Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
"Ordering" of Complex Plane I have heard that the Complex Numbers do not form an ordered field, so you can't say that one number is larger or smaller than another. I've always been fine with this, but I recently wondered what happens if one describes size as the distance along a space-filling curve? If the real numbe... | You can definitely put an order $\prec$ on the complex numbers. In fact, the order can be a well-order, by the well ordering theorem. The question is whether such an order "is useful."
Here are some things we probably want in the order $\prec$:
*
*if $0 \prec \alpha, \beta$, then $0 \prec \alpha \beta$.
*if $\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1620004",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "8",
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Projection and positive element in C$^*$-algebras Let $A$ be a $C^*$-algebra, $p\in A$ a projection.
Assume that $a$ is a element in $ \text{Ball}(A_+)$ such that $a\leq p$.
Q: May I say $ap=pa$? Why?
| You have
$$
0\leq(1-p)a(1-p)\leq(1-p)p(1-p)=0.
$$
Thus $$0=(1-p)a(1-p)=(a^{1/2}(1-p))^*(a^{1/2}(1-p)),$$
and $a^{1/2}(1-p)=0$, from where $a(1-p)=0$; so $a=ap$. Taking adjoints, $a=pa$. If follows that $a=pap$.
| {
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"timestamp": "2023-03-29T00:00:00",
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What is an example of a non-zero "ring pseudo-homomorphism"? By "pseudo ring homomorphism", I mean a map $f: R \to S$ satisfying all ring homomorphism axioms except for $f(1_R)=f(1_S)$.
Even if we let this last condition drop, there are only two ring pseudo-homomorphisms from $\Bbb Z$ to $\Bbb Z[\omega]$ where $\omega$... | Every commutative example takes the following form. Suppose $f : R \to S$ is a non-unital ring homomorphism between two commutative rings. Then $f(1_R)$ is some idempotent $m \in S$, as Jendrik Stelzner remarks. $mS$ is a "non-unital" subring of $S$ (it's a subring except that its unit is $m$, not $1_S$), and $f$ is a ... | {
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Prove that $\sum_{n=1}^\infty \frac{1}{n(n-x)}$ converges absolutely and uniformly on each [a,b]\N Let $a<b$, $a,b\in R$ Prove that the series $\sum_{n=1}^\infty \frac{1}{n(n-x)}$ converges absolutely and uniformly on [a,b]\N
What is [a,b]\N? I am really confused. For absolute convergence I know we take the absolute o... | Fact: Suppose $f_n$ is a sequence of real-valued functions defined on a set $E.$ Let $N\in \mathbb N.$ If $\sum_{n=N}^{\infty} f_n$ converges uniformly on $E,$ then $\sum_{n=1}^{\infty} f_n$ converges uniformly on $E.$ I'll use this below.
In our problem, it's enough to show that the series converges uniformly on $[-N,... | {
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Intersection of all open intervals of the form $(-\frac{1}{n}, \frac{1}{n})$ is $\{0\}$? In Mathematical Analysis by Apostol he mentions that the "Intersection of all open intervals of the form $(-\frac{1}{n}, \frac{1}{n})$ is $\{0\}$"
Obviously this is a super basic question but I thought that an open interval does no... | $\cap_{n=1}(-\frac{1}{n}, \frac{1}{n}) = {0}$ is equivalent to the statement that for all $n \in \mathbb{N}, 0 \in (-\frac{1}{n}, \frac{1}{n})$, which is obviously true.
As $n \rightarrow\infty$, $(-\frac{1}{n}, \frac{1}{n})$ approaches the zero, but never reaches the zero
| {
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Find function f(x) Find function f(x), where:
$$f(3)=3$$
$$f'(3)=3$$
$$f'(4)=4$$
$$f''(3) = \nexists$$
How to find function like this in general? What steps should I do?
| Take a function $|x|$ which has no derivative at 0. Integrate it to get a function $x\cdot|x|$ (yes, $1\over2$ is missing, but who cares) which has no second derivative at 0. Shift it to get $(x-3)|x-3|$ which has no second derivative at 3. Add some $const\cdot x^2$ to get $f'(4)-f'(3)=1$. Then add some $const\cdot x$ ... | {
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Are $\frac{\pi}{e}$ or $\frac{e}{\pi}$ irrational? Is it clear whether $\displaystyle \frac{\pi}{e}$ or $\displaystyle \frac{e}{\pi}$ are irrational or not?
If not, then there would exist $q,p\in \mathbb{Z}$ such that $$p\cdot \pi = q\cdot e$$
| A QUITE SIMPLE REMARK $$\begin{cases}\frac{e}{\pi}=t\\e+\pi=s\end{cases}\qquad (*)$$ would imply $$\pi=\frac{s}{t+1}\\e=\frac{st}{t+1}$$ Consequently and least one of $t$ and $s$ in (*) must be trascendental.
| {
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How to evaluate $\lim _{x\to 0}\left(\frac{x\left(\sqrt{3e^x+e^{3x^2}}-2\right)}{4-\left(\cos x+1\right)^2}\right)$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Can you explain the method and the steps used?
$$\lim _{x\to 0}\left(\frac{x\left(\sqrt{3e^x+e^{3x^2}}... | You get in the denominator
$$
2^2-(1+\cos x)^2=(3+\cos x)·(1-\cos x)=(3+\cos x)·2 \sin^2(x/2)
$$
so that you can reduce the limit to
$$
\lim_{x\to 0}\frac{x^2}{(3+\cos x)·(1-\cos x)}·\lim_{x\to 0}\frac{\sqrt{3e^x+e^{3x^2}}−2}{x}
\\
=\frac12·\lim_{x\to 0}\frac1{\sqrt{3e^x+e^{3x^2}}+2}·\lim_{x\to 0}\frac{3(e^x-1)+(e^{3x^... | {
"language": "en",
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Sudoku grid guaranteed to be solvable? I want to generate random sudoku grids.
My approach is to fill the three 3x3 groups on a diagonal of the grid, each with the numbers 1-9 randomly shuffled. That looks e.g. like the grid below:
+-------+-------+-------+
| 5 6 4 | · · · | · · · |
| 1 7 2 | · · · | · · · |
| 9 8 3 |... | WLOG the top block is in the canonical order, because we can just relabel, as in 5 -> 1, 6 -> 2, … in your example. I'm not going to make that replacement through the rest of your grid, but will just pretend you started off like that.
+-------+-------+-------+
| 1 2 3 | · · · | · · · |
| 4 5 6 | · · · | · · · |
| 7 8 9... | {
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"timestamp": "2023-03-29T00:00:00",
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Does $f_n(x)=\frac{x}{n}\log(\frac{x}{n})$ converge uniformly in $(0,1)$? $f_n(x)=\frac{x}{n}\log\frac{x}{n}$
I tried to expand the bounds to $[0,1]$ and prove that the hypotheses of Dini's uniform convergence criterion are satisfied, but I'm not even sure I can expand the bounds since $f_n(0)$ isn't defined.
| Hint:
$$
f_n(x)=\frac{x\log x}{n}-\frac{\log n}{n}\,x
$$
and $\lim_{x\to0^+}x\log x=0$.
| {
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How to calculate $\int_a^bx^2 dx$ using summation? So for this case, we divide it to $n$ partitions and so the width of each partition is $\frac{b-a}{n}$ and the height is $f(x)$.
\begin{align}
x_0&=a\\
x_1&=a+\frac{b-a}{n}\\
&\ldots\\
x_{i-1}&=a+(i-1)\frac{b-a}{n}\\
x_i&=a+i\frac{b-a}{n}
\end{align}
So I pick left ... | Hint: $$\sum_{i=1}^{n}(i-1)^2 = \sum_{i=1}^{n}(i^2-2i+1) = \sum_{i=1}^{n}i^2 - 2\sum_{i=1}^{n}i + \sum_{i=1}^{n}1$$
You should be able to take it from here.
| {
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Integrate: $\int \frac{\sin(x)}{9+16\sin(2x)}\,\text{d}x$. Integrate: $$\int \frac{\sin(x)}{9+16\sin(2x)}\,\text{d}x.$$
I tried the substitution method ($\sin(x) = t$) and ended up getting $\int \frac{t}{9+32t-32t^3}\,\text{d}t$. Don't know how to proceed further.
Also tried adding and substracting $\cos(x)$ in the nu... | To attack this integral, we will need to make use of the following facts:
$$(\sin{x} + \cos{x})^2 = 1+\sin{2x}$$
$$(\sin{x} - \cos{x})^2 = 1-\sin{2x}$$
$$\text{d}(\sin{x}+\cos{x}) = (\cos{x}-\sin{x})\text{d}x$$
$$\text{d}(\sin{x}-\cos{x}) = (\cos{x}+\sin{x})\text{d}x$$
Now, consider the denominator.
It can be rewritten... | {
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The sum of infinitely many $c$s is $c$ implies $c = 0$. This seems like an obvious claim, but I would like to be able to prove this rigorously.
Suppose I have $c \in \mathbb{R}$ satisfying
$$\lim_{n \to \infty}\sum_{i=1}^{n}c = c\text{.}$$
How does it follow that $c = 0$?
What I think I shouldn't do: $$c\lim_{n \to \in... | We'll assume that $c \in \mathbb{R}$.
$$\lim_{n\to\infty}\sum_{i=1}^{n}c=c$$
Subtract $2c$
$$\lim_{n\to\infty}\sum_{i=1}^{n-2}c=-c$$
$$-\lim_{n\to\infty}\sum_{i=1}^{n-2}c=\lim_{n\to\infty}\sum_{i=1}^{n}c$$
But we know that $$\lim_{n\to\infty}\sum_{i=1}^{n-2}c = \lim_{n\to\infty}\sum_{i=1}^{n}c$$
Thus
$$
-\lim_{n\to\inf... | {
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Prove that G is a cyclic group
Suppose that $|G| = pq$ where $p$ and $q$ are primes such that $p < q$ and $p$ does not divide $q − 1$. Prove that $G$ is a cyclic group.
A cyclic group is a group that has a unique generator element, so is the way to go with this to find that element? Or am I missing something?
| A precedent question shows Prove that G has a normal Sylow p-subgroup that $G$ has a Sylow normal group $N$ of order $p$, Consider $m:G\rightarrow G/N$ it defines an extension $1\rightarrow N\rightarrow G\rightarrow G/N\rightarrow 1$ this extension splits since the theorem of Sylow implies that $G$ has a $q$-subgroup, ... | {
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Continuity between topological spaces Let $X,Y$ be topological spaces, and $f:X\to Y$. I am trying to show that, if "$A$ closed in $Y\Rightarrow$ $f^{-1}(A)$ closed in $X$", then we have sequential continuity:
$$x_n\to x\;\text{in }X\Rightarrow f(x_n)\to f(x)\;\text{in }Y$$
I am stuck because this is "in the wrong dire... | It is maybe easier to think about proving the contrapositive rather than proving it directly. Suppose $x_n\to x$ but $f(x_n)\not\to f(x)$ for some sequence $(x_n)$ in $X$. You want to somehow get from this a closed set $A\subseteq Y$ such that $f^{-1}(A)$ is not closed. To do this, look at what it means for $f(x_n)$... | {
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How can you find $m$ in $mx^2+(m-3)x+1=0 $ so that there is only one solution How can you find $m$ in $$mx^2+(m-3)x+1=0 $$ so that there is only one solution.
I tried to solve it by quadratic equation but I end up with two solutions.
So I want it know that is there a way so that I'll only get one solution the end.
Than... | Let $x=\alpha$ be one solution of the given quadratic equation: $mx^2+(m-3)x+1=0$ then $\alpha$ & $\alpha$ be the roots of the given equation $$\text{sum of roots}=\alpha+\alpha=-\frac{m-3}{m}$$$$\alpha=\frac{3-m}{2m}\tag 1$$
$$\text{product of roots}=\alpha\cdot \alpha=\frac{1}{m}$$$$\alpha^2=\frac{1}{m}\tag 2$$
subst... | {
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Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$ Assume $b>0$. Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$.
We know if $b>1$ the integrand can be bounded and it's just an integral $\int_{1}^{\infty} ... | Hint: For $0 < b \leqslant 1$
$$\int_1^c |x^{-b}e^{\sin x} \sin 2x| \, dx > e^{-1}\int_1^c \frac{|\sin 2x|}{x} \, dx = \frac{1}{e}\int_2^{2c} \frac{|\sin x|}{x} \, dx $$
Show the integral on the RHS is not finite as $c \to \infty$ using:
$$\int_{n \pi}^{(n+1) \pi} \frac{| \sin x|}{x}\, dx \geqslant \frac{2}{(n+1) \pi}$... | {
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Limit of sum of periodic function Let $f_1,f_2,...,f_n$ are periodic functions,if $\lim\limits_{x\rightarrow\infty}\sum_{i=1}^n f_i(x)$ is existent and bounded.
How to show $\sum_{i=1}^n f_i(x)\equiv C$ ?
$C$ is a constant.
| (this assumes continuity of the $f_i$, still thinking of how to remove it)
Suppose $F(x):=\sum_i f_i(x)\to C$, where $f_i(x+p_i)=f_i(x)$ for each $i$.
Now, find an sequence of positive reals $h_N$ increasing to $\infty$ which is simultaneously close to $p_i\mathbb{Z}$ for all $i$, i.e.
$$h_N=a_i^{(N)}p_i+\varepsilon_N^... | {
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Non-associative commutative binary operation Is there an example of a non-associative, commutative binary operation?
What about a non-associative, commutative binary operation with identity and inverses?
The only example of a non-associative binary operation I have in mind is the commutator/Lie bracket. But it is not c... | The simplest example of a nonassociative commutative binary operation (but lacking an identity element) is the two-element structure $\{a,b\}$ with $aa=b$ and $ab=ba=bb=a;$ note that $a=bb=(aa)b\ne a(ab)=aa=b.$ This is the NAND (or NOR) operation of propositional logic, where $a=\text{true}$ and $b=\text{false}$ (or vi... | {
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Supremum proof simple I got stuck on this problem and can't figure it out, I hope somebody can help me, I also wrote my attempt. Thanks in advance!!
Question: Let $(a_n)$ be a convergent sequence in $\mathbb{R}$. $a_n
\to a^*$. Let $A=\{a_n | n \in \mathbb{N}\}$. I have to show that:
$\sup A \geq a^*$
My attempt:... | Let $b$ be an upper bound for $A$.
Then $a_n \le b$ for all $n$, and so $\displaystyle a^*=\lim_{n\to\infty} a_n\le b$.
Since $\sup A$ is an upper bound for $A$, we can take $b=\sup A$ and conclude that $a^* \le \sup A$.
| {
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Proving that if a function is a metric then it is symmetric and non negative I am trying to prove that given a metric d using only the properties that it $d(a,b)=0 iff a=b$ and $d(a,c)\le d(a,b)+d(b,c)$ that $d(a,b)=d(b,a)$ and $d(a,b) \gt 0$ I understand that it is part of the definition in most texts but it is left ... | Note that $d:\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ defined as $d(a,b)=0$ is such that :
*
*$\forall a\in\mathbb{R}, d(a,a)=0,$
*$\forall a,b,c\in\mathbb{R},d(a,c)=0\leq d(a,b)+d(a,c)=0$
but you have not that $a\neq b\implies d(a,b)>0,$ so this point should be an axiom.
For the symmetry part, consider $d':\{0,1... | {
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Power series for $(a+x)^{-1}$ Is it possible to write the following expression in terms of power series?
$$ (a+x)^{-1}=\sum\limits_{k = - \infty }^\infty {{b_k}{x^k}} $$
where $0 < a < 1$ and $0 < x < 1$.
| It can be turned into geometric series:
$$\frac{1}{a+x} = \frac{1}{a} \cdot \frac{1}{1-\left(-\frac{x}{a}\right)} = \frac{1}{a}\cdot\sum\limits_{k=0}^{\infty}\left(-\frac{x}{a}\right)^k$$
which converges whenever $|x| < |a|$.
| {
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If $G$ is a finite group s.t. $|G|=4$, is it abelian ? If $G$ is a finite group s.t. $|G|=4$, is it abelian ? To me it's isomorphic to $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$ or $\mathbb Z/4\mathbb Z$, but a friend of me said that they are not the only group of order 4, and there exist some non-abelian. Do yo... | If $G$ is cyclic, $G$ is abelian. And if $G$ is not cyclic, the order of all elements are equal to $2$. And a group whose all elements are of order $2$ is abelian.
In any cases, a group of order $4$ is abelian.
| {
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An automorphism that is not inner. Consider the group $G=SL_3(\mathbb{C})$. I want to show that the automorphism $\phi$ of $G$ given by $\phi(x)=(x^{-1})^T$ is not inner. Probably I should do this by contradiction, i can show that if $\phi(x)=RxR^{-1}$, then $R^T R$ lies in the centre of $SL_3$. How can I proceed to ob... | The operator $\operatorname{tr}$ is invariant under conjugation, but $\operatorname{tr} \phi(\lambda) \not = \operatorname{tr} \lambda$ for constant $\lambda\not = 0, \pm 1$.
| {
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Is $n−m$ always the largest possible number of linearly independent vectors in this vector space? Fix linear independent vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let $S$ be the vector space of such that
$S:=$ {$x∈ \mathbb{R}^n:a_i⋅x=0∀1≤i≤m$} .
The vector space $S$ always has at least $n − m$ linearly independen... | Consider the map from $R^n$ to $R^m$ defined by
$$
v \mapsto (v \cdot a_1, v \cdot a_2, \ldots, v \cdot a_m).
$$
What does the rank-plus-nullity theorem tell you about this linear transformation?
| {
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What is a contraction on a space $(X,d)$? I have been reading some proofs on the elementary theorems of differential equations. One such proof uses the concept of a "contraction". See the definition below.
Definition 4 Let $(X,d)$ be a space equipped with a distance function $d$. A function $\Phi:X\to X$ from $X$ to i... | No, a contraction is not a metric(aka distance function). What a contraction does is bring every pair of points closer together(in the implicit metric). For instance, $f(x)=x/2$ is a contraction in the euclidean metric of $\mathbb{R}$.
| {
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For any integer $n$ greater than $1$,how many prime numbers are there greater than $n!+1$ and less than $n!+n$? For any integer $n$ greater than $1$,how many prime numbers are there greater than $n!+1$ and less than $n!+n$ ?
By trying different values of $n$ for $n=2,3,4,5,6$ I get a feeling that the number of primes i... | Hint: Let $a = n! + 7$. Does $7 \mid n!$? Does $7|7$? Does $7 \mid n! + 7$? Is $n! + 7$ prime?
If it isn't what number greater than $n! + 1$ and less than or equal to $n! + n$ might be?
Can you factor anything out of the number $n! + k$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How was Zeno's paradox solved using the limits of infinite series? This is a not necessarily the exact paradox Zeno is thought to have come up with, but it's similar enough:
A man (In this photo, a dot 1) is to walk a distance of one unit from where he's standing to the wall. He, however, is to walk half the distance f... | If the subject walks at a constant speed, you don't need calculus to calculate exactly when he will arrive at his destination. Just use the simple speed-equals-distance-over-time formula. This formula was unknown to the ancient Greek philosophers. They had no notion of actually measuring speed. It wouldn't be for anoth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1623415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 3
} |
Financial Mathematics, interest problem Determine the amount of interest earned from time $t=2$ to $t=4$ if $240$ is invested at $t=1$ and an additional $300$ is invested at $t=3$.
Given, $a(1)=1.2$, $a(2)=1.5$, $a(3)=2.0$, $a(4)=3.0$
I tried finding $A(0)$ using the equation $A(1)=A(0).a(1)$. Hence $A(0)=200$
Hence I ... | Answer:
$$a_{(2-4)} = \frac{a(4)}{a(2)}$$
$$A(2) = A(1).\frac{a(2)}{a(1)}$$
$$A(2) =240\times\frac{1.5}{1.2} = 300$$
$$A(4) = 300\times\frac{3}{1.5} = 600$$
$$A'(4) = A'(3)\frac{a(4)}{a(3)}$$
$$A'(4) = 300\times\frac{3}{2}$$
$$A'(4) = 450$$
$$I_1 = A(4)-A(2) = 600-300 = 300$$
$$I_2 = A'(4) - A'(3) = 450-300 = 150$$
$$I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1623535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sin(x): surjective and non-surjective with different codomain? Statement that $\operatorname{sin}(x)$ not surjective with codomain $\mathbb R$ and surjective with codomain $[-1,1]$ found here:
*
*Non-surjective: $\mathbb{R}\rightarrow\mathbb{R}: x\mapsto\operatorname{sin}(x)$
*Surjective: $\mathbb{R}\rightarrow[-1,... | $f(x)=\sin(x)$ is not a surjection from $\Bbb R\to \Bbb R$
$f(x)=\sin(x)$ is a surjection from $\Bbb R\to [-1,1]$
The wiki page tells you that in the case of $\Bbb R\to [-1,1]$ that $\sin(x)$ is a surjection but is not an injection. It is a surjection because every possible output has a preimage. Suppose $y\in [-1,1]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1623629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Solving an integral using euler method Hy. Can someone please explain me how can I resolve $\int_{0}^{\infty} e^{-(sI-A)t}\,dt$ ? I must have the final result $(sI-A)^{-1}$. I think it has the euler form, for beta integral but I don't know how can I get to that result.
| If you wrote the integral correctly, just use the fact that
$$\int e^{kt}=k^{-1}e^{kt}+C$$
where, of course, $k$ is a constant and $C$ is an arbitrary constant. There is no need for a fancy integral such as the beta integral. Finding the definite integral from that is easy, since the limit is easy.
All this, of course,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1623704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing there is a prime in a ring extension using Nakayama's lemma Here's the problem that I'm working on:
if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, Undergraduate Commutative Algebra, Exercise 4.12(i))
There was a... | For the first part, the contradiction comes from the fact that there exists $r\in A$ such $r=1+p, p\in P$ and $rB=0$ this implies $(1+p)A=0$ so for every $a\in A, (1+p)a=a+pa=0$ which implies $a\in P$ contradiction since $P$ is different of $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1623811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove that $3^x + 3^{x-2}$ ends with $0$ for any integer $x > 1$ I think that $3^x+3^{x-2}$ ends in a $0$ (i.e. is divisible by $10$) $\forall x \in \Bbb Z, x > 1$.
Examples:
$3^2+3^{2-2}=9+1=10 \\
3^3+3^{3-2}=27+3=30 \\
3^4+3^{4-2}=81+9=90 .$
In fact, I wrote a quick Python program and left it on overnight, it reporte... | $3^{x} + 3^{x - 2} = 3^{x}(1 + 3^{-2}) = 3^{x}({10}/{9}) = (3^{x}/9)(10)$ The value is always a multiple of 10 since for x ≥ 2 we always have an integer value. The proof is complete.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1623892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove $x^TAx = 0$ $\implies$ A is skew-symmetric We know for a skew-symmetric matrix A, $x^TAx = 0$. But is the converse statement true, i.e. does $x^TAx = 0$ imply A is skew-symmetric? If yes, then how to prove it?
| $x^T A x = 0\ \forall x$ can be written as:
$\sum_i \sum_j x_j a_{ij} x_i = 0\ \forall x$
That is:
$x_1^2 a_{1,1} + x_1 a_{1,2} x_2 + ... = 0$
The only way to have $x_i x_j$ cancel out $x_j x_i$ is by: $a_{ij} = -a_{ji}$, and the remaining terms $x_i^2 a_{ii}$ must be zero (for all $x$), so we must have $a_{ii}=0$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1623934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Expanding an expression in a certain field If $\mathbb F_2$ is a field of characteristic $2$, then we have $x+x=y+y=z+z=0$ for all $x,y,z \in \mathbb F_2$. When I expand $(x+y)(y+z)(z+x)$, I get
\begin{align}
(x+y)(y+z)(z+x) &= xz^2+y^2z+yz^2+x^2y+x^2z+xy^2+2xyz \\
&= xz^2+y^2z+yz^2+x^2y+x^2z+xy^2+(x+x)yz \\
&= xz^2+... | In $\mathbb{F}_2$, we also have $x^2=x$ and so on. So your expression will result in $0$. Another way to look at it is that at least two of $x,y$ or $z$ will take the same value, in which case at least one of $x+y, y+z, x+z$ will be $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1624070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this f(z) function analytic? Is $f(z) = z^2\sin z$ an analytic function for $z \in \mathbb{C}$ ?
$z = x + iy $
I really don't know how to split this at this format $f(x,y) = P(x,y) + iQ(x,y)$, so I can prove that if this function is analytic with neccesary Cauchy–Riemann equations. Which are $ P_x = Q_y$ and $P_y =... | Using Cauchy-Riemann here is unnecessary work. It's easier (even if you haven't done so before) to show that $\sin z$ is holomorphic (analytic), and that the product of two holomorphic functions is again holomorphic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1624163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Reduced row echelon form and linear independence Let's say I have the set of vectors $S = \{v_1, v_2, ..., v_n\}$ where $v_j \in R^m$, $v_j = (a_{1j}, a_{2j}, ..., a_{mj})$.
If the matrix formed by each of the vectors $A=[v_1, v_2, ..., v_n]$ looks like this (I believe), which is not a square matrix:
$$A = \begin{pmatr... | Yes, if you can convert the matrix into reduced row echelon form(or even just row echelon form) without a row of $0$s,then the vectors are linearly independent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1624238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
Factorial grow faster than Exponential - permutation case It is said that factorial grows faster than exponential, but in the case of permutation:
permutation with repetition = n^n
permutation without repetition = n!
And of course permutation with repetition has a bigger "space" than permutation without repetit... | The $n^n$ function is not an exponential as the basis varies.
To prove $n!=o(n^n)$, let's evaluate the ratio:
$$\frac{n!}{n^n}=\frac1n\cdot\frac2n\cdots\frac{n-1}n\cdot \frac nn <\frac 1n$$
since each factor $\dfrac kn$ is less than $1$ if $k<n$. This inequality implies $\;\displaystyle\lim_{n\to\infty}\frac{n!}{n^n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1624332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Need clarifying on basic derivatives of natural log/e So here's the question:
Find the derivative: $ y= e^{\cos(x)}$
Hint: This is a combination of the chain rule and the natural log. The derivative is $(\ln a)(a^{f(x)}) * f'(x)$
So normally without using the hint I would use the chain rule and solve it like this:
... | It is not a coincidence as you predicted but a special case for $e^{f(x)}$ since $ln(e) = 1$. For example: if the question were $y = a^{cos(x)}$, your answer would be $\frac{dy}{dx} = (ln(a))(a^{cos(x)})(-sin(x))$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1624400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $A=\sin^{20}\theta +\cos^{48}\theta $ then identify the correct option.
If $A=\sin^{20}\theta +\cos^{48}\theta $, then for all values $\theta$
a) $A\geq 1$
b) $ 0< A\leq 1$
c) $1<A< 3$
d) None of these
$0 \leq \sin^{20}\theta \leq 1$
$0 \leq \cos^{48}\theta \leq 1 $
So I think it is $d.)$ , but I am confus... | Hint: $0 \leq \sin^2\theta \leq 1$, $0 \leq \cos^2\theta \leq 1$, and $\sin^2\theta + \cos^2\theta = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1624535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Proving the surprising limit: $\lim\limits_{n \to 0} \frac{x^{n}-y^{n}}{n}$ $=$ log$\frac{x}{y}$ A few months ago, while at school, my classmate asked me this curious question: What does $\frac{x^{n}-y^{n}}{n}$ tend to as $n$ tends to $0$? I thought for a few minutes, became impatient, and asked "What?" His reply, log$... | Hint:
$$
\begin{align}
\lim_{n\to0}\frac{x^n-1}{n}
&=\log(x)\lim_{n\to0}\frac{e^{n\log(x)}-1}{n\log(x)}\\
&=\log(x)\lim_{u\to0}\frac{e^u-1}u\\[6pt]
&=\log(x)
\end{align}
$$
Subtract
Swapping Integral and Limit
Let $\lambda_{\rm{min}}=\min\left\{1,y^{-5/4}\right\}$ and $\lambda_{\rm{max}}=\max\left\{1,x^{-5/4}\right\}$... | {
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"answer_id": 1
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context free grammar accepted and generated language problems I'm having problems completing the following questions, I am able to attempt them but don't know if they are correct. Any help would be much appreciated.
Answer the following questions for a context free grammar $G=(N,\Sigma,P,S)$, where $N,\Sigma, P, S$ rep... | $$S\rightarrow aAa|bAb\\
A\rightarrow aAa|bAb|B\\
B\rightarrow aB|bB|\varepsilon$$
$(1)$ You want the word $baabab$ so $S\to bAb\to baAab\to baaBab\to baaBbab\to baabab$
| {
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"question_score": "1",
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"answer_id": 1
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Is the axiom $g1 = g$ essential for a group action A group $G$ acts on a set $\Omega$ if
(1) $\omega\cdot 1_G = \omega$
(2) $(\omega \cdot g)\cdot h = \omega \cdot (gh)$
for all $\omega \in \Omega$ and $g,h \in G$. But is (1) really essential, what if we drop (1) and just require (2), then we must have
$$
\omega\cdot... | Let $\Omega = \{1,2\}$, take $G$ to be any group and set $x \cdot g = 1$ for all $x \in \Omega$ and $g \in G$. Then of course $(x \cdot g) \cdot h = 1 \cdot h = 1 = x \cdot (gh)$, but $2 \cdot 1_G = 1 \neq 2$ .
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Limits using Epsilon Delta definition $$\lim_{x \to 3} x^{3}=27 $$
$$ 0<|x-3|<\delta$$
$$|x-3||x^{2}+3x+9|< \varepsilon$$
My teacher said that delta must be in the interval [0,3] but I don't think it's correct.
What is the correct approach to solve this kind of questions?
Thanks in advance.
| take$ \delta = min(1,\frac{\varepsilon}{37})$
then $|x-3||x^{2}+3x+9|< 37|x-3| < \epsilon $
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An integral arising from Kepler's problem $\frac{1}{2\pi}\int_0^{2\pi}\ \frac{\sin^2\theta\ \text{d}\theta}{(1 + \epsilon\cos\theta)^2}$ I'm dealing with this integral in my spare time, since days and days, and it's really interesting. I'll provide to write what I tried until now, and I would really appreciate some hel... | I am assuming $0 < \epsilon < 1$ in the following. The cases
$\epsilon = 0$ and $\epsilon = 1$ have to be handled separately,
see below.
There is a small error in your calculation, the root of
$2z + \epsilon(z^2+1) = 0$ are
$$
z_1 = \frac{-1-\sqrt{1 - \epsilon^2}}{\epsilon} \, , \quad z_2 = \frac{-1+\sqrt{1 - \epsilon... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Degeneracies of simplex $y$ which appears as any face of some simplex $x$ Let $K$ be simplicial set and $d_i:K_n\rightarrow K_{n-1}$, $s_i: K_n\rightarrow K_{n+1}$ ($i = 0,...,n$) face and degeneracy maps respectively.
Suppose we have some $x\in K_n$ with $d_0x = ... = d_n x = y$. How to prove that $s_0y=...=s_{n-1}y... | Could you expand on your motivation for this question? I think I have found a counterexample, essentially because if this identity was true, it would have to be an easy consequence of the simplicial identities.
The functor $F$ that sends a simplicial set $K_*$ to the $n$-simplices satisfying this condition is corepres... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1625233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate this integral $\int \frac{x^2}{4x^4+25}dx$. I have to calculate this integral $\int \frac{x^2}{4x^4+25}dx$.
I dont have any idea about that.
I thought about parts integration.
Thanks.
| Hint:
$$\int \frac { x^{ 2 } }{ 4x^{ 4 }+25 } dx=\int { \frac { x^{ 2 } }{ 4x^{ 4 }-20{ x }^{ 2 }+25-20{ x }^{ 2 } } dx } =\int { \frac { { x }^{ 2 } }{ { \left( 2{ x }^{ 2 }-5 \right) }^{ 2 }-20{ x }^{ 2 } } dx } =\int { \frac { x^{ 2 }dx }{ \left( 2x^{ 2 }-2\sqrt { 5 } x-5 \right) \left( 2x^{ 2 }+2\sqrt { 5 } x-5 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1625320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do you solve B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$? How do you solve B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$ ?
$A = \left.\frac{s^2-s}{s^2+1} \right\vert_{s=-1} = \frac{1-(-1)}{1+1}=1$
| Notice, $$\frac{s-1}{s+1}\frac{s}{s^2+1}=\frac{A}{s+1}+\frac{Bs+C}{s^2+1}$$
$$s^2-s=(A+B)s^2+(B+C)s+(A+C)$$
comparing the corresponding coefficients on both the sides, one should get $$A+B=1\tag 1$$
$$B+C=-1\tag 2$$
$$A+C=0\tag 3$$
on solving (1), (2) & (3), one can easily get $\color{red}{A=1}, \color{blue}{B=0}, \co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1625419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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How to determine (and explain) the sum of angles without measuring? Below is a photo of the angles/triangles in which I am working on determining the sum of the angles without measuring. The angles are a,b,c,d,e,f.
I understand that angles are formed my intersecting lines and I see many intersecting lines in this imag... | Call the angles of the three outer triangles $a, b, g$; $c, d , h$; and $e, f, i$, respectively. Then
$$a + b + c + d + e + f = (a + b + g) + (c + d + h) + (e + f + i) - (g + h + i) = 3 \times 180^{\circ} - 180^{\circ} = 360^{\circ},$$
where we've used the fact that $g, h, i$ are also the angles of the inner triangle.... | {
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"timestamp": "2023-03-29T00:00:00",
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Continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $ Give an example of a continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $
If $f \in L^1([a,b]), a< b$ that would mean that $$\int_{a}^{b}|f(x)|dx < \infty \tex... | How about $f(x)\equiv1$ for the first one, since then $\int_a^bf(x)dx=b-a<\infty$, but clearly $f\not\in L^1(\Bbb R)$. For the second, pick a function that oscillates enough that cancellation allows integrability but not absolute integrability. The answer below has a nice example.
EDIT: If one of $a$ or $b$ is infinite... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is... Problem :
If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is
(a) $2^6$ (b) $2^{12}$ (c) $2^{16}$ (d)$2^{20}$
My approach :
$f(x) = ^{7-x}C_x = \frac{(7-x)!}{x!(7-2x)!} $
But having n... | Assuming that $x \in \mathbb{Z}$. The binomial coefficient $\binom{7-x}{x}$ is defined for $0 \leq x \leq 7$ and $x \leq 7-x$. Thus $x \in \{0,1,2,3\}$. Now you can find possible values of this coefficient. And this gives the size of set $A$.
Any reflexive relation $R$ should have pairs of the form $(x,x)$ for all $x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1625844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Roots of $f(x)+g(x)$ Question : Let $p,q,r,s \in \mathbb R$ such that $pr=2(q+s)$. Show that either $f(x)=x^2+px+q=0$ or $g(x)=x^2+rx+s=0$ has real roots .
My method :
To the contrary suppose that both $f(x)$ and $g(x)$ has no real roots.
Now observe that $f(x)+g(x)=2x^2+(p+r)x+(q+s)=0$
Since $pr=2(q+s)$ , $f(x)+g(x)... | (1). You did not state that $p\ne r$ and it is not needed. (2).Supposing that neither $f$ nor g has a real root, I do NOT observe that $f(x)+g(x)=0.$ Instead I observe that if neither $f$ nor $g$ has a real root then $\forall x\;(f(x)+g(x)>0).\;$ (3). I agree that $$\forall x\;[f(x)+g(x)=(x-p/2)(x-r/2)]$$ which implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1625953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Mensuration and similarity Cone P has a volume of 108cm^3
Calculate the volume of 2nd come , Q , whose radius is double that of cone P and its height is one-third that of cone p
Here's my working ....
$$V_Q=\frac13 \pi (2r)^2 \cdot \frac{h}{3}\\
= \frac{4}{9} \pi r^2 h\\
= \frac{1}{3}... | Hint:
In your equation, you already have $$V_Q = \left(\frac{1}{3}\pi r^2h\right) \cdot 43$$
You also know what the volume of cone $P$ is, and you know that it is equal to $\frac{1}{3}\pi r^2 h$
| {
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"url": "https://math.stackexchange.com/questions/1626089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Limit $\lim_{x \to 0} \dfrac{\sin(1-\cos x)}{ x} $ $$\lim_{x \to 0} \dfrac{\sin(1-\cos x)}{ x} $$
What is wrong with this argument: as $x$ approaches zero, both $x$ and $(1-\cos x)$ approaches $0$.
So the limit is $1$ .
*
*How can we prove that they approaches zero at same rate?
*This is not about solving the limi... | By your argument, $$\lim_{x\to 0}\frac{x}{x}$$ is also $0$ because as $x$ approaches zero, both $x$ and $x$ approach $0$, so the limit is $0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Awodey's Category Theory Exercise 9.9.2 I am having problems with this question in Awodey's Category Theory book p.248:
*Show that every monoid M admits a surjection from a free monoid $F(X) → M$, by considering the counit of the free $\dashv$ forgetful adjunction.
Let $f : F(X)→M$ be a monoid homomorphism and... |
Let $f : F(X)→M$ be a monoid homomorphism and $m \in M$. We want to show that there is $x \in F(X)$ such that $f(F(x))=m$. But now I'm kind of stuck.
This looks like you're trying to show that every monoid homomorphism from a free monoid is surjective, but you only have to show that there exists a surjection from a f... | {
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Why does the sum of two linearly independent solutions of a second order homogeneous ODE give a general solution? The following is a short extract from the book I am reading:
If given a Homogeneous ODE:
$$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}+5\frac{\mathrm{d} y}{\mathrm{d}x}+4y=0\tag{1}$$
Letting
$$D=\frac{\m... | If you recall from linear algebra, abstract functional spaces can be considered as vector spaces. We define the zero function to serve as the zero vector, and pointwise addition/multiplication as the vector space operations.
For a collection of normal vectors, to show linear independence, we want to show none of the ch... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Interpreting functions for poset I'm stuck at a question with the following expression of poset $(\mathcal{N}, \sqsubseteq)$:
$$\forall x, y \in \mathcal{N}: x \sqsubseteq y \mbox{ iff } (x = y \vee x = \bot_{\mathcal{N}} \vee y = \top_{\mathcal{N}}).$$
$\mathcal{N} = \mathcal{N} \bigcup\{ \bot_{\mathcal{N}} , \top_{\m... | This poset is the set of all natural numbers with equality relation, so any two distinct elements $x, y \in \mathcal{N}$ are incomparable, together with the least $\bot_\mathcal{N}$ and the greatest $\top_\mathcal{N}$ elements. To find its height you should find the largest cardinality chain, that is a subset in which ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a rational surjection $\Bbb N\to\Bbb Q$? The question is in the title. Is there a one-dimensional rational function $f\in\Bbb R(X)$ which restricts to $\Bbb N\to\Bbb Q$, which is a surjection onto $\Bbb Q$? My guess is no.
Expanding the scope a little (at the expense of precision), are there any "nice" functio... | This is an alternative argument based on Arnaud's suggestion. Without loss of generality, you can assume the rational function $f$ is $\frac{p}{q}$ where $n = \deg(p) \ge \deg(q) = m$ (otherwise trade $f$ in for $\frac{1}{f}$). But then (using $(\frac{p}{q})' = \frac{p'q - pq'}{q^2}$) you have:
$$
f'(x) = \frac{(n-m)p_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Generator matrix of $E_5$
Let $E_5$ denote the binary even weight code of length 5. Write down a generator matrix of $E_5$.
So I know the length $n = 5$ is the number of rows in the generator matrix and the number of columns will be the dimension of $E_5$.
So I firstly need to know how to work out the dimension and I... | Think of how many degrees of freedom you have for this code. How many bits can you fill in however you want, and still be able to use the remaining bits to make the word "even"?
| {
"language": "en",
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Suppose $a \in \mathbb{Z}.$ Prove that $5 \mid 2^na$ implies $5 \mid a$ for any $n \in \mathbb{N}$ This question is supposed to be solved by induction, however I'm unsure of where to get my base case from exactly, because the question is asking about both $a$ and $n$. I started with my base case being $n = 1$, but then... | $1)$ assume $5\nmid a$
$2)$ by the Division Algorithm $a=5q+r$, $\space\space r\lt 5$,$r<q\space\space\space\space\space\space\space 1)$
$3)$ $2^na=2^n(5q+r)=2^n(5q)+2^nr=5(2^nq)+2^nr$, $\space\space r<5\space\space\space\space\space\space\space 2)$
$4)$ $5\nmid 2^na\space\space\space\space\space\space\space1),2),3)$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$3^x-2^y=17$ where $x$ and $y$ are both positive integers First, I was wondering what would the solution(s) be to the equation $3^x-2^y=17$ such that $x$ and $y$ are both positive integers. I couldn't find any small possibilities. Is there a proof that such a solution exists? How about the general case $3^x-2^y=z$? Is ... | It is not true that there is a solution for every odd $z$ not divisible by $3$. For example, $3^x-2^y=13$ is impossible. For we can check by hand that there is no solution if $y=1$ or $y=2$. And if $y\ge 3$, then $13+2^y\equiv 5\pmod{8}$. But powers of $3$ are congruent to $1$ or $3$ modulo $8$. There are infinitely ... | {
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$|\int\limits_a^b f(t)dt| \leq \int\limits_a^b |f(t)|dt$, where $f(t) \in \mathbb C$ Let $f: [a,b] \to \mathbb C, t \to f(t) = \text {Re } f(t) + i\text{ Im } f(t)$. Suppose $f$ is continuous.
Let $\int\limits_a^b f(t)dt = \rho e^{i\theta}$, where $\rho \geq 0$ is the module. Then:
$$|\int\limits_a^b f(t)dt| = \rho = e... | For the first step, up to the sign, note that $\rho$ is real. Since it is equal to $e^{-i\theta}\int\limits_a^b f(t)dt$, the last expression is a real number and so $$\rho=e^{-i\theta}\int\limits_a^b f(t)dt= \text{Re }e^{-i\theta}\int\limits_a^b f(t)dt.$$
For the second step, write $z=x-iy$ and note that $$\text{Re }z=... | {
"language": "en",
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Does a finite first-order theory which has a model always have a finite model? I'm curious whether this is true or not:
Let T be a finite, first-order theory. If T has a model, then T has a finite model.
I would assume the answer is 'yes', but I wanted to make sure I haven't missed something obvious. The reason I bel... | Let the language contain a single binary predicate symbol, to be interpreted as "less than."
Consider the theory $T$ with the following axioms:
1) There are at least two objects.
2) The relation $L$ is a strict total order.
3) For any $x$ and $y$ such that $L(x,y)$, there exists a $t$ such that $L(x,t)$ and $L(t,y)$.
T... | {
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Show that $\sum_{n=0}^\infty |a_n|<\infty.$
Let $\{a_n\}$ be a real sequence. Suppose for every convergent real sequence $\{s_n\}$, the series $\sum_{n=0}^\infty a_n s_n$ converges. Show that $\sum_{n=0}^\infty |a_n|<\infty.$
Mt attempt:
Consider the sequence $\{s_n\}$ where $s_n=1$ for all $n$. Then we have that $\s... | Just elaborating Patrick's comment. Fix $N$. Let
$$
s_n =\begin{cases}
|a_n|/a_n &\text{if } n\leq N \text{ and } a_n\neq 0 \\
1 &\text{if } n\leq N \text{ and } a_n=0 \\
1 &\text{if } n> N
\end{cases}
$$
It is clear that $s_n$ is a convergent sequence. So by the hypothesis,
$$
\sum_{n=1}^{\infty} s_n a_n = \sum_{n=... | {
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Proving $\frac{1}{\sqrt{1-x}} \le e^x$ on $[0,1/2]$. Is there a simple way to prove $$\frac{1}{\sqrt{1-x}} \le e^x$$ on $x \in [0,1/2]$?
Some of my observations from plots, etc.:
*
*Equality is attained at $x=0$ and near $x=0.8$.
*The derivative is positive at $x=0$, and zero just after $x=0.5$. [I don't know how t... | If $f(x)=(1-x)e^{2x}$, then $f'(x)=(1-2x)e^{2x}=0$ when $x=\frac{1}{2}$. Drawing a graph/checking the second derivative shows it to be a maximum, whence $1=f(0)\le f(x)\le f(1/2)=\frac{e}{2}$ on $[0,\frac{1}{2}]$. We thus have:
$$1\le(1-x)e^{2x}\le\frac{e}{2}$$
$$\implies \frac{1}{1-x}\le e^{2x}\le\frac{e}{2(1-x)}$$
$$... | {
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Prove with use of derivative How to prove this inequality using derivative ?
For each $x>4$ ,
$$\displaystyle \sqrt[3]{x} - \sqrt[3]{4} < \sqrt[3]{x-4} $$
| If $f(x)=x^\frac{1}{3}$ we must show that $$ f(x) < f(x-4) + f(4)
$$
Note that for any $a,\ b>0$ if we have $f(a+b)< f(a) + f(b)$, then
we are done. Note that this is equivalent to $$ f(a+b)-f(a) < f(b) -
f(0) $$
In further this is
equivalent to $$ \int_a^{a+b} f'(x)dx < \int_0^b f'(x) dx $$
So $f'(x) = \frac{1}{3} \f... | {
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When $n$ is odd, $\langle (123),(12...n)\rangle$ generates $A_{n}$ Working with the knowledge that the set of 3-cycles generates $A_{n}$, the basic idea is to express any 3-cycle as a word in $(123)$ and $(12...n)$ when $n$ is odd.
Not knowing how to progress, I decided to work with the concrete group $A_{5}$. I've lis... | Note that
\begin{align*}
(1\ 2\ \dots\ n)(1\ 2\ 3)(1\ 2\ \dots\ n)^{-1} &= (2\ 3\ 4)\\
(1\ 2\ \dots\ n)(2\ 3\ 4)(1\ 2\ \dots\ n)^{-1} &= (3\ 4\ 5)\\
&\ \ \vdots
\end{align*}
More generally, given a permutation $\sigma$ and a cycle $(a_1\ a_2\ \dots\ a_k)$, we have
$$\sigma(a_1\ a_2\ \dots\ a_k)\sigma^{-1} = (\sigma(a_... | {
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Prove that $f:\mathbb{R}^n\to\mathbb{R}$ is continuous
Let $f:\mathbb{R}^n\to\mathbb{R}$ such that for every continuous curve, $\gamma:[0,1]\to\mathbb{R}^n$: $f\circ\gamma$ is continuous. Prove that $f$ is continuous.
So I know we shall prove it by a contradiction. Let's assume that $f$ isn't continuous at $x_0$. The... | Construct a curve "joining the points" $x_1$, $x_2$,... with $\gamma(0) = x_1$, $\gamma(1/2) = x_2$,...
| {
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The order of pole of $\frac{1}{(2\cos z -2 +z^2)^2} $at $z=0$ What is the order of the pole of :
$$\frac{1}{(2\cos z -2 +z^2)^2}$$
at $z=0$
This is what I did :
$$\cos z = \frac{e^{iz}-e^{-iz}}{2} $$
then :
$$ \implies \frac{1}{(e^{iz}-e^{-iz}+z^2-2)^2} $$
How do I continue?
| Hint : Let , $f(z)=2\cos z-2+z^2$. Find the order of zero of $f$.
$f'(z)=-2\sin z+2z$.
$f''(z)=-2\cos z+2$
$f'''(z)=2\sin z$
$f^{iv}(z)=2\cos z$.
Clearly , $f(0)=f'(0)=f''(0)=f'''(0)=0$ , but $f^{iv}(z)\not= 0$. So $f$ has a zero of order $4$.
So , $(2\cos z-2+z^2)^2$ has a zero of order $8$ at $z=0$.
| {
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Is this statement about this continuous function correct? Is it true that if $f^2$ is continuous, then $\lvert f \rvert$ is continuous?
Can I say that since $f^2 \geq 0$ then $\lvert f \rvert = \sqrt{f^2}$ and that indicates that $\lvert f \rvert$ is also continuous beacuse of the arithmetic property of continuous func... | Yes, the square root function is continuous in its domain, and composition of continuous functions is continuous.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Infinite exponentials We can read a lot of about convergence of series or Infinite products.
E.g. for series.
Following series
$$\sum_{i=1}^\infty a_i$$
is convergent when
$$\lim_{n\rightarrow\infty}a_n=0$$
and
*
*D'Alembert's criterion
$$\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|<1$$
2. Cauchy's c... | I think you can find in WP's "tetration" the criterion for the powertower over the reals, which was established by L. Euler (but independently by others)
Over the complex numbers the range of convergence of the infinite tetration was determined by W. Thron(1957) "Convergence of Infinite Exponentials with C... | {
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Number of strings over a set $A$ How can I calculate the number of strings of length $10$ over the set $A=\{a,b,c,d,e\}$ that begin with either $a$ or $c$ and have at least one $b$ ?
Is it accomplished through some sort of combinatorial logic coupled with discrete mathematics?
| Yes, you are correct. Use constructive counting. Begin by selecting the first letter in the string, which you said could be either $A$ or $C.$ There are $2$ ways to do this.
Now our problem becomes: construct a string of length $9$ with at least one $B.$ We count this with complementary counting - how many strings can ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Huber loss vs l1 loss From a robust statistics perspective are there any advantages of the Huber loss vs. L1 loss (apart from differentiability at the origin) ? Specifically, if I don't care about gradients (for e.g. when using tree based methods), does Huber loss offer any other advantages vis-a-vis robustness ?
More... | The Huber function is less sensitive to small errors than the $\ell_1$ norm, but becomes linear in the error for large errors. To visualize this, notice that function $| \cdot |$ accentuates (i.e. becomes sensitive to) points near to the origin as compared to Huber (which would in fact be quadratic in this region). The... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding dual basis to some given basis Can someone briefly explain me how to find dual basis to some given basis?
Let's say we've got $n$-dimensional linear space with given basis $(e_1,e_2,...,e_n)$. How to find $(e_1^*, e_2^*, ...., e_n^*)$?
| Let $V$ be a vector space over a field $k$ with basis $e_1, ... e_n$. The dual $V^{\ast}$ is defined to be the vector space consisting of all linear transformations from $V$ to $k$. The definitions of addition and scalar multiplication in $V^{\ast}$ are obvious, as is the fact that $V^{\ast}$ is actually a vector sp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to use xor properly I need to know how to use XOR properly on more than two variables.
I have following example.
a xor b xor c
Now, the way I understand it is that:
$a$ xor $b$ = $a$ * not $b$ + not $a$ * $b$
That part is fairly simple for me to understand but how does the next part work? I imagine it goes like thi... | NOTE: Just so you know, a XOR b is the same as $a \oplus b$. Also, here's a shortcut to XOR: If $a$ and $b$ are different, then $a \oplus b=1$. Otherwise, if they're the same, $a \oplus b=0$. Your formula works as a definition for XOR, but when you're calculating XOR in your head, I think this rule of thumb is easier.
... | {
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"timestamp": "2023-03-29T00:00:00",
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How to find the antiderivative of this function $\frac{1}{1+x^4}$? How to find the antiderivative of the function $f(x)=\frac{1}{1+x^4}$?
My math professor suggested to use the method of partial fractions, but it doesn't seem to work, because the denominator cannot be factored at all.
Attempting to integrate with other... | It's easy to see factorization in the first line.
\begin{align*}
\int \frac{1}{1+x^{4}}\mathrm{d}x &=\int \left ( \frac{\dfrac{\sqrt{2}}{4}x+\dfrac{1}{2}}{x^{2}+\sqrt{2}x+1}+\frac{-\dfrac{\sqrt{2}}{4}x+\dfrac{1}{2}}{x^{2}-\sqrt{2}x+1} \right )\mathrm{d}x\\
&=\int \left [ \frac{\dfrac{\sqrt{2}}{4}\left ( x+\dfrac{\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1628530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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prove there is a unique set $X$ such that every set $Y$, $Y∪X = Y$ For this proof. It seems obvious that $X=∅$ such that for every set $Y$, $Y∪X = Y$ since the $Y∪X$ is just Y. How should I go about this?
Let there be sets $X,Z$
Since $Y∪X = Y$ then,
{$x| x ∈ Y ∨ x ∈ X$} = {$x| x ∈ Y$}
Since $Y∪Z = Y$ then,
{$x| x ∈ Y... | You need one more step after your last equation: If $\{x :x\in Y\lor x\in Z\}=Y=\{x:x\in Y\lor x\in X\}$ holds for every $Y$, then it holds when $Y=X$: $$X=\{x:x\in X\lor x\in X\}=\{x: x\in X\lor x\in Z\}.$$ And it holds when $Y=Z$: $$Z=\{x:x\in Z\lor x\in Z\}=\{x:x\in Z\lor x\in X\}.$$ Comparing the RHS of these, we h... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate the limit $\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+...+\frac{1}{\sqrt{n^2+n}}\right)$ I have to calculate the following limit:
$$\lim_{n\to\infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+...+\frac{1}{\sqrt{n^2+n}}\right)$$
I believe the limit equals $1$, and I think I can prove it with... | Note that we can write the term of interest as
$$\sum_{k=1}^n\frac{1}{\sqrt{n^2+k}}=\frac1n\sum_{k=1}^n\frac{1}{\sqrt{1+k/n^2}}$$
To evaluate the limit we can expand the summand using the binomial theorem as
$$\frac{1}{\sqrt{1+k/n^2}}=1-\frac{k}{2n^2} +O(k^2/n^4)$$
Therefore, we have
$$\begin{align}
\lim_{n\to \infty}\... | {
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"timestamp": "2023-03-29T00:00:00",
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Does a right circular cone only consists of pair of straight lines, hyperbolas, parabola, circles and ellipses? I was reading about the conic sections and that a conic section includes pair of straight lines, ellipses, hyperbola, circles and parabola.
Are all these 5 components enough to form a right circular cone or a... | A cone is a surface and as such cannot consist of a finite number of curves.
A conic section is the intersection of a cone with a plane, so it is a planar curve. Depending on the relative positions of the cone and the plane, the section can be a single or a double line, an ellipse, a parabola or an hyperbola (the circl... | {
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Please explain, "Asymmetric is stronger than simply not symmetric". In some textbook I found a statement like, "Asymmetric is stronger than simply not symmetric".
But as I try to perceive this statement, both appear to be same to me.
For example, parentof is an asymmetric relation. If $A$ is a parentof $B$, $B$ can not... | It is just the standard negation of quantifiers: Asking that a relation is never true is stronger than the negation of the relation being always true.
A relation being symmetric means that all pairs can be inverted.
The negation of this is that some pairs cannot be inverted.
Asymmetric means that all pairs cannot be in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1628966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 6,
"answer_id": 0
} |
Solving a system of ODE that arose in solving Burgers' equation Consider the Burgers' equation $$\partial_t u = \alpha u\partial_xu$$ Intend to solve this using Fourier Galerkin method. So When I convert this into $N$th Fourier partial sum, I get a system of ODE's as follows $$\partial_t u_m = \sum_{k=-N}^{k=N}iku_ku_{... | Analytic solving of the PDE :
$$\partial_t u - \alpha u\partial_xu=0$$
Thanks to the method of characteristics :
$$\frac{dt}{1}=\frac{dx}{-\alpha u}=\frac{du}{0}$$
On the characteristic curves :
\begin{cases}
u=c_1\\
\alpha c_1 dt +dx=0\:\:\rightarrow\:\: \alpha u t +x=c_2\\
\end{cases}
The general solutio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Logic with numbers Nick wrote each of the numbers from 1 to 9 in the cells of the 3x3 table below. Only 4 of the numbers can be seen in the figure. Nick noticed that for the number 5, the sum of the numbers in the neighboring cells is equal to 13 (neighboring cells are cells that share a side). He noticed that the same... | The number in the middle, shaded square abuts four numbers whose sum is at least $5+6+7+8=26$, so cannot be either the $5$ or the $6$. Hence the $5$ and $6$ each go in a side square, abutting whatever's in the middle and two corner squares. Since the total abutting sums both equal $13$, the corner abutting sums for t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Chenge weight function in shifted orthogonality We know that in Chebyshev orthogonal polynomial the weight function is $$\frac{1}{\sqrt{1-x^2}}$$ in interval $[-1,1]$. Do in shifted chebyshev orthogonal as example for interval $[0,1]$ the weight function changed?
| What you need is a scaling function $K(x)$ so that
$$
\left< T_n, T_m \right>
= \int_0^1 T_n(x) T_m(x) K(x) dx
= 0 \quad \text{for } n \ne m,
$$
which is the orthogonality criterion.
In the standard case, picking $$K(x) = \frac{1}{\sqrt{1-x^2}},$$ guarantees
$$
\left< T_n, T_m \right>
= \int_{-1}^1 T_n(x) T_m(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Trig Integration, how did they get this answer?
Here is the full question for context:
And here is the full answer given:
I don't understand how they go from the first part in the red box to the next, I don't have a clue.
| It's a $u$ substitution where they take $u = \sin(x)$. If $u = \sin(x)$, then $\mathrm{d}u = \cos(x) \;\mathrm{d}x$. So we have
$$\begin{align}
\int {\sin^{2n}(x)\cos(x)} \;\mathrm{d}x \;\;&=\;\;
\int {u^{2n}} \;\mathrm{d}u \\\;\;&=\;\;
\frac{u^{2n+1}}{2n+1} \;\;=\;\;
\frac{\sin^{2n+1}(x)}{2n+1}
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $\psi:\mathbb{R}^3\rightarrow\mathbb{R}$ given by $\psi(x,y,z) = z$ is linear Show that $\psi:\mathbb{R}^3\rightarrow\mathbb{R}$ given by $\psi(x,y,z) = z$ is linear
I know that to be linear it must satisfy:
(i) $\psi(x+y+z)=\psi(x)+\psi(y)+\psi(z)$
(ii) $\psi(ax)=a\psi(x)$
So, this is how far I've gotten:
(... | You are the right way. But note that you can simplify what you have done. To show that it is linear you need to check
$$\psi((x_1,y_1,z_1)+(x_2,y_2,z_2))=\psi(x_1,y_1,z_1)+\psi(x_2,y_2,z_2)$$ and
$$\psi(a(x,y,z))=a\psi(x,y,z).$$ Now, it is
$$\psi((x_1,y_1,z_1)+(x_2,y_2,z_2))=\psi(x_1+x_2,y_1+y_2,z_1+z_2)=z_1+z_2=\psi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers? How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
| Consider a Ferrer's diagram of $n$ into at most $r$ parts.
$ooooooo\\
oooooo\\
ooo\\
o$
For example this is a partition of $17$ into $4$ parts - $17=7+6+3+1$.
So in this example we know $r\ge4$. This means we can insert a first column of size exactly $r$ to get a partition of $n+r$ with exactly $r$ parts.
We also need ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Primitive Root Mod P I have been able to answer all of the parts to the question apart from part (v).
Any tips on how?
I assume I have to show that $ m=2^n $, but I am unsure as to how.
I can't imagine it is very difficult/long.
Thanks :)
| Theorem: If $a\in\mathbb Z$, $p$ is prime, $\gcd(a,p)=1$, and for all prime divisors $q$ of $p-1$ we have $a^{(p-1)/q}\not\equiv 1\pmod{p}$, then $a$ is a primitive root mod $p$.
First, if $p=2^{2^n}+1$ with $n\ge 2$, then $\gcd(5,p)=1$, because by Fermat's Little Theorem $$2^{2^n}\equiv \left(2^{2^{n-2}}\right)^4\equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
If $E\backslash C$ is relatively open in $E$ then $C$ is relatively closed in $E$ Given that $E\subset \mathbb{R}^n$ and $C\subset E$, prove that if $E\backslash C$ is relatively open in $E$ then $C$ is relatively closed in $E$
The best I've so far been able to arrive at is the following:
We have that, by definition, t... | Note that $E\cap C=C$.
So, following your deduction, if $E\setminus C=A\cap E$ for some open set $A$, then $C=A^c\cap E$ showing that $C$ is relatively closed in $E$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the value of : $\lim_{x\to\infty}\frac{\sqrt{x+\sqrt{x+\sqrt{x}}}}{x}$ I saw some resolutions here like $\sqrt{x+\sqrt{x+\sqrt{x}}}- \sqrt{x}$, but I couldn't get the point to find
$\lim_{x\to\infty}\frac{\sqrt{x+\sqrt{x+\sqrt{x}}}}{x}$.
I tried $\frac{1}{x}.(\sqrt{x+\sqrt{x+\sqrt{x}}})=\frac{\sqrt{x}}{x}\left(\s... | For large $x$ you have $\sqrt x<x$ and so
$$
\sqrt{x+\sqrt{x+\sqrt{x}}}<\sqrt{x+\sqrt{2x}}<\sqrt{3x}.
$$
Since $\sqrt{x+\sqrt{x+\sqrt{x}}}/x<\sqrt{3x}/x\to0$ when $x\to\infty$, your limit is zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
True of false: The sum of this infinite series. I'm fairly sure it is false, but I'm not quite sure about which test I should use to prove it.
$$\sum_{n=2}^\infty \ln\left(\frac{n-1}{n}\right) = -1 $$
I think using the integral test should work, but it may get kind of messy, so I'm looking for som advice. All I have to... | It is actually very easy. See that
$$f(N)=\sum_{n=2}^N\ln \left(\frac{n-1}{n}\right)=
\sum_{n=2}^N\ln \left(1-\frac{1}{n}\right)$$
Is decreasing.
Now $f(3)\simeq-1.09861>f(+\infty)$. So $f(+\infty)\neq -1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1629944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is $S^{\infty}$ contractible? Recently I was reading this post:
Unit sphere in $\mathbb{R}^\infty$ is contractible?
Then a doubt came across to me: why I can't consider the linear homotopy $H:I\times S^{\infty}\to S^{\infty}$ given by the restriction of
$$H_t= \dfrac{F_t}{|F_t|}$$
to the sphere $S^{\infty},$ where $F... | The current answer already explains why the proposed homotopy cannot work. Let me give a geometric interpretation of the two-step homotopy on the linked answer.
Trying to contract $S^\infty$ to $(1, 0, 0,\cdots)$ directly using straightline homotopy cannot possibly work: The situation is the same as that of trying to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove that if a set is nowhere dense iff the complement of the closure of the set is dense. I am able to prove the iff in the forward direction. But, I am having trouble proving the statement in other direction. I am trying to use the definition of dense, but I am not getting anywhere with it.
| Assume that complement of (clA) is dense.
Then cl[complement of (clA)] = X.
So complement of (int clA) = X.
Thus int clA is empty.
Hence A is nowhere dense.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Parallel Translation is Path Independent iff Manifold is Flat Problem. Let $M$ be a smooth Riemannian manifold and $\nabla$ be the Levi-Civita connection. Then the following are equivalent
*
*$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z\equiv 0$
*For all $p,q\in M$, parallel translation along a curv... | For the direction: $2) \Rightarrow 1)$ use this formula, which connects the curvature tensor to parallel transport along small closed loops.
$1) \Rightarrow 2):$ (which as pointed by Anthony Carapetis hols only locally):
Any flat mnaifold is locally isometric to Euclidean space, so you can transfer the parallel transpo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Is this a mistake on my part or theirs? I'm not sure if I'm the one making the mistake, or my math book. It looks like the negative sign completely disappeared.
$$\frac{3x^2}{-\sqrt{18}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3x^2\sqrt{2}}{-\sqrt{36}} = \frac{3x^2\sqrt{2}}{6} = \frac{x^2\sqrt{2}}{2}$$
Here is the orig... | Be cautious about the negative sign,
$$\frac{3x^{2}}{-\sqrt{18}} \times \frac{\sqrt{2}}{\sqrt{2}}=- \frac{3x^{2}}{\sqrt{18}} \times \frac{\sqrt{2}}{\sqrt{2}} =- \frac{3x^{2}}{3\sqrt{2}} = -\frac{x^{2}}{\sqrt{2}}$$
Maybe, there is a typo in the textbook.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$ If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$
Using the condition for common root, $$(3c-5b)(b-3a)=(c-5a)^2$$
$$3bc-9ac-5b^2+15ab=c^2+25a^... | The polynomial $x^2+3x+5$ is irreducible over $\mathbb{Q}$. Hence, if $x^2+3x+5$ and $ax^2+bx+c$, with $a,b,c\in\mathbb{Q}$, have a common root, they must be proportional. That is, $$ax^2+bx+c=a\left(x^2+3x+5\right)\,.$$
The problem would be more challenging if $x^2+3x+5$ is replaced by $x^2+3x+2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Show that the set of Hermitian matrices forms a real vector space How can I "show that the Hermitian Matrices form a real Vector Space"? I'm assuming this means the set of all Hermitian matrices.
I understand how a hermitian matrix containing complex numbers can be closed under scalar multiplication by multiplying it b... | Hint:
i give you the intuition for $2\times 2$ matrices that you can extend to the general case.
An Hermitian $2\times 2$ matrix has the form:
$$
\begin{bmatrix}
a&b\\
\bar b&c
\end{bmatrix}
$$
with: $a,c \in \mathbb{R}$ and $b\in \mathbb{C}$ ($\bar b $ is the complex conjugate of $b$).
Now you can see that, for two ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
A triple of pythagorean triples with an extra property I'm trying to prove the non-existance of three positive integers $x,y,z$ with $x\geq z$ such that\begin{align}
(x-z)^2+y^2 &\text{ is a perfect square,}\\
x^2+y^2 &\text{ is a perfect square,}\\
(x+z)^2+y^2 &\text{ is a perfect square.}
\end{align}
I failed tying t... | It is impossible that non-existance because there are infinitely many counterexamples. In fact, we must have by the Pythagorean triples
$$x-z=t^2-s^2;\space y=2ts\qquad (*)$$ $$x=t_1^2-s_1^2; \space y=2t_1s_1$$ $$x+z=t_2^2-s_2^2;\space \space y=2t_2s_2\qquad (**)$$ so we have from $(*)$ and $(**)$ $$x=\frac{t^2-s^2+t_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Calculate (or estimate) $S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$. Let $x\in\mathbb R$, $x>1$ and
$$S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$$
where $\zeta(x)$ is the Riemann zeta function. Calculate (or estimate) $S(x)$.
| What one could do fairly easy (pure heuristics) is to obtain an expansion for large $x$.
The Riemann Zeta function can be approximated in this limit by $\zeta(z)\sim 1+\frac{1}{2^z}$
Therefore our sum reads
$$
S(x)=\sum_{k=1}^{\infty}\frac{\zeta(kx)}{k!}\sim_{x\rightarrow\infty}\sum_{k=1}^{\infty}\frac{1}{k!}+\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Homogeneous coordinate rings of product of two projective varieties In Ex 3.15, chapter I of Hartshorne's "Algebraic Geometry", we have shown that $ A(X \times Y) \cong A(X) \otimes A(Y)$, when X and Y are affine varieties. Is the same statement true for projective varieties, i.e. if X and Y are projective varieties th... | Let $X \subset \mathbb{P}^n, Y \subset \mathbb{P}^m$ be projective varieties with homogeneous coordinate rings $S(X)$ and $S(Y)$ respectively. If we take the equations that define $X$, we see that they are homogeneous polynomials in $n+1$ variables. Naturally, they define an affine variety of $\mathbb{A}^{n+1}$, which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1630899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the nature of this improper integral? Consider this improper integral of first kind:
$$\int_0^{+\infty}{\frac{t\ln t}{{(t^2+1)}^{\alpha}}}\,{dt}, \quad \alpha\in\mathbb R$$
Its required to find the nature of this improper integral.
We know that the problem of this integral is only at $+\infty$ so we must study ... | If $a \le 1$, then your estimates give that the integrand is greater than
$$t^{1 - 2\alpha} \ge t^{-1}$$
and the integral is divergent.
If $a > 1$, then estimate the logarithm away with a very small power of $t$ - for example, adjusting the constants carefully yields for any $\beta > 0$ the estimate
$$\ln t < C t^{\bet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1631028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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