu(A)=1$.
D2 = a set of full outer measure is a subse... | D1 and D2 are not equivalent in general.
Consider $X=[0,1]$ and $\mu=$ Lebesgue outer measure
By this thread, there is a non-measurable set $V$ with $\mu(V)=1$, so $V$ is a set of full measure in the sense of D1. However, as you pointed out, any set of full measure in the sense of D2 must be measurable, so $V$ is not ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why does there not exist a particular finite set of congruences that forms a partition of the set of integers There is a paragragh that I saw in an article and could not really understand. The article is Covering Systems of Congruences, J. Fabrykowski and T. Smotzer, Mathematics Magazine Vol. 78, No. 3 (Jun., 2005), pp... | The answer is yes. There are several.
This question gives one example: Prove {0 mod 2, 0 mod 3, 1 mod 4, 1 mod 6, 11 mod 12} is a covering system
$0 \mod 2, 0 \mod 3, 1\mod 4, 1\mod 6, 11 \mod 12$.
This paper is all about them https://arxiv.org/pdf/1705.04372.pdf
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Maximum Value of $g(x)=(8+x)^3(8-x)^4$ I think there can not be a maximum value of this, as if I plug $x=$1000, it will increase the value of the function in leaps and bounds. The answer says that the maximum value will occur at $x=-8/7$. What am I missing here?
| We have that
$$g(x)=(8+x)^3(8-x)^4\implies g'(x)=3(8+x)^2(8-x)^4-4(8+x)^3(8-x)^3=0$$
$$3(8+x)^2(8-x)^4=4(8+x)^3(8-x)^3$$
which is true when
*
*$x=\pm 8$
*$3(8-x)=4(8+x) \implies 7x=-8 \implies x=-\frac87$
and the latter is a local maximum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to show a class of structures is not axiomatizable? For example, let $F$ be a field and $L$ be the language of $F$-vector space.
(1) Prove that the class of finite dimensional $F$-vector space is not axiomatizable.
(2) Prove that if $F$ is infinite then the class of infinite dimensional $F$-vector space is not axio... | Hint : for all these questions, you can use the ascending Löwenheim-Skolem theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3016857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Every finite group has a chief series A chief series in a group $G$ is a series of normal subgroups such that
$1=N_0 \triangleleft N_1 \triangleleft ... \triangleleft N_n=G$,
for which each factor $N_{i+1} / N_i$ is a minimal (non-trivial) normal subgroup of $G/N_i$.
I am trying to prove that every finite group has at ... | We'll prove by induction on size of $G$. If $|G|$ is 1 or 2, then $\{0\} \triangleleft G$ is a chief serie. Let $G$ be a group of size $n$ and let $N$ be a minimal normal subgroup of $G$, if $N$ is $\{0\}$, then $N \triangleleft G$ is a chief series. So suppose $N$ is not $\{0\}$, then by induction, $G/N$ has a chief ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3017195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to select distribution? — Binomial, Poisson, ... How do I go about finding which distribution I need to use for my exercise?
I have the following exercise:
Compute the probability that within a group of 5 students exactly two
are born on a Sunday.
What gives me a hint on what probability distribution that is?
| The Guide from the previous answer is helpful. It should help you pinpoint the exact distribution to choose.
Sometimes the problem might not be easily convertible to a distribution. In those cases you can always go back to thinking in terms of basic probabilities.
How many total ways can the student birthday be arran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3017322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Distance an arbitrary point is found along a given vector Say I have a vector in 2D space defined by two points $(x_1, y_1)$ and $(x_2, y_2)$: $$\vec{v}=(x_2 - x_1, y_2 - y_1)$$ I would like to find how far along that vector an arbitrary point $(x_3, y_3)$ is. This very woolly language$^*$, so I've attempted to create ... | You are right in calling this a projection. If $(x_1,y_1)$ is the origin, then you can project ${\bf u} = (x_3,y_3)$ onto $\bf v$ thus:
$${\rm proj}_{\bf v}{\bf u} = \frac{\bf u \cdot v}{\bf v \cdot v}{\bf v}.$$
If $(x_1,y_1)$ is not the origin, then just shift the frame of reference to make it so.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3017416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
LaSalle for time varying systems I am looking for an explanation, why LaSalles theorem is in general not applicable to time varying systems. Can someone provide an example system with
$$
\dot{x}(t) = A(t)x(t) \tag{1}
$$
I.e., why can't LaSalles theorem be used if I have a Lyapunov function $V(t, x)$ for the system $(1)... | Consider the system
$$
\left\{\begin{array}{lll}\tag{1}
\dot x&=&0\\
\dot y&=&0\\
\end{array}\right.
$$
and the function $V(t,x,y)=e^{-t}(x^2+y^2)$. The directional derivative is negative definite:
$$
\dot V= -e^{-t}(x^2+y^2)+e^{-t}(2x\dot x+2y\dot y)=-e^{-t}(x^2+y^2),
$$
but the solutions of (1) do not approach the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3017681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $x \in A$ why does it not follow that $x \in A-B$? I understand that if $x \in A-B$ then $x \in A$ because $x \notin B$ but why doesn't the reverse hold true?
| Because $A\setminus B$ only includes elements in $A$ which are not also in $B$. Put differently, $A$ can be partitioned as
$$
A = (A\setminus B )\cup (A\cap B)
$$
and if $x\in A\cap B$, then $x\in A$ but $x\notin A\setminus B$.
For instance, take $A=B=\{1\}$ for a trivial counterexample. Then $A\setminus B = \emptyset... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3017931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $\lim_{r\to s} \|D_rf-D_sf\|_p = 0,\,r,\,s > 0$ Full Question: Let $D_r$ be the dilation operator $D_rf(x) = f(rx)$ on $L^p(\mathbb{R}^d),\, 1 \leq p < \infty$. Show that $\lim_{r\to s} \|D_rf-D_sf\|_p = 0,\,r,\,s > 0$
I was told to use $\int f(rx)d\lambda^d(x) = |r|^{-d}\int f(x)d\lambda^d(x)$.
So I was thin... | Given $\epsilon >0$ there exists $g \in C_c(\mathbb R^{d})$ such that $\|f-g\|<\epsilon$. DCT tells you that the result is true with $g$ in place of $f$. Now $\|f(rx)-f(x)\| \leq \|g(rx)-g(x)\|+\|f(x)-g(x)\|+\|f(rx)-g(rx)\|$ and $\|f(rx)-g(rx)\|=r^{-d} \|f(x)-g(x)\|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Doubt in passing in the Riemann mapping theorem
I have a question, maybe silly in the passage marked in red. I understood everything up to this part. Why does $H'(0) > 0$ imply $e^{i \theta} = 1$? Is the Schwarz Lemma being used?
| From $H=F \circ G^{-1}$ we get with the chain rule and the rule for the derivative of $G^{-1}$ that
$$ H'(0)= \frac{F'(z_0)}{G'(z_0)}.$$
Since $F'(z_0), G'(z_0)>0$, it follows that $H'(0)>0$.
From $H(z)=e^{i \theta}z$, we get $H'(0)=e^{i \theta}.$
Furthermore we have: $e^{i \theta}>0 \iff e^{i \theta}=1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does this limit $lim_{n\to\infty}\sum_{i=0}^n 1/n \sqrt{1 - i^2/n^2}$ converge to $\pi/4$? While trying to find an approximate area of a quarter of a circle by splicing it into small rectangles and summing their areas I've reached a point where I have this formula:
$$\sum_{i=0}^n 1/n \sqrt{1 - i^2/n^2}$$
Writing quick ... | The sum is nothing but a Riemann sum for $\int_0^{1}\sqrt{1-t^{2}}\, dt$. You can evaluate this by making the substitution $t=\sin\, \theta$ and using the formula $2 \cos ^{2}\, \theta =1+\cos\, (2\theta)$ and you will get $\pi /4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Sigma-algebra for countable sample space Let $E=\{\{\omega \} : \omega \in \Omega \}$.
$\sigma ( E) = \{ A \subseteq \Omega : A \ is \ countable \ or \ A^c \ is \ countable \} $ is a $\sigma$-algebra generated by the set $E$ i.e. the smallest $\sigma$-algebra containing $E$ (already proved).
Prove:
$\sigma (E)$ is ... | If $\Omega$ is not countable then it can be written as disjoint union of two uncountable subsets $\Omega_1,\Omega_2$.
See here for a proof of that.
So then $\Omega_1,\Omega_2\notin \sigma(E)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Trace of symmetric matrix equals sum eigenvalues I need to show that if $\mathbf{S}$ is symmetric, then it's trace sums to the sum of the eigenvalues. But I don't know how to show this. Can anybody give me a hint?
P.S. Shame on my google skills, buy I really can't find any pages on this specific issue. Not with the ass... | If $S$ is a symmetric matrix then $S$ has a spectral decomposition as $S=PDP'$ where $D$ is the diagonal matrix consisting the eigenvalues of $S$ and $P$ is orthogonal. Then $tr(S)=tr(PDP')=tr(DP'P)=tr(D)=\sum \text{eigen values of } S.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
What is derivative of $\sin ax$ where $a$ is a constant? What is the derivative of $\sin a x$ where $a$ is a constant.
Actually, I'm studying Physics and not so well-versed with calculus. So, I have studied the basic rules of calculus but am stuck here.
I somewhat know about the product rule but don't get what to do i... | Derivative of $\sin(ax) = a \cos(ax)$ by Chain Rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove by induction $7^n$ is an odd number for every natural number n I have proved the base case already of n=1 where $7 = 2p+1$
Then I assumed $n=k$ for $7^k = 2p+1$ for $k \in N$ and $p \in N$
To prove $7^{k+1} = 2p+1$ I have these steps so far:
$7(7^k) = 2p+1$
$7(2p+1) = 2p+1$
$14p+7 = 2p+1$
I am not sure how to p... | Just rewrite your sum as
$$14p+7=2(7p+3)+1$$
Also, it's not a good idea to use $p$ for all of your steps. If $7^k=2p+1$, then you need to show that $7^{k+1}=2q+1$ for some integer $q$. Otherwise the equality is not actually equal since you are using the same expression as your desired form for different valued expres... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Normal operator in Hilbert Complex space share an eigenvalue. Everyone, I get stuck in an exercise of Functional Analysis.
Let $T \in B(H)$ (H a complex Hilbert space) and $T^*$ adjoint of $T$. Supose $T$ is a normal operator.
1) Prove that $Ker(T)= Ker(T^*) = R(T)^\perp$ - I've finished this.
2) Using previous proof. ... | $T$ is normal means $T^*T=TT^*$, which is equivalent to
$$
\|Tx\| = \|T^*x\|,\;\;\; x\in H.
$$
So $\mathcal{N}(T)=\mathcal{N}(T^*)$ follows. The sum of normal operators is normal, and any scalar times a normal operator is normal. And the identity $I$ is normal. So, if $T$ is normal, then so is $\alpha I-T$ for any ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If a first derivative doesn't exist at a certain point, is it not a critical point? Say $f'(x)$ of a function was $(x+2) \over (x+3)$. If $x = -2$ then $f'(x)$ = 0, which means that at this point, there would be a local min or max. But what if $x = -3$? It doesn't exist for $f'(x)$, so do we just ignore it? State it DN... | I think the easiest explanation at a basic calculus level is to simply say that no, it is not a critical point. Hence if we want to do problems regarding optimization for a function which does not have its derivative (or the function itself) defined everywhere, then you would want to consider the "shape" of that functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Is the set $\{|f(0)|: \int_{0}^{1}|f(t)|dt\le1\}$ bounded? Let $x_0 \in [0,1]$ and define $T:C[0,1] \rightarrow \mathbb{R}$ by $T_{x_0}(f)=f(x_0)$. Let $||\cdot||_1$ be a norm on $C[0,1]$. Is $T_0$ bounded or not? That is, is the set
$$
\left\{|T_{0}(f)|:||f||_1 \leq 1\right\}=\{|f(0)|:||f||_1 \leq 1,f \in C[0,1]\}
$$... | For every $a>0$, the function
$$f_a(x)=\frac{2 a e^{-a^2 x^2}}{\sqrt{\pi} \textrm{erf}(a)}$$ with the error function $\textrm{erf}(a)=\frac{2}{\pi}\int_0^a e^{-t^2}dt$ is in the set $\{|f(0)|: \int_{0}^{1}|f(t)|dt=1\}$ and evaluates to $f_a(0)=\frac{2 a }{\sqrt{\pi} \textrm{erf}(a)}$. Because $\textrm{erf}(a) \rightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Proof that $\frac{1+x^2}{n^2} \geq 1-e^{-x^2/n^2}$ for all $x,n \in \mathbb{R}$ I'm looking for a simple proof that $\frac{1+x^2}{n^2} \geq 1-e^{-x^2/n^2}$ for all $x,n \in \mathbb{R}$.
My first attempt was to express the exponential as a Taylor series:
$$\frac{1+x^2}{n^2} \geq \frac{x^2}{n^2}-\frac{1}{2!}\frac{x^4}{n^... | Set $y=x^2/n^2$. Then you want to show that
$$
\frac{1}{n^2}+y\ge 1-e^{-y}
$$
Note that $y\ge0$. A standard process is to consider
$$
f(y)=\frac{1}{n^2}+y-1+e^{-y}
$$
and note that $f(0)=1/n^2>0$. Also
$$
f'(y)=1-e^{-y}=\frac{e^y-1}{e^y}>0
$$
for $y>0$. Therefore the function $f$ is strictly increasing over $[0,\infty)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Is there a surface on which a hexagon can have all right angles? So I was watching a video that features astronomer and topologist Cliff Stoll talking about how figures that aren't quadrilaterals can have all their angles equal 90 degrees on different surfaces. For example, on a sphere, you can create a triangle that h... | You would need a surface of negative curvature.
It is best to use a hyperbolic plane for this, where you can easily fit any regular n-gon with given angles as long as the sum of its external angles is greater than 360 degrees. The problem is that the hyperbolic plane does not fit in Euclidean space.
The pseudosphere i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 3,
"answer_id": 0
} |
Show that $\int_{0}^{2\pi} \cos^2(x) dx = \int_{0}^{2\pi} \sin^2(x) dx$ I'm trying to follow the argument in the image below, which aims to show that: $\int_0^{2\pi} \cos^2(x) dx = \int_0^{2\pi} \sin^2(x) dx$.
I believe this on an intuitive level, and I understand that it uses the periodicity of the sine and cosine fun... | There are two ways to check the validity of the claim without changing the variable $x$:
1) Consider their difference:
$$\int_0^{2\pi} \cos^2(x) dx - \int_0^{2\pi} \sin^2(x) dx = \int_0^{2\pi} \cos 2xdx=-\frac12\sin 2x|_0^{2\pi}=0.$$
2) Use the half angle formula:
$$\begin{align}\int_0^{2\pi} \cos^2(x) dx &= \int_0^{2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proof explanation: Calculate a spectrum of a pair of commuting operators According to the following paper of Taylor:
J. L. Taylor, A joint spectrum for several commuting operators, J. Functional Anal. 6(1970), 172-191.
we have
Let $A= \begin{pmatrix}0&1\\1&0\end{pmatrix}$ and $I= \begin{pmatrix}1&0\\0&1\end{pmatrix... | Chose a basis so that $A\equiv\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. Now note that
$$(I-1)\oplus (A-1)\equiv0\oplus\begin{pmatrix}0&0\\0&-2\end{pmatrix},\qquad (I-1)\oplus (A+1)\equiv0\oplus\begin{pmatrix}2&0\\0&0\end{pmatrix}$$
both fail to be injective maps $\Bbb C^2\to\Bbb C^2\oplus \Bbb C^2$. For that reason you a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Given the subspace of $L^2$ made by constant functions, characterize its orthogonal complement
Let $X=\{v\in L^2([-1,1]):v \text{ is constant a.e.}\}$ be a subspace of $L^2([-1,1])$, characterize $X^{\perp}$.
I don't really know what characterize means, I know that: $X^{\perp}=\{f\in L^2([-1,1]):\langle f,v \rangle=0... | While the word "characterize" is somewhat ambiguous, since the characterization of a set may be given in many ways (enumeration of elements, "in words", via those satisfying a given proposition, or a union/intersection of some described sets), I am sure whoever has asked this question(person/text) will have expected th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3019926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$f$ is T periodic and $f(x) + f'(x) \ge 0 \Rightarrow f(x) \ge 0$
Let $f: \Bbb R \to \Bbb R$ be a function such that $f'(x)$ exists and is continuous over $\Bbb R$. Moreover, let there be a $T > 0$ such that $f(x + T) = f(x)$ for all $x \in \Bbb R$ and let $f(x) + f'(x)\ge 0$ for all $x \in \Bbb R$.
Show that $f(x) \g... | $ f \ne 0$
$f(x)=0 \Rightarrow f'(x) \ge 0$, hence $f$ can cross the X-axis at most once. periodicity means that it cannot cross at all.
if $f$ is non-positive then $f' \ge 0$ so $f$ is monotone increasing. again this contradicts periodicity
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3020024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 6
} |
Addition of Logarithmic Equation
According to the basic rules of $\log$, I'm solving both $\log$ terms as for first: base is $3$, $N$ is $9$ so exponent is calculated as $2$, and same for other term. But I'm confused with this '$x$'. Adding both terms $\log$ according to my logic will result in $4$. But I know I'm doi... | HINT
Recall that
$$\log_3 (9\cdot x)=\log_3 9+\log_3 x$$
then what about $\log_2 (4\cdot x)$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3020153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Identities similar to $\arctan(x)+\arctan(1/x)=\pi/2$ The $\arctan(x)+\arctan(1/x)=\pi/2$ (for $x>0$) identity can be solved by taking the derivative of the left hand side, showing it is $0$, and then plugging in, say, $x=1$ to get its constant value $\pi/2$.
Are there any other (nontrivial) identities which can be sol... | The proof for $x>0$ can be obtained as follows
*
*let $\alpha=\arctan x \quad x\in\left(0,\frac{\pi}2\right)$
then
$$\tan\left(\frac{\pi}2-\alpha\right)=\frac1{\tan \alpha}=\frac1x \implies \frac{\pi}2-\alpha =\arctan \frac1x$$
Another similar identity is
$$\arcsin x + \arccos x=\frac{\pi}2 \quad \forall x\in[-1,1]$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3020249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 4
} |
How to convert a straight line into polar coordinates? The straight line $y=mx+b$ can be expressed in polar coordinates as:
$$\rho=x\cos(\theta) + y\sin(\theta)$$
Where $(\rho,\theta)$ defines a vector from the origin to the nearest point on the line. Thus the Hough transform of a straight line in $x-y$ space is a poin... | $$\frac{\left|c\right|}{\sqrt{a^2+b^2}}$$
Gives you the normal distance from the origin to the straight line which is $\rho$
So if you multiply $$ax+by=c$$
by $\rho/c$ you get
$$mx+ny=\rho$$ where
$m=\cos(\theta)$ and $n=\sin(\theta)$ and getting $\theta$ given this should be easy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3020367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
What's the difference between $f(x)=\sqrt{x^2+9}$ and $k(x^2+9)=\sqrt{x^2+9}$? Let's say we 've got a function $f(x)=\sqrt{x^2+9}$, which is a composite function. $f(x)=\sqrt{g(x)}$ and $g(x)=x^2+9$.
When we have a function like $h(x)=x$, we are allowed to set $x$ to $x+9$ and have $h(x+9)=x+9$.
So why do we need $g(x)... | A function is a mapping from elements of a domain set to elements of a range set (also called the "codomain"). Let's consider your example of $k(x) = \sqrt{x}$. This actually is an incomplete definition of a function; you also need to specify the domain and range. So suppose $k$ takes nonnegative real numbers to non... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3020494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Why does $z^n-1=0$ have at max n solutions? $z\in\mathbb{C}$ I know that there is a Theorem which says that a Polynom of Degree n has at most n Solutions, however we have not proved it yet in our class. Is there Maybe another explaination for this Special case?
| There's an explanation if you represent them in polar co-ordinates and consider that multiplying two complex numbers involves adding their arguments (ie angles) and multiplying their magnitudes (distances from the origin). It turns out they need to have $1$ as their magnitude and be multiples of$\frac{360°}{n}$ apart o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3020762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
Prove that $G$ is an open interval for two-valued continuous function $f$
Suppose $G\subset \mathbb R$ is a non-empty open set such that the function $f:G \rightarrow \{0,1\}$ is a two-valued function and is continuous. Show that any two-valued function on $G$ is a constant if and only if the set $G$ is an open interv... | Theorem: A metric space $X$ is connected if and only if any continuous function $f:X\to \{0,1\}$ is constant.
Proof:Suppose $X$ is connected and $f:X\to \{0,1\}$ is continuous. If $f$ is not a constant function, then $f$ is onto. Let $A=f^{-1}(0)$ and $B=f^{-1}(1)$. Then $A\cup B=X$,and $A,B\neq \emptyset$. Also note ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3020937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to show that if $f^2(x)$ is uniformly continous function then f is uniformly continous $f:R\to [0,\infty)$ is function such that $f^2(x)$ is uniformly continuous on R then I have to show that f is uniformly continuous ?
My attempt :
$|f^2(x)-f^2(y)|<\epsilon$ for $|x-y|<\delta$
then
$|f(x)-f(y)<\epsilon/|f(x)+f(y... | I think the result holds for all $f\in C(\mathbb{R})$. You could use mean value theorem: if $f(x)f(y)<0$ then there is some $z$ between $x$ and $y$ such that $f(z)=0.$ The details is omitted.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3021073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$1,2, \ldots, n$ are permuted. None of the numbers $1,2,3$ are adjacent and $n>4$. The numbers $1, 2, \ldots, n$ are permuted. How many different permutations exist such that none of the numbers $1, 2, 3$ are adjacent when $n>4$?
Solution:
$4,5, \ldots, n$ can be shuffled in $(n-3)!$ ways and $3!$ ways to arrange $1,... | We can count the cases
*
*the permutations are n!
from which we need to eliminate the ways
*
*we can arrange $1,2,3$ adjacent that is: $3!(n-2)(n-3)!$
*we can arrange exactly a pair adjacent that is: $3!(n-3)![(n-3)(n-4)+2(n-3)]=3!(n-3)!(n-3)(n-2)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3021175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Proper condition on the dihedral group Is there a theream which is a condition on $n\in\mathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups?
After spending some time figuring a condition I tried to find some similar thread but didn't find any.
| Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, \forall n \ge3$.
By your question, $D_n \le D_n$. So true for $\forall n\ge3$.
But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$.
Hence when $n$ is composite we get a required subgroup.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3021319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A finite field $F$ such that $F[\sqrt{2}]$ and $F[\sqrt{3}]$ are not isomorphic (as fields) Assuming $F$ is a finite field such that $F[\sqrt{2}]$ and $F[\sqrt{3}]$ are both fields, I am trying to prove that they must both be isomorphic. Or is there a counterexample? Is there a counterexample where $F$ has prime order?... | If neither $2$ nor $3$ are squares in the finite field $F$, then both $F[\sqrt{2}]$ and $F[\sqrt{3}]$ are extensions of $F$ having degree $2$, so both have $|F|^2$ elements. Hence they're isomorphic: if $p$ is a prime, then any two fields of cardinality $p^n$ ($n>0$) are isomorphic, because both are the splitting field... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3021494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Obtain the leading order uniform approximation of the solution Obtain the leading order uniform approximation of the solution to $ \epsilon y′′-x^2y′-y=0$.
The boundary conditions are $y(0)=y(1)=1$.
Since $a(x)<0$ the boundary layer is at $x=1$.
The outer solution will be of the form $y(x; \epsilon) = y_0(x) + \epsilon... | This is a complicated problem, but if you're careful it does work out. Firstly, let's write out the equation assuming there is a boundary layer or width $\epsilon^\alpha$ at $x_0$, so let $X=(x-x_0)/\epsilon^\alpha$ and $Y(X)=y(x)$. We get
$$ \epsilon^{1-2\alpha}Y_{XX}-\epsilon^{-\alpha}(\epsilon^\alpha X+x_0)^2Y_X-Y=0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3021588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Values of $x$ satisfying $\sin x\cdot\cos^3 x>\sin^3x\cdot\cos x$
For what values of $x$ between $0$ and $\pi$ does the inequality $\sin x\cdot\cos^3 x>\sin^3x\cdot\cos x$ hold?
My Attempt
$$
\sin x\cos x\cdot(\cos^2x-\sin^2x)=\frac{1}{2}\cdot\sin2x\cdot\cos2x=\frac{1}{4}\cdot\sin4x>0\implies\sin4x>0\\
x\in(0,\pi)\im... | As an alternative for a full solution we can consider two cases
*
*$\sin x \cos x >0$ that is $x\in(0,\pi/2)\cup(\pi,3\pi/2)$
$$\sin x\cdot\cos^3 x>\sin^3x\cdot\cos x \iff\cos^2x>\sin^2x \iff2\sin^2 x<1$$
$$-\frac{\sqrt 2}2<\sin x<0 \,\land\, 0<\sin x<\frac{\sqrt 2}2 \iff \color{red}{x\in(0,\pi/4)}\cup(\pi,5\pi/4)$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3021679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
About closed graph of an unbounded operator I am working on problems related to the closed graph of an unbounded operator. There is a proposition:
Let $X,Y$ be Banach spaces and let $A:\mathrm{dom}(A)\to Y$ be linear and defined on a linear subspace $\mathrm{dom}(A)\subset X$. Prove that the graph of $A$ is a closed s... | Let $(x_n, Ax_n)$ be a Cauchy sequence in $\operatorname{graph}(A)$. Then, by definition of the graph norm, $(x_n)_n$ is a Cauchy sequence in $\operatorname{dom}(A)$.
Since $\operatorname{dom}(A)$ is a Banach space w.r.t. the graph norm, $(x_n)_n$ converges to some $x \in \operatorname{dom}(A)$ w.r.t. the graph norm. T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3021781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving the convergence of the sequence defined by $x_1=3$ and $x_{n+1}=\frac{1}{4-x_n}$ Consider the sequence defined by
$$x_1=3 \quad\text{and}\quad x_{n+1}=\dfrac{1}{4-x_n}$$
I can calculate limit by assuming limit exist and solving quadratic equation, but I first wanted to give existence of limit.
I tried to show ... | It can be approached in a graphical manner:
*
*Draw the graph of $y = \frac{1}{4-x}$ to scale while marking the essentials.
*Asymptote at $x=4$; Value at $x = 3$ is $1$.
*Comparing it to the previous value of the sequence would require the plot of $y=x$ on the same axes.
*Mark the intersection as $x=2-\sqrt3$ wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3022296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Four dimensional cross product of THREE vectors There are many MSE posts about how to define a cross product in $\mathbb{R^4}$. It is impossible to define a cross product of two vectors in $\mathbb{R^4}$, since there are infinitely many directions perpendicular to those two vectors, and we don't know which direction to... | The short answer is yes. One way is to take the formal determinant
$$\left|\begin{matrix}e_1&e_2&e_3&e_4\\
a_1&a_2&a_3&a_4\\
b_1&b_2&b_3&b_4\\
c_1&c_2&c_3&c_4\\
\end{matrix}\right|$$
where $e_1,\ldots,e_4$ are the standard unit vectors, and $a=\sum a_ie_i$
etc., are the three vectors.
Or you can rephrase this in terms
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3022394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Lagrange notation: $f^{(0)}(x)$? Using Lagrange notation, is $f^{(0)}(x)=f(x)$? Is this standard notation, or would one have to define $f^{(0)}(x)=f(x)$ first, before using it?
Aside: the context of the question is whether to include the first term within the summation when expressing the Taylor series and hence start... | A definition of something is always good before using it, as happend in the wikipedia article:
The derivative of order zero of $f$ is defined to be $f$ itself.
So you can start summation at $n=0$ in Taylor series after refering to this definition. But you can put your mind at rest. Most mathematicians would expect ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3022524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Lower bound bound for the Ramsey number $R_k(3,3,...,3)$ The question is:
Show that $R_k(3,3,...,3)\geq 2^k+1$. The upper bound part of this problem has been proved in the link How to obtain lower and upper bounds for Ramsey number $R_k (3,3,\dots,3)$, however the lower bound is not clearly shown procedurally because I... | Following the hint in the link: let $n = 2^k$ and consider the complete graph on the set $\{0,1\}^k$ and colour the edge between $(x_1,\ldots,x_k) \neq (y_1,\ldots y_k)$ by the colour $c = \min(i: x_i \neq y_i)\in \{1,\ldots,k\}$.
It's clear we cannot have a triangle of a fixed colour $c$: suppose
$ (x_1,\ldots,x_k),... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3022677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Minimum value of the given function
Minimum value of $$\sqrt{2x^2+2x+1} +\sqrt{2x^2-10x+13}$$ is $\sqrt{\alpha}$ then $\alpha$ is________ .
Attempt
Wrote the equation as sum of distances from point A(-1,2) and point B(2,5) as
$$\sqrt{(x+1)^2 +(x+2-2)^2} +\sqrt{(x-2)^2 + (x+2-5)^2}$$
Hence the point lies on the line ... | There are a number of ways to do this, including brute force and calculus, but since you already found out the rather nice geometric interpretation as the sum of the distances from the given points, let's do that.
A few things will come in handy here.
*
*The line on which $A$ and $B$ lie is $y=x+3$, which is parall... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3022822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Finding the minimal number of members I've been working on the following problem
For every issue in the Blue's association, a commission with 10 members (belonging the Blue's) is formed in order to solve the problem. The only condition is
There can't be two commissions having more than one member in common
The Blue'... | Let $i$ denote each member of Blue's association and assume that there are $N$ members in total, that is, $i=1,2,\cdots, N.$ And let $j,k=1,2,\ldots, 40$ denote each of 40 commission. We will show that $N$ is at least $82$.
Consider the set
$$
S=\{(i,j,k)\;|\;1\leq i\leq N, 1\leq j<k\leq 40, i\text{ belongs to }j,k\te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 3
} |
What are the complex roots of $x^3-1$? What are the complex roots of $x^3-1$?
Work I've done so far:
I've set $x = a + bi$. Since $x^3-1=0$, I set $x^3 = (a+bi)^3=1$.
This gives me the following:
(1) $(-ab^2 + a^3) + (2ab^2 + 2a^2b + a^2b - b^3)i$
Which means that I set $(-ab^2 + a^3) = a(a^2-b^2)= 1$ which is also equ... | Hint:
The simpler way is to factorize:
$$
x^3-1=(x-1)(x^2+x+1)
$$
can you find all the roots?
Anyway, your algebra is wrong because:
$$
(a+ib)^3=a^3+3a^2(ib)+3a(ib)^2+(ib)^3=a^3-3ab^2+i(3a^2b-b^3)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Right triangle geometry problem Right triangle $\Delta ABC$ ($\angle ACB=90°$). The following is constructed: from point $C$ altitude $CD$, angle bisector $CL$ of $\angle ACB$, angle bisector $DK$ of $\angle ADC$, angle bisector $DN$ of $\angle BDC$.
$D, L$ lie on $AB$, $K$ lies on $AC$, $N$ lies on $BC$.
Prove that $C... | First, since $CD \perp AB$ and $DK$ and $DN$ are angle bisectors to the right angles
$\angle \, ADC$ and $\angle \, BDC$, then $$\angle \, KDN = \angle \, KDC + \angle \, NDC = 45^{\circ} + 45^{\circ} = 90^{\circ}$$
However, $\angle \, KCN = 90^{\circ}$ so the quadrilateral $CKDN$ is inscribed in a circle.
Next, prove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Showing that $\int_0^\pi\frac{\cos n\theta}{\cos\theta-\cos\theta_0}d\theta=\pi\frac{\sin n\theta_0}{\sin\theta_0}$ I am reading Debnath & Bhatta "Integral Transforms and Their Applications, 3rd". They cited one example from Zayed "Handbook of Function and Generalized Function Transformations" and stated an integral (E... | Are those integrals even well-defined? Let $\theta_0$ be such that $\cos(\theta_0)=1/2$. For instance let $\theta_0=\frac{\pi}{3}$. Take $n=1$. Now
$$\int_0^\pi\frac{\cos(n\theta_0)}{\cos\theta-\cos(\theta_0)}\;d\theta=\int_0^\pi\frac{1/2}{\cos\theta-1/2}\;d\theta.$$
This integral is actually an improper one, as $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is there a name for this relation: for all $x$ there is $y$ such that $xRy$, and for all $x,y,z$, if $xRy$ and $xRz$, then $y=z$? Suppose for all $x$ there is $y$ such that $xRy$, and for all $x,y,z$, if $xRy$ and $xRz$, then $y=z$.
Does there exist such a binary relation $R$ on some set such that the above properties ... | One set of examples is functions from $A$ to $B$. For this let $xRy$ mean that $(x,y) \in f$ [using the "function as ordered pairs" formulation]. Then your first requirement expresses that $f$ produces an output $f(x)$ for each $x$ in $A,$ while your second expresses that $f$ is a function.
There may be more examples... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
derivative of inverse matrix by itself Let $A$ be a matrix, supposedly $k\times k$ matrix.
I know that
$$\frac{\partial A^{-1}}{\partial A} = -A^{-2} $$
I do not know how I am supposed to obtain the following results using this fact. I want to know the step of
$$\frac{\partial a^\top A^{-1} b}{\partial A} = -(A^\top)... | Start with the defining equation for the matrix inverse and find its differential.
$$\eqalign{
I &= A^{-1}A \\
0 &= dA^{-1}\,A + A^{-1}\,dA \\
dA^{-1} &= -A^{-1}\,dA\,A^{-1} \\
}$$
Next note the gradient of a matrix with respect to itself.
$$
{\mathcal H}_{ijkl}
= \frac{\partial A_{ij}}{\partial A_{kl}}
= \delta_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Prove $\sum_{i=1}^n a_i$ = $\sum_{i=2}^{n+1} a_{i-1}$ Given $\sum_{i=1}^n a_i$ = $\sum_{i=2}^{n+1} a_{i-1}$
How would you show this true for all n ∈ N and $a_1, a_2, . . . , a_n$ ∈ R?
I know it is obviously true because i would just use a substitution like i=j-1 then summing j-1 from 2 to n+1 gives the same result but ... | Prove that
$F(n):= \sum_{i=1}^{n}a_i-\sum_{i=2}^{n+1}a_{i-1}= 0$ , for $n \in \mathbb{Z^+}$ by induction.
1) $n=1$√.
2) Hypothesis $F(n)=0$.
3) Step for $n+1$.
$F(n+1)=$
$\sum_{i=1}^{n+1}a_i - \sum_{i=2}^{n+2}a_{i-1}=$
$\sum_{i=1}^{n}a_i +a_{n+1}$
$- \sum_{i=2}^{n+1}a_{i-1}- a_{n+1}=$
$F(n)+(a_{n+1}-a_{n+1})=0$,
since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Variation of distributing $k$ balls into $n$ distinguishable boxes I am interested in understanding a variation problem of distributing balls into boxes. It seems to be not any of the individual case mentioned in twelvefold way classification.
The problem is described as follows:
In total there are $K$ balls, there ar... | This is a straightforward application of the principle of inclusion-exclusion. First, count up all the ways to put all the balls into boxes, ignoring the condition that no box can be empty. This is simply
$$
\prod_{i=1}^I \binom{N}{n_i}
$$
Next, you have to subtract out the "bad" cases where some box is empty. For each... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3023944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Why is $\lim_{x\to -\infty}\sqrt{x^2+5x+3}+x = \lim_{x\to -\infty}(-x)+x=\lim_{x\to -\infty}0 = 0$ not correct? $\lim_{x\to -\infty}\sqrt{x^2+5x+3}+x = \lim_{x\to -\infty}(-x)+x=\lim_{x\to -\infty}0 = 0$
Apparently, the 2nd step is illegal here. Probably because for $x=-\infty$ I'd get $(+\infty-\infty)$ which is not p... | The first step is illegal, not the second. This is because $\sqrt{x^2 + 5x + 3} \not\equiv -x$.
Also addressing your comment:
The idea behind $\lim_{x \to -\infty} \sqrt{x^2 + 5x + 3} = \dots = \lim_{x \to -\infty} −x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $\lim_{x \to -\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 2
} |
Basic division and multiplication We know that 2 x 2 /2 can be solved by removing '2' from the denominator and numerator, we can't do the same if the operation was addition, These "rules" have been established based on the understanding of these operations and how these interact with each other.
Up until now, I've just... |
$\dfrac {2 \times 2}{ 2 }$ can be solved by removing $2$ from the numerator and the denominator and it gives the same result as $\dfrac 4 2$, how? Is it an axiom?
No, it is not an axiom. But, as for every rule for operating with numbers, it is obviously justified by axioms.
Regarding the rationals (i.e. fractions) th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Finding magnitude of a complex number
$$z = \dfrac{2+2i}{4-2i}$$
$$|z| = ? $$
My attempt:
$$\dfrac{(2+2i)(4+2i)}{(4-2i)(4+2i)} = \dfrac{4+12i}{20} = \dfrac{4}{20}+\dfrac{12}{20}i = \dfrac{1}{5} + \dfrac{3}{5}i$$
Now taking its magnitude and we have that
$$|z| = \sqrt{\biggr (\dfrac 1 5 \biggr ) ^2 +\biggr (\dfrac 3 5... | Yes, you are. You can do it also like this: $$\Big|{2+2i\over 4-2i}\Big|=\Big|{1+i\over 2-i}\Big|={|1+i|\over |2-i|}= {\sqrt{2}\over \sqrt{5}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Triple Integrals Help Suppose $E$ is the sphere $x^2 + y^2 + z^2 = 1$ whose density at each point is proportional to the distance from the origin. Find an expression for the mass of $E$ as a Triple Integral and explain why it's difficult to compute
I believe it is difficult to compute because the region is a sphere and... | For an arbitrary density $\rho$, the mass is expressible as a triple integral in spherical polar coordinates, viz. $\int_0^{2\pi}d\phi\int_0^\pi d\theta\sin\theta\int_0^1 \rho(r,\,\theta,\,\phi)r^2 dr$. If $\rho$ only depends on $r$ we can first integrate out the angles, giving $4\pi\int_0^1\rho(r)r^2 dr$. The choice $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Fourth point of intersection of two conics Five points in general position define a unique conic section. Let $Q_1$ be a conic through points $A,B,C,E_1,F_1$ and likewise $Q_2$ through $A,B,C,E_2,F_2$. Two conics (over an algebraically closed field) generally intersect in four points, so these two will have a fourth po... | Let the conics be $ABCE^+F^+$ and $ABCE^-F^-$, and let the fourth point of intersection be $D$. Write $D$, and the $E$s and $F$s, using rampantly-reciprocated barycentric coordinates
$$D = \left(\;\frac1{a} : \frac1{b} : \frac1{c}\;\right) \qquad E^\pm = \left(\;\frac{1}{a_\circ^\pm} : \frac{1}{b_\circ^\pm} : \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Let $f'$ be continuous. If $f$ is uniformly continuous then $f'$ is uniformly continuous? Is this, in general, true? I need this in order to prove that a polynomial is uniformly continuous in $\mathbb{R}$ if and only if $p(x)$ is of degree less or equal to 1. If $p(x)$ is of degree less or equal to 1, is easy to prove ... | This is a modification of the example by Guido A wherein the function is defined on the whole line: take $f(x)=\sqrt {\frac {\pi} 2 +\tan^{-1}x}$. [Note that $f'$ is unbounded].
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
decomposition group and inertia group, the minimal polynomial,surjectivity of the map $D_{M/P}\rightarrow Gal$ Can anyone explain the underlined sentence?
For notation, A:Dedekind domain, K=Frac(A), L/K:Galois extension, B:The integral closure of A in L, M:A maximal ideal of B, P:The intersection of M and A (hence the... | Let me expand on the highlighted part:
$g(y)$ is the min. polynomial of $\overline{\alpha}$ over $A/P$, so it has to divide the polynomial $\overline{f}(y)=f(y) \,\mathrm{mod}\, P$, since $\overline{\alpha}$ is a root of $\overline{f}(y)$ (and $\overline{f}(y)$ is nonzero, take $f(y)$ monic). From this and the expressi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3025036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Sylow $3$-subgroups of an order $180$ group My task is to show that if $G$ is a group with order $180 =2^23^25$ with $36$ Sylow $5$-subgroups, then there are two Sylow $3$-subgroups $H$ and $K$ such that $|H \cap K| = 3$. The number of Sylow $3$ groups $n_3$ is either $1$, $4$, or $10$ since $ n_3 \equiv 1 \pmod 3$ an... | Since no group of order 180 has $n_5=36$, any conclusion follows vacuously. The normal 5-complement would give a nontrivial normal subgroup of an $A_5$ composition factor, (the only candidate for a non-abelian composition factor) so if it existed the group would have to be solvable. But then the counting portion of P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3025215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
If A complement is the union of two separated sets, prove that the union of those separated sets with A is connected. Let $A$ be a connected subset of a connected metric space $(X,d)$.
Assume $A^{c}$ is the union of two separated sets $B$ and $C$.
Prove that $A \cup B$ and $A \cup C$ are connected.
Attempt
Proving $A \... | Isn't the complement of $A \cup B$ just C? And I think it's easy to see that C is open.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3025311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Words of weight 5 in in Ternary Golay Code I'm not really good at doing this type of exercices. But I'd like to know how to prove that ther are 132 words of weight 5 in the Ternary Golay Code. I am not allowed to use the weight enumerator.
I tried to ask the same question in the global code $GF(3)^{11}$ but not succeed... | I assume that you are expected to answer this question using only the (big) piece of information that the ternary Golay code $G$ is a perfect code with covering radius $\rho=2$.
An attack (filling in the details as the OP solved the problem themself):
*
*The number of vectors of weight three in the space $GF(3)^{11}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3025523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.