Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Constructing 6-fold cover of $S^1 \vee S^1$ with deck transformation group $\cong S_6$ So i'm thinking that this will be a cover space of maximum possible symmetry. Will a "necklace" of 6 circles work? Any tips appreciated
| The comments are correct that this construction is impossible, however the reason is flawed.
First of all, the action of the deck transformation group $G$ of a connected covering map $f : X \to Y$ a free action, meaning that the action of a nontrivial element of $G$ has no fixed points --- so no, a deck transformation ... | {
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How to adjust the diagonal so that a matrix is on the stability threshold? I am working on the stability of food webs, which can be represented by a Jacobian matrix showing the interaction strengths between species. I know that a matrix is locally stable if all real parts of the eigenvalues are negative, and I know how... | To be on the boundary of instability, you don't need all eigenvalues to be zero, you just need the largest occuring real part to be zero (so that the “rightmost” eigenvalue(s) is (are) on the imaginary axis).
This you can do by adding $cI$ to your matrix (where I is the identity matrix, and $c$ is a suitable constant);... | {
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"url": "https://math.stackexchange.com/questions/3163345",
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I'm facing problem with inhomogeneous equation. Let $u(x,t)$ be a function that satisfies the PDE $$u_{xx}-u_{t t} = e^x+6t,$$ $x\in \mathbb{R}, t>0,$ and the initial conditions $$u(x,0)=\sin(x) ,u_{t}(x,0)=0$$ for every $x\in \mathbb{R}.$
Then what is the value of $u(\pi/2,\pi/2)?$
I'm not getting how to solve an inh... | Let $\begin{cases}p=x+t\\q=x-t\end{cases}$ ,
Then $u_x=u_pp_x+u_qq_x=u_p+u_q$
$u_{xx}=(u_p+u_q)_x=(u_p+u_q)_pp_x+(u_p+u_q)_qq_x=u_{pp}+u_{pq}+u_{pq}+u_{qq}=u_{pp}+2u_{pq}+u_{qq}$
$u_t=u_pp_t+u_qq_t=u_p-u_q$
$u_{tt}=(u_p-u_q)_t=(u_p-u_q)_pp_t+(u_p-u_q)_qq_t=u_{pp}-u_{pq}-u_{pq}+u_{qq}=u_{pp}-2u_{pq}+u_{qq}$
$\therefore ... | {
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"timestamp": "2023-03-29T00:00:00",
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normal subgroup basis of $\mathrm{GL}(n, \mathbb{Q}_{p})$ Consider the group $G=\mathrm{GL}(n, \mathbb{Q}_p)$. This group is locally compact and totally disconnected, and we have a basis of open subgroups given by
$$
K(p^m) = \left\{ \begin{pmatrix} a&b\\c&d\end{pmatrix} \equiv \begin{pmatrix}1&0\\0&1\end{pmatrix} mod... | $\DeclareMathOperator{GL}{GL}\DeclareMathOperator{SL}{SL}\DeclareMathOperator{PSL}{PSL}$Here is another, more abstract proof. It uses some more powerful facts, but on the upside, requires pretty much no computations (if you use these facts as black boxes).
Instead of showing that $\GL_n$ does not have a basis of open n... | {
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For which $a>0$ series is convergent?
For which $a>0$ series $$\sum { \left(2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n} \right) \right)^a } $$ $(n \in \mathbb N)$ is convergent?
My try:From Taylor theorem I know that:$$a_{n}={ \left(2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{... | The term that is $o(n^{-8})$ is smaller, for all sufficiently large $n$, than $Cn^{-8}$, for at least some well-chosen value of $C$.
Making use of that $C$, you can finish the proof by noting that for sufficiently large $n$, $\frac7{72n^6} > Cn^{-8}$ since for large enough $n$, $$n^2 > \frac{72C}{7}$$.
| {
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Finding canonical names for equivalence classes Given $x,y\subset \omega$, define the equivalence relation
$$x=^*y\iff x\Delta y\text{ is finite.}$$
Let $M$ be a ctm of ZFC and $G$ an $M$-generic filter over $P$=Fn$(\omega\times\omega,2)$. Put $g=\bigcup G:\omega\times\omega\to2$ and let
$$a_i=\{n<\omega: g(i,n)=1\}$$
... | Yes, of course. Let's focus on adding just a single Cohen real $\dot a$.
If $s$ is a finite binary sequence, $\dot a^s$ would be $\{(\check n,p)\mid p\subseteq s\lor s\subseteq p\land p(n)=1\}$.
Now $[\dot a]=\{\dot a^s\mid s\in 2^{<\omega}\}^\bullet$, where $\{\dot x_i\mid i\in I\}^\bullet$ is the name given by $\{(\d... | {
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Integers $n$ satsifying $\frac{1}{\sin \frac{3\pi}{n}}=\frac{1}{\sin \frac{5\pi}{n}}$
If $\displaystyle \frac{1}{\sin \frac{3\pi}{n}}=\frac{1}{\sin \frac{5\pi}{n}},n\in \mathbb{Z}$, then number of $n$ satisfies given equation ,is
What I tried:
Let $\displaystyle \frac{\pi}{n}=x$ and equation is $\sin 5x=\sin 3x$
$\di... | Since $-n$ is a solution if $n$ is a solution, it suffices to look for solutions with $n\gt1$. Since $\sin x$ is strictly increasing for $0\le x\le\pi/2$, we cannot have $\sin(3\pi/n)=\sin(5\pi/n)$ if $n\ge10$, so it suffices to consider $2\le n\le9$.
From $\sin x=\sin(\pi-x)$, we have
$$\sin\left(5\pi\over n\right)=\s... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving an alternating infinite series to be divergent I am trying to prove $$\sum_{n=0}^\infty \frac{(-4)^{3n}}{5^{n-1}}$$ is a divergent series. I know that it increases as the series progresses regardless of sign, so it must be divergent, but im not sure how to prove it.
Usually with an alternating infinite series ... | Since $\lim_{n\to\infty}\left\lvert\frac{(-4)^{3n}}{5^{n-1}}\right\rvert=\infty$, you don't have $\lim_{n\to\infty}\frac{(-4)^{3n}}{5^{n-1}}=0$, and therefore the series diverges.
| {
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Evaluating: $\lim_{n \to 0} \prod_{\substack{i=nk \\k \in \Bbb Z_{\geq 0}}}^{2-n} \left( 2-i \right) $ How would we evaluate:
$$\lim_{n \to 0} \prod_{\substack{i=nk \\k \in \Bbb Z_{\geq 0}}}^{2-n} \left( 2-i \right) $$
Is it possible to evaluate this manually? Or do we have to make a program to get an approximation?
E... | The product can be written in an equivalent form as:
$$\Pi(n)=\prod_{k=0}^{\frac{2-n}{n}} (2-kn)=\prod_{k=0}^{\frac{2-n}{n}} -n(k-\frac{2}{n})=\prod_{k=0}^{\frac{2-n}{n}} -n\prod_{k=0}^{\frac{2-n}{n}} (k-\frac{2}{n})=$$
$$=2(-n)^{\frac{2-n}{n}} \left(\frac{n-2}{n}\right)_{\frac{2-n}{n}}$$
This may be helpful.
| {
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Can $5^n+1$ be sum of two squares? I want to determine whether or not $5^n+1$ , $n\in\mathbb{N}$ can be written as sum of two squares.
Obviously, the real problem is when $n$ is odd. I am aware of the known results about numbers written as sum if squares, but I couldn't apply them here. We can see that when $n$ is odd,... | Say $x^2 + y^2 = 5^n + 1$.
Looking mod 2, we get that $x^2 + y^2 = 0 \mod 2$, so $x$ and $y$ have the same parity, and looking mod 4, we get that $x^2 + y^2 = 2 \mod 4$, so $x$ and $y$ must both be odd.
Say $x = 2r + 1$ and $y = 2s+ 1$, so $$x^2 + y^2 = (2r + 1)^2 + (2s+1)^2 = 4(r^2 + r + s^2 + s) + 2 = 5^n + 1$$
or
$$... | {
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Proof Verification: $\epsilon(\sigma)=\epsilon(\sigma^{-1}) \, \, \forall{}\sigma\in{}S_n$ Not certain whether my proof is right, would appreciate it if I could get some feedback on it. Also, the epsilon here is the sign function of the permutation so $\epsilon=sgn$
Proof:
Since the mapping $ \, \, \epsilon:S_n\righ... | When you say $n=m-2$, that should be $n\equiv m-2\mod 2$. The value of $\epsilon(\sigma)$ doesn't tell you what that exponent is, only that it's even or odd.
In fact, nothing about this depends on that "number of inversions" formula. It's entirely a consequence of $\epsilon$ being a homomorphism to the two-element grou... | {
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Find angles in trapezoid with sides' lengths equal $15$, $7$, $7$, $8$
What are the interior angles in the trapezoid with sides whose lengths equal $15$, $7$, $7$, $8$ (sides with lengths $15$ and $7$ are parallel)?
I found this problem in elementary school's problem book. We should not use any trigonometry. Is ther... | If we slide the two slant sides together, reducing the two parallel bases by equal amounts, we end up with a triangle with sides $7$, $8$, and $8$ - the two slanted sides and the difference between the parallel sides.
The angles in this isosceles triangle are $\arccos\left(\frac{3.5}{8}\right)\approx 64^\circ$, $\arcco... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\sum_{k=0}^n {{2n + 1}\choose {2k + 1}}$ Evaluate $$ \sum_{k=0}^n {{2n + 1}\choose {2k + 1}} $$
I'm really stuck on this one, no idea how to progress. My best guess is to somehow get it into the form of $ n\choose k $ and then take that summation and work with that. Or maybe binomial theorem, but I'm very exp... | Hint:
$$2\sum_{k=0}^n\binom{2n+1}{2k+1}a^{2n-2k}b^{2k+1}=(a+b)^{2n+1}-(a-b)^{2n+1}=?$$
Set $a=b=1$
| {
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An integration that wolfram cannot help me. $$\int e^{x\sin x+\cos x}\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2x}dx$$
I noted the fact that $\frac{d(x\cos x)}{dx}=-x\sin x+\cos x$ but I cannot apply the substitution on it.
| Here is the best solution I can do
Compute the following:
\begin{align}
& \int e^{x\sin x+\cos x}\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2x}dx \\
& =\int e^{x\sin x+\cos x} \cdot x^2 \cos x dx+ \int e^{x\sin x+\cos x} \cdot \frac{-x\sin x+\cos x}{x^2\cos^2x}dx \\
\end{align}
Remark:
$$\frac{d(x\sin x+\cos x)}{dx}=x ... | {
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Idempotents and cyclic codes Let $C_1 = \langle e_1(x) \rangle$, $C_2 = \langle e_2(x) \rangle$ cyclic codes, where $e_1(x)$ and $e_2(x)$ are idempotents.
I know what cyclic codes and idempotents are, but why can one deduce the following: $C_1 \subset C_2 \Leftrightarrow e_1(x) e_2(x) = e_1(x)$?
| I believe you are talking about an idempotent $e=e(x)\in F[x]/(x^n-1)$. The fact you are speaking of is actually true for idempotents in any commutative ring with identity.
Suppose $e,f$ are two idempotents in a commutative ring $R$.
Then it is elementary to show that $(e)\cap (1-e)=\{0\}$ and $(f)\cap (1-f)=\{0\}$.
N... | {
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Solve: $\|u+v\| \le \|u\| + \|v\|$ with $\|x\| = \left( \sqrt{|x_1|} + \sqrt{|x_2|} \right)^2$ I was given the following task:
Check if $x\rightarrow \left(\sqrt{|x_1|} + \sqrt{|x_2|}\right)^2$ is a norm on $\mathbb{R}^2$.
I've already shown that
$$\|x\| \ge 0\qquad \|x\| = 0 \Leftrightarrow x = 0$$
$$\|\alpha x\| =... | Let $u(a^2,b^2)$ and $v(c^2,d^2),$ where $a$, $b$, $c$ and $d$ are positives.
Thus, we need to prove that
$$(a+b)^2+(c+d)^2\geq\left(\sqrt{a^2+c^2}+\sqrt{b^2+d^2}\right)^2$$ or
$$ab+cd\geq\sqrt{(a^2+c^2)(b^2+d^2)},$$ which can be wrong by C-S.
| {
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Leibniz convergence test to compute the limit Can we compute the limit of a series using Leibniz test?
This is the problem I am struggling with:
" Let $(a_{n})_{n \ge 1}$ be a sequence of natural numbers, $a_{n} \ge 2$ ,
Let $b_{n} = 1 - \frac{1}{a_{1}} + \frac{1}{a_{1}a_{2}} +- \dots + (-1)^n\frac{1}{a_1a_2...a_n}$ ... | Here is my solution, but I am not sure if it is correct.
Suppose that $b_n \rightarrow \frac{p}{q}$ , where $p,q \in \mathbb{Z^*}$.
For any $\epsilon > 0$ , we have $|b_n - \frac{p}{q}| < \epsilon$ , and by choosing $\epsilon := \frac{\epsilon}{|q|}$ , we get:
$|1 - \frac{1}{a_1} + \frac{1}{a_1a_2} + \dots (-1)^n\frac{... | {
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Which is the maximum value of $k^2 \binom{n}{k}$, for $k$ and $n$ integers?
Given an integer $n$, what is the maximum value that $k^2 \binom{n}{k}$ can take, for $k$ integer?
I've done the case of which $\binom{n}{k}$ is maximum, but for this one I don't see where to begin with. Any hint?
| Hint: if $f(k) = k^2 {n \choose k}$, then
$$ \frac{f(k+1)}{f(k)} = \frac{(k+1)(n-k)}{k^2} $$
When is this $> 1$ or $< 1$?
| {
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Independence of coin flips
A fair coin is tossed three times in succession. If at least one of
the tosses has resulted in Heads, what is the probability that at
least one of the tosses resulted in Tails?
My argument and answer: The coin was flipped thrice, and one of them was heads. So we have two unknown trials.... | Well, let's look at the problem. It's asking the odds of flipping any tail given that you flipped at least one head, or $P(T>0|X>0)$ Using the definition of conditional probability, we can say that $P(T>0|X>0)=P(T>0,X>0)/P(X>0)$. Then, using some identities, we have that $P(T>0,X>0)=1-P(T=0)-P(X=0)$ and $P(X>0)=1-P(X=... | {
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The direct sum of two finitely generated algebras is finitely generated Let $X$ and $Y$ be sets of variables. Let $R$ be a commutative ring. Suppose $S$ and $T$ are finitely generated $R$-algebras. Then $S\cong R[X]/I$ and $T\cong R[Y]/J$ for some $R[X]$-ideal $I$ and $R[Y]$-ideal $J$.
Why is $S\bigoplus T$ a finitely... | What do you mean by $\oplus$ when you are working with $R$-algebras, or more generally with commutative rings? See this wiki section.
If you are writing $\oplus$ to mean the coproduct (in the category of $R$-algebras), then you ought to write $\otimes_R$ instead, and your equation is correct. If you mean the product, t... | {
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Duality between Ideal and Filters on a poset I am studying Ideals and Filters on a poset, and it is clear to me that this are dual notions in the sense that reversing inclusions in the definition we can get the one from the other, or in the sense that taking complements of subsets belomging to a structure we obtain the... | There is a nice duality between filters and ideals in topology, but then we have to go via Boolean algebras and Stone spaces:
If we have a Boolean algebra (BA) $B$ (a bounded distributed and complemented lattice essentially, so with operations $\land, \lor, \lnot$ and a $0$ and $1$.) on the set $S(B)$ of its ultrafilt... | {
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Why is this map $S^2\to S^1$ nullhomotopic? I know that $\pi_2(S^1)=0$ since $S^1$ has $\mathbb{R}$ as universal cover, which is contractible. However, I have a map $S^2\to S^1$ that I can't intuitively see why it is nullhomotopic (it has to be, since otherwise it would represent a nontrivial element of $\pi_2(S^1)$).
... | Imagine poking the sphere inwards from $x=0$ and $x=1$ so that under projection to the $x$-axis it doesn't reach all the way around the interval $[0,1]$. Performing this in $\mathbb R^3$ exhibits a homotopy between two embeddings of $S^2$. From this point it should be clear that following this homotopy with the map you... | {
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What are the chances of my server giving an error given $N$ daily users? This probability question is based on a real problem: My server gives an error if it gets hit by more than $5$ requests in one second. If I have $N$ daily users, and each one sends an average of $M$ requests to the server per day (assuming each re... | The probability that a specific user sends a request in a specific second is $p=\frac{M}{86,400}$. Then the number of requests received by the server in a specific second, noted $X$, follows a binomial law, $X\sim\mathcal{B}(N,p)$.
So $\mathbb{P}\{X\geq 5\}$ gives you the probability to have an error for a specific sec... | {
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Why is $\cos(36°) = \frac{\phi}{2}$ where $\phi$ is the golden ratio? A recent question had a comment which intrigued me, so I went to Wolfram Alpha and put in $\cos(36°)$ and it was half the golden ratio? Why is $\cos(36^\circ) = \frac{\phi}{2}$? Is there a nice geometric proof? I have never heard this before and it'... | Because it's $\frac{\sin 72^\circ}{2\sin 36^\circ}=\frac{\sin 108^\circ}{2\sin 36^\circ}$, which by the sine rule, applied to a side-side-diagonal isosceles triangle of a rectangular pentagon, is $\frac{\varphi}{2}$.
| {
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Finding system of linear equations starting from parametric solution I need to find a system of two linear equations in variables $x_1$, $x_2$ and $x_3$ from a solution vector of the form $x_1=t$, $x_2=1+t$ and $x_3=2-t$, but I'm not sure where to start.
| Each individual implicit equation $ax_1+bx_2+cx_3+d=0$ represents a plane with normal $\mathbf n = (a,b,c)^T$. A line $t\mathbf v+\mathbf p_0$ that lies on this plane is perpendicular to $\mathbf n$, i.e., $\mathbf n\cdot\mathbf v=0$, which in this case produces the constraint $a+b-c=0$. Obviously, any point on the lin... | {
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OR equality constraint for binary integer program I am trying to find a way to implement an OR equality constraint in a Binary Integer Program. For example, say I want to add the following logical condition to the program:
$$x_1+x_2+x_3+x_4+x_5 = 1\; \text{OR}\; x_1+x_2+x_3+x_4+x_5 = 3$$
$$\textbf{x}\in \mathbb{B}^5$... | You can do it with one additional binary variable $y$:
$$\sum_{i=1}^5 x_i = 1 + 2y.$$
| {
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Alternating series test question. I'm reading this in my text:
So the alternating series test says:
i) is about the sequence decreasing
ii) is about the limit of the b term going to 0
I'm confused about why we need to do anything more once we find out that ii) isn't satisfied in example 2. What does it mean that we'r... | If conditions (i) and (ii) are satisfied, then you conclude that the series ${\bf converges}$.
If one of the conditions fails, then you cannot conclude that the series ${\bf diverges}$
The limit $(a_n)$ does not exist because it converges to two different values. In fact, if $n$ is even, then sequence $a_n$ converges ... | {
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"question_score": "1",
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"answer_id": 2
} |
If A is an m by n matrix, prove that the set of vectors b that are not in C(A) forms a subspace. If A is an m by n matrix, prove that the set of vectors b that are not in C(A) forms a subspace.
I would like to first understand if I am interpreting the question correctly. My understanding of this question is that I nee... | Your result is wrong. Maybe this picture can help you figure out.
Since every subspace must contain the zero vector, noted by $\underline{0}$.
We know that $C(A)$ is a subspace for $\mathbb{R}^m$, so $\underline{0}\in C(A) \subseteq \mathbb{R}^m$, that means the zero vector are inside the column space and also $\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving $\;\ln k \geq \int_{k-\frac{1}{2}}^{k+ \frac{1}{2}}\ln x dx$ I'm trying to prove $$\ln k \geq \int_{k-\frac{1}{2}}^{k+ \frac{1}{2}}\ln x dx$$
In other words, I'm trying to show why the area of the rectangle with height $\ln k$ and width $1$ bounds the area under the graph of $f(x)=\ln x$ in the interval $[k-\f... | Logarithm is a concave function, by Jensen inequality,
$$\ln E\left( U\right) \ge E\left( \ln (U)\right)$$
where $U \sim Uni\left( k-\frac12, k+\frac12\right)$.
$$\ln k \ge \int_{k-\frac12}^{k+\frac12} \ln (x)\, dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Weak convergence in $\ell_p$ I have failed to prove the following statement:
Let $1<p< \infty$. Then $f_n \to f$ weakly in $\ell_p$ if and only if $\mathrm{sup} \| f_n \|_p < \infty$ and $f_n \to f$ pointwise.
Any help will be appreciated.
| Suppose $f_n \rightharpoonup f$: use PUB to prove boundedness, and to prove the pointwise convergence, test against the standard basis vectors $e_j = (0,0, \ldots 0, 1, 0,0, \ldots)$ (the $1$ in the $j$-th slot).
Suppose boundedness and pointwise convergence. Fix any $\xi \in \ell^q$, where $q$ is the Holder conjugate ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Integration by parts for $u \in H^{1}$ and $v\in H^{1}_{0}$ Let $\Omega$ be a smoothly, open bounded domain in $\mathbb{R}^{n}$. Assume that $u\in W^{1,2}\left(\Omega\right)$ and $v\in W^{1,2}_{0}\left(\Omega\right)$. Is the integration by parts always true, that is
$$ \int_{\Omega}\left(\partial_{i}u\right) v + \left(... | The above formula above is an application of the Gauss-Green (divergence) theorem to the vector
$$
\overline{uv}=\left.
\begin{pmatrix}
uv\\
uv\\
\vdots\\
uv
\end{pmatrix}\quad\right\}\text{ $n$ rows}
$$
As a matter of fact,
$$
\begin{split}
\nabla\cdot\overline{uv}&=
\left(\frac{\partial}{\partial x_1}, \frac{\partial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding the fundamental group of a subspace of $\mathbb{R}^4$ Let $C^{\times} := \mathbb{C} \backslash \{0\}$ and $Z := \{(w, z) \in \left(\mathbb{C}^{\times}\right)^2 \ | \ w^n = z\}$ for some fixed $n \geq 1$. I'm trying to find the fundamental group of $Z$. My toolbox consists pretty much out of Seifert-van Kampen a... | We can write $Z = \{(w, w^n) \mid w \in \mathbb{C}^{\times} \}$. Let $\alpha : \mathbb{C}^{\times} \to \mathbb{C}^{\times}, \alpha(w) = w^n$. We see that $Z$ is nothing else than the graph of $\alpha$.
But for any contiunuous map $\phi : X \to Y$ between topological spaces $X,Y$ the graph $G(\phi) = \{ (x,\phi(x)) \mid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove tautology using propositional equivalence and the laws of logic determine q ∧ ( p → ¬q) → ¬p
q ∧ ( ¬p ∨ ¬q) → ¬p
(q ∧ ¬p)∨ (q ∧ ¬q) → ¬p
(q ∧ ¬p)∨ F → ¬p
i dont know how to solve this further. Kind of leaves me confused
what would be the next step
| First, a term like $P \lor F$ is equivalent to just $P$. So, as the next step you get:
$(q \land \neg p) \to \neg p$
And now rewrite this second implication just as you did the first. That is, the next step is:
$\neg (q \land \neg p) \lor \neg p$
Now do DeMorgan and you're almost there!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Partial fractions, disagreement with Wolfram Alpha On my math homework, I have this problem, and WolframAlpha says that $A=-\frac{1}{7}$ and $B=\frac{1}{7}$. However, while solving the problem on my own, I found a restriction that $A \neq -B$. How is it that this answer works? The problem in question is:
$$\dfrac{1}{x^... | Here's the full derivation: $$\frac{1}{x^2-3x-10}=\frac{A}{x+2}+\frac{B}{x-5}\implies$$ $$(x+2)(x-5)\left(\frac{1}{x^2-3x-10}\right)=(x+2)(x-5)\left(\frac{A}{x+2}+\frac{B}{x-5}\right)\implies$$ $$1=A(x-5)+B(x+2)=A(x)-5A+B(x)+2B=x(A+B)+1(2B-5A)$$
Hence we need $A+B=0\space\text{and}\space2B-5A=1$. So substitute $A=-B$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Show that $f(z)=\frac{1}{2\pi}\int\limits_0^{2\pi}f\left(\frac{e^{i\theta}+z}{1+\overline{z}e^{i\theta}}\right)d\theta$
Let $f$ be analytic on domain $\Omega$ which contains the closed unit
disk $\overline{\mathbb{D}}$. Show that
(a)
$$f(0)=\frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})d\theta$$
(b) Use part (a) t... | Consider the function $g_z(w) = \frac{z+w}{1+\overline{z}w}$ for $|z|<1$ and $|w|\leq 1$. When $|w| = 1$, observe that
$$|g_z(w)| = \left| \frac{z+w}{1+\overline{z}w} \right| = \left| \frac{z+w}{1+\overline{z}w} \right| \cdot \left| \frac{1}{\overline{w}} \right| = \left| \frac{z+w}{\overline{w} + \overline{z}} \right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Using Rolle's theorem to show $e^x=1+x$ has only one real root
Applying Rolle's Theorem, prove that the given equation has only one root:
$$e^x=1+x$$
By inspection, we can say that $x=0$ is one root of the equation. But how can we use Rolle's theorem to prove this root is unique?
| Let $f(x) = e^x - 1 - x$, and we observe that $f(0)=0$. $f$ is also obviously continuous and differentiable over the real numbers (if you wish to verify that in detail, you can do that separately).
Suppose there exists a second root $b \neq 0$ such that $f(0) = f(b) = 0$. Then there exists some $c \in (0,b)$ (or $(b,0)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3169097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
$X$=number of successes before 2nd failure in a seq of independent Bernoulli trials. pmf of $X$ and $\mathbb E[X]$
Let random variable $X$ denote the number of successes before the 2nd failure of a sequence of independent Bernoulli(p) trials. I need to describe the pmf of $X$ and calculate the expected value $\mathbb ... | Let $X_{1}$ denote the number of successes before the first failure
and let $X_{2}$ denote the number of successes between the first
failure and the second failure.
Then $X_{1}$and $X_{2}$ are independent and identically distributed
with $P\left(X_{i}=k\right)=p^{k}\left(1-p\right)$ for $i=1,2$.
Using the first hint fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3169296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to understand fibres of morphisms of schemes. Let $f:X\to Y$ be a morphism of schemes, and let $k(y)$ to be the residue field of the point $y$. The fibre of the morphism $f$ over the point $y$ is defined to be the scheme $X_y=X\times Spec(k(y))$.
It's said that $X_y$ is homeomorphic to $f^{-1}(y)$. If we consider a... | The question is of a local nature so let's assume $X=Spec(B)$ and $Y=Spec (A)$ are affine schemes. Then $f: Spec$ $ B\rightarrow Spec$ $A$ corresponds to a ring homomorphism $ g: A \rightarrow B$ . Let $y$ correspond to the prime ideal $p$ in $A$ . Then $X_y= X \times_{ Y} Spec (k(y)) = Spec ( B \otimes_A \frac { A_p}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3169457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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If $g \circ f=g$, prove that $g$ is constant Let $f:\mathbb{R} \to [0,1]$ be a monotonic function so that $|f(x) - f(y) | <|x-y|$, $\forall x, y \in \mathbb{R} $, $x\neq y$. If $g:\mathbb{R} \to \mathbb{R} $ is a continuous function and $g \circ f=g$, prove that $g$ is constant.
This problem also previously asked to pr... | Denote by $c$ the fixed point of $f$.
Let $a \in \mathbb R$ be arbitrary. Then
$$g(a)=g(f(a))=g(f^2(a))=....=g(f^n(a))=...$$
The sequence $x_n =f^n(a)$ converges to the fixed point $c$ of $f$. Therefore by continuity of $g$
$$g(a)=g(x_n) \to g(c)$$
This shows that $g(a)=g(c)$ for all $a \in \mathbb R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3169756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $a^{2x-1} = b^{1-3y}$ and $a^{3x-1} = b^{2y-2}$, show $13xy = 7x +5y -3.$ If $a^{2x-1} = b^{1-3y}$ and $a^{3x-1} = b^{2y-2}$, show $13xy = 7x +5y -3.$
I apologize in advance if this forum finds this question trivial but I am desperate for any help, will appreciate it. Thanks.
| As suggested in J. W. Tanner's comment to the question, assuming that $2x - 1 \neq 0$ and $3x - 1 \neq 0$, then taking appropriate roots of both sides gives that
$$a^{2x-1} = b^{1-3y} \; \Rightarrow a = b^{\frac{1-3y}{2x-1}} \tag{1}\label{eq1}$$
$$a^{3x-1} = b^{2y-2} \; \Rightarrow a = b^{\frac{2y-2}{3x-1}} \tag{2}\lab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3169927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to integrate a product of three terms? (includes an exponential and trig. function) Given a periodic signal with $T=2$, I need to find the complex Fourier coefficient $C_n$, using the following formula:
$$C_n = \frac 1 T \int_\frac {-T} 2^\frac T 2 f(t) \, e^{-jnwt} dt$$
where
$$f(t)=t^2 \cos(3 \pi t)$$
My integral... | In my humble opinion, I think that the easiest is to consider two integrals
$$A=\int t^2 \cos(3 \pi t) \, e^{-inwt}\, dt\qquad \text{and}\qquad B=\int t^2 \sin(3 \pi t) \, e^{-inwt} \,dt$$ and use
$$C=A+iB=\int t^2 e^{i 3 \pi t} \, e^{-inwt}\, dt=\int t^2 e^{i (3 \pi -n \omega )t}\,dt$$
To make life easier, let $$x=i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3170033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integral roots of cubic equation $x^3-27x+k=0$ The number of integers $k$ for which the equation $x^3-27x+k=0$ has atleast two distinct integer roots is
(A)$1$
(B)$2$
(C)$3 $
(D)$4$
My Attempt: The condition for cubic $x^3+ax+b=0$to have $3$ real roots happens to be $4a^3+27b^2\leq0$. But how to go about finding cond... | Suppose $x^3 - 27x + k = 0$ has distinct integer roots $a$ and $b$; then
$$
a^3 - 27a = b^3 - 27b,
$$
or
$$
a^3 - b^3 = 27(a - b).
$$
Since, by hypothesis, $a\ne b$, a factor of $a-b$ can be removed, resulting in
$$
a^2 + ab + b^2 = 27.
$$
After multiplying by $4$, this can be rearranged into
$$
(2a + b)^2 + 3b^2 = 108... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3170175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
On the integrals $\int_{-1}^0 \sqrt[2n+1]{x-\sqrt[2n+1]x} \mathrm dx$ Playing with integrals of type,
$$
I(n)=\int_{-1}^0 \sqrt[2n+1]{x-\sqrt[2n+1]x} \mathrm dx,
$$
$$
n \in \mathbb{N}
$$
I got two interesting results for the limiting cases $n=1$ and $n \to \infty$:
$$
\lim_{n \to \infty} I(n) = 1
$$
The second result... | For $n \in \mathbb{N}$ we have
\begin{align}
I (n) &= \int \limits_{-1}^0 \left[x - x^{\frac{1}{2n+1}}\right]^{\frac{1}{2n+1}} \mathrm{d} x \stackrel{x = -y}{=} \int \limits_0^1 \left[y^{\frac{1}{2n+1}} - y\right]^{\frac{1}{2n+1}} \mathrm{d} y = \int \limits_0^1 y^{\frac{1}{(2n+1)^2}}\left[1 - y^{\frac{2n}{2n+1}}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3170351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Level-set of constant rank It is well-known the regular value theorem holds. Does this generalization also hold? I can not think of a counter-example.
$f:M\to N$ be a smooth map of two smooth manifolds of arbitrary dimension $m,n$. For a level set $S:=f^{-1}(q)\neq\emptyset$, if $\text{rank}(f)_p\equiv r$ for all $p\i... | It does not hold: consider $S^1=\{(x,y)\in\mathbb{R}^2;x^2+y^2=1\}$ and the map $p_1:S^1\to\mathbb{R}$ defined by $p_1(x,y)=y$. Then $p_1^{-1}(1)=\{(0,1)\}$ and $p_1$ has rank $0$ on $(0,1)$ (if you parametrize by $\theta\mapsto(\cos(\theta,\sin(\theta))$, then$(0,1)$ has coordinate $\frac{\pi}{2}$ and
$$\frac{\partia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3170519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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The canonical open set which is equal to a set with the BP modulo meager sets is regular open (Kechris) A set $U$ in a topological space $X$ is called regular open iff $U=(\overline{U})^°$.
Exercise $(8.30)$ (Kechris, "Classical Descriptive Set Theory")
Prove that
$$U(A)=\bigcup \{U\,\text{open}\mid U\Vdash A\}$$
... | The aforementioned hint is that $U(A)\Vdash A$. As you indicate, this allows to show $(\overline{U(A)})^°\Vdash A$.
By the hint and its definition, $U(A)$ it is the largest open set $U$ such that $U\Vdash A$. On the other hand, $U(A)\subseteq (\overline{U(A)})^°$. Hence $U(A) = (\overline{U(A)})^°$ and therefore it is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3170630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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$2$-dimensional Runge-Kutta for system of polynomial ODEs I have just started getting into ODEs, and have come across the Runge-Kutta method for numerically solving them. However, in playing around with them to model hypothetical situations, I came across the equations:
$$\begin{aligned} \dot x &= x - x^2 - y\\ \dot y ... | The Runge-Kutta formulas for a system of differential equations are really the same as for a single equation, it's just that your dependent variable is a vector rather than a scalar. Write your system as
$$\dfrac{dX}{dt} = F(t, X(t))$$
where $X = (x, y)$. If you're using the classical fourth-order Runge-Kutta with s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3170716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove that $\sum_{n=0}^{\infty}\frac{x^n}{n!}$ is not uniformly convergent on $(0, +\infty)\ni x$.
Prove that $\sum_{n=0}^{\infty}\frac{x^n}{n!}$ is not uniformly
convergent on $(0, +\infty)\ni x$.
I wanted to do it by applying Cauchy's test and coming to a contradiction (i.e. $\epsilon>$ positive constant):
$$\eps... | Much easier if we can use that the sum is the exponential: the partial sum is a polynomial and the exponential grows faster than any polinomial, so for any $n$
$$\sup_{x\in(0,\infty)}\left|e^x - \sum_{k=0}^n\frac{x^k}{k!}\right| = \infty$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3170836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Calculating the Kernel, dimension of linear equations, real numbers and galois field Given these problems below, how would one calculate the result?
By intuition, I have managed to solve two of them but cannot crack the last one.
Btw, I am not sure that the approach of my intuition is the right one.
I am interested ... | All you need here is the dimension theorem: if $f:U\to V$ is any linear map, then
$$\dim\ker f+\dim\mathrm{im} f\,=\, \dim U$$
Choose bases $u_1,\dots, u_k$ for $\ker f$ and $v_1,\dots, v_r$ for $\mathrm{im} f$ and arbitrary preimages $w_j$ of $v_j$. Then show $u_1,\dots, u_k, w_1,\dots, w_r$ is a basis of $U$.
This... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3171032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$F(x) = L(1, \chi ) \log x + O(1)$ I wish to prove $$F(x) = L(1, \chi ) \log x + O(1)$$
when $A(n) = \sum_{d|n} \chi (d)$ and $F(x) = \sum_{n \leq x} \frac{A(n)}{n}$
I started of course by substituting $A(n)$ in $F(x)$, which becomes a horrible double sum.
Knowing that: $L(1, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}... | Let $\chi(n)$ be $q$-periodic and $\sum_{n=1}^q \chi(n)=0$. For any $a \le q$ $$ \sum_{n \le x, n \equiv a \bmod q} \frac1n = \frac{\log(x)}q+C(a)+O(1/x)$$
$$\sum_{n \le x} \frac{\chi(n)}{n} = \sum_{a=1}^q \chi(a)\sum_{n \le x, n \equiv a \bmod q} \frac1n = \sum_{a=1}^q \chi(a) ( \frac{\log(x)}q+C(a)+O(1/x)) \\= \sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3171117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Showing that $ \int_0^\pi \frac{\sin{(kx)}}{\sin(x)} d x = \pi $ for odd integer $k$ I am trying to show that
$$ \int_0^\pi \frac{\sin{(kx)}}{\sin(x)} d x = \pi $$
for odd integer $k$.
It seems like this could be done using multiple angle formulae, but I'm stuck.
I can get to
$$ \int_0^\pi \frac{\sin{((2N+1)x)}}{... | Note that $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$
Then $$\frac{\sin(kx)}{\sin(x)}=\frac{e^{ikx}-e^{-ikx}}{e^{ix}-e^{-ix}}=\frac{(e^{ix})^k-(e^{-ix})^k}{e^{ix}-e^{-ix}}$$
Using (n is odd)$$a^n-b^n=(a-b) \sum _{j=0}^{n-1} a^j b^{n-1-j}$$
We have $$\frac{\sin(kx)}{\sin(x)}=\sum _{j=0}^{k-1} (e^{ix})^j (e^{-ix})^{k-1-j}=\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3171236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 2
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How to show that the given matrix has non-zero determinant Given $p,q$ to be primes where $p<q$ .
Show that the following marix has non-zero determinant,
\begin{bmatrix}
1&2 & 2 & 2 &\dotso & 2\\
2&q-p+1 & 1 & 1 &\dotso & 1\\
2& 1 & q-p+1 & 1 & \dotso & 1\\2&1 & 1 & q-p+1 &\dotso & 1 \\ \dotso &\dotso & \dotso & ... | If you substract the $2$-times the first row from all other rows, then you see that the determinant of the full matrix is equal to the determinant of a matrix with diagonal entries $q-p-3$ and off-diagonal entries $-3$. This matrix can be written as
$$
(q-p)I - 3 E,
$$
where $E$ is the matrix with all entries one. The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3171488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
About the Radius of Convergence of $ \sum_{n\ge 0}a_n z^n $
Fix $ \delta>0 $ and let
$$ \Omega=\{ z\in\mathbb C:|z|<1 \}\cup\{ z\in\mathbb C:|z-1|<2\delta \} .$$
Assume that $ f(z) $ is a holomorphic function on $ \Omega $ whihc has a Taylor series expansion $ \sum_{n\ge 0}a_nz^n $ at $ z=0 $ such that $ a_n $ is ... | Note that if $|z| < 1+\delta$, \begin{align}\sum{|a_nz^n|} &\le \sum{a_n(1+\delta)^n}\\
&=\sum_{n \ge 0}\left(a_n\sum_{k=0}^{n}\binom{n}{k}{\delta}^k\right)\\
&=\sum_{k\ge 0}\left({\delta}^k\sum_{n\ge k}a_n\binom{n}{k}\right)\\
&=\sum_{k=0}^\infty\frac{f^{(k)}(1)}{k!}\,\delta^k < \infty,\end{align} hence $f(z)$ has a T... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does 1+1≠0 mean? I am using Schaum's outline of linear algebra in which there is a result that the notion of alternating and skew-symmetric bilinear forms is equivalent provided that 1+1≠0, i.e a bilinear form f satisfying:
f(v,v)=0 satisfies f(u,v)=-f(v,u) and
f(u,v)=-f(v,u) satisfies f(v,v)=0
provided that 1+... | If you are working over the field $\mathbb F_2$, then you'll have $1+1=0$. More generally, the fields for which this equality holds are called fields with characteristic $2$. So, that book assumes that we are not working over such a field.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Weak solutions to $\Delta u=f$ are in $W^{2,2}$ I believe the following statement is true.
Let $\Omega$ be a smoothly, bounded domain in $\mathbb{R}^{n}$.
The statement:
Let $u\in H^{1}(\Omega)$ so that
there exists $f\in L^{2}(\Omega) \;s.t.\int_{\Omega}\nabla u\nabla \varphi=\int_{\Omega} f\varphi, \forall \varphi... | Please check a general elliptic regularity result ($L^p$ version) on:
Dauge, Monique. Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions, 1988
Theorem 20.10 (together with the explanation of notation above the theorem), essentially we have the following regularity result:
$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Mathemathical model equation PDE I am studying models in DPE an the professor give us this problem:
$\begin{cases}u_t+au_x=f(x,t)\\ u(x,0)=g(x)\end{cases}$
I've studied the transport equation and the Burger's equation. About heat equation, a little. Hoewever, this one is not included in those kinds of problems.
I wil... | We can find the characteristics with this system of ODE's:
$\dfrac{dt}{1}=\dfrac{dx}{a}=\dfrac{du}{f(x,t)}$
From the first proportion, $x=c_1+at$. Now:
$\dfrac{dt}{1}=\dfrac{du}{f(at+c_1,t)}$ or
$f(at+c_1,t)dt=du$ Integrating,
$$u(x,t)=\int_0^tf(ar+c_1,r)dr+c_2=\int_0^tf(ar+x-at,r)dr+c_2$$
For the general solution we h... | {
"language": "en",
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"source": "stackexchange",
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Finding values for K such that the roots of the quadratic are strictly imaginary: $x^2+\left(K^2+3K-7\right)x+K$ $x^2+\left(K^2+3K-7\right)x+K$
I'd like to know the general approach needed to find out how to find solutions for K when I want the roots of this equation to have a specific property, such as strictly imagi... | Hint: The roots of a monic quadratic are strictly imaginary iff it is of the form $(x-bi)(x+bi)=x^2+b^2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Baby rudin theorem 8.8 how he found the two inequalities (56 and the last)?
Thanks!
| The last inequality is achieved as follows.
$ |Q(re^{i\theta})|$
$=|1+b_k r^k e^{ik\theta}+b_{k+1}r^{k+1}e^{i(k+1)\theta}+ \dots +b_{n}r^{n}e^{in\theta}|$
$\le|1+b_k r^k e^{ik\theta}|+|b_{k+1}r^{k+1}e^{i(k+1)\theta}|+ \dots +|b_{n}r^{n}e^{in\theta}|$
$=1-r^k |b_k|+|b_{k+1}|r^{k+1}+ \dots +|b_{n}|r^{n}$
$=1-r^k$ {$|b_k|... | {
"language": "en",
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"source": "stackexchange",
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Grillet's "Abstract Algebra", p. 148, ex. 3: Noetherian subring of a ring
Let $R$ be a Noetherian subring of a commutative ring $S$. Suppose that $S = (R\cup\{b_1,...,b_m\})$ for some $b_1,...,b_m \in S_n$. Then $S$ is Noetherian.
I'm not sure how to approach this exercise. One idea was to take an ideal $J$ of $S$ an... | Hint: try finding a surjective ring homomorphism $R[x_1,\dots,x_m]\to S$, and then use the fact that $R[x_1,\dots,x_m]$ is Noetherian by the Hilbert basis theorem
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Using a generating function for piggy bank problem A piggy bank contains 45 loonies and 25 toonies. How
many ways can the coins be divided so that Jamie gets no loonies, Julie gets no toonies but
at least 10 loonies, and Brenda gets an odd number of toonies? Use generating functions to
solve the problem. (Note that you... | To answer your first question, your method is essentially correct. If you have $a$ ways of distributing the loonies and $b$ ways of distributing the toonies, then you have $ab$ ways of doing both. See also the rule of product and this video.
To answer your biggest question, you know that each loony will go to one of tw... | {
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"source": "stackexchange",
"question_score": "1",
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probability of subset contained in dice rolls of custom dice with duplicate symbols I play a game where you roll 24 dice, 6 of 4 different colors with each color having a different set of numbers and math operations
these are the dice colors and possible symbols, there are 6 dice of each color, 24 total:
red = 0, 1, 2,... | The probability is $$\frac{635545571166992339}{2369190669160808448} \approx 0.268.$$
You can compute this by dynamic programming.
For a given multiset of symbols $S$ and a set of dice $D$, we will compute the probability $p(S,D)$ that all symbols in $S$ come out in a throw of $D$ as follows:
*
*Base case: $D$ is emp... | {
"language": "en",
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"source": "stackexchange",
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Why does $\frac{X - aZ}{Z}$ have a double pole at the point $(0 : 1 : 0)$ and not just one (Divisors)? If I have an Elliptic Curve E and the function $\frac{X - aZ}{Z}$, I would have expected the divisor to be, defining a point $P = (a,b)$ and $-P = (a,-b)$, $div(f) = [P] + [-P] - [\infty]$.
Instead the correct solutio... | A rational function $f(x) := x - a$, for example, expressed in homogeneous coordinates is $f(X/Z) = (X - aZ)/Z$ which has a single zero in the numerator and a single pole given by the denominator. Thus a simple zero and a simple pole always appear together. In general, when there are multiple zeros "up to multiplicity"... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there an orientable $3$-manifold with non-vanishing $w_2$? In the case that $M$ is a closed orientable $3$-manifold, using Wu's formula we can show $w_1(M) =0 \implies w_2(M) =0$, and so $w_3 = w_1w_2 + Sq^1 w_2 = 0$ (or you can use the fact that $\chi(M)=0$ for closed orientable manifolds with odd dimension). It ca... | All orientable three-manifolds $M$ are parallelizable. If you just want to deduce the noncompact case from the closed one, this requires little machinery.
Basically, you find first an exhaustion of $M$ with connected compact manifolds with boundary $M_k$. Then you inductively construct linearly independent vector field... | {
"language": "en",
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Second Order Differential Equation Assistance Hi Maths stack exchange!
I’m doing this question for homework,
$$y′′+4y′+4y=0$$
I managed to find the auxiliary equation.
$$y=(A+Bx)e^{-2x}$$
The issue is when I was looking at the solutions of what to do next, it said this should be the next line.
$$y'(x)=Be^{-2x}+(-2)(A+... | You have the correct idea so far, they got that line by differentiation using the chain rule. If we take your progress so far $y(x)=(A+Bx)e^x$ and differentiate it with respect to $x$ we get: $$\frac{d}{dx}(y(x))=\frac{d}{dx}(Ae^{-2x}+Bxe^{-2x})=\frac{d}{dx}(Ae^{-2x})+\frac{d}{dx}(Bxe^{-2x})$$ From this we can split it... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If A is nowhere dense, then its complement X \ A is dense in X. Let X a topological space. A is nowhere dense in
if the interior of the closure of A is empty.
I have to prove that if A is nowhere dense, then its complement X \ A is dense in X.
I tried to prove it, even looking a similar question but I could not prov... | If $A$ is nowhere dense, then $int(\bar{A})=\emptyset$. This means that it contains no (non-empty) open sets. Let $U\subset X$ be an open set. Since $A$ is nowhere dense, then $U$ is not a subset of $A$, which means - $U\cap A^c\neq\emptyset$. This means that $A^c$ meets any open set, and it is therefore dense in $X$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Graphing the function $(-2)^x$ When I wanted to graph $y=(-2)^x$ many graphing calculator apps refused to plot it. TI-Nspire CAS plotted it as shown in the first picture. I think the plot is not correct as only the envelopes should be there with no values between the envelopes as shown in the second picture and the $(-... | The graph of $f(x)=(-2)^x$ is problematic for real numbers $x$. Think about what happens when $x=\frac{1}{2}$. Then $f(x)=(-2)^{\frac 1 2}=\sqrt{-2}$. Can you see why this is a problem to graph?
The graph of $f(x)=(-2)^x$ only makes sense for integer values of $x$. Also, as zwim pointed out in the comments, your second... | {
"language": "en",
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An interesting inequality with condition If $a,b,c$ positive reals and $\frac{a}{b+c} \ge 2$ I have to prove that
$(ab+bc+ca)\left(\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2}+\frac{1}{(a+b)^2} \right)\geq \frac{49}{18}$
We may assume that $a\geq b \geq c.$ Firstly, let's show that
$\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2}+\frac{1}{... | This can be solved in a brute force way:
$$\frac{a}{b+c}\ge2\implies a=2b+2c+x$$
..where $x$ is some non-negative value. The inequality:
$$(ab+bc+ca)\left(\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2}+\frac{1}{(a+b)^2} \right)-\frac{49}{18}\ge0$$
...becomes:
$$((2b+2c+x)b+bc+c(2b+2c+x))\left(\frac{1}{(b+c)^2}+\frac{1}{(2b+3c+x)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3173875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the probability of rolling ten different, standard 6-sided dice simultaneously and obtaining a sum of 30? Find the probability of rolling ten different, standard 6-sided dice simultaneously and obtaining a sum of 30?
I started to answer this question by setting up an equation like this:
x1+x2+...+x10=30
with 0 les... | Say that $n$ represents the number of dices, $x$ the total sum and $f(n, x)$ the total number of different ways in which this sum can be obtained.
We have the following recurrence relation:
$$f(n,x)=\sum_{i=1}^6 f(n-1, x-i)\tag{1}$$
...which baciscally says that you can calculate $f$ by assuming that the first dice can... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the maximum likelihood estimator for Pareto distribution and a unbiased estimator Let $X_1,...X_n$ be a random sample from the Pareto distribution with parameters $\alpha$ and $\theta$, where $\alpha$ is known.
Find the maximum likelihood estimator for $\theta$ and say if it is unbiased, if not find an unbiased e... | You've got some notation errors and the work is a bit sloppy, but it is essentially the correct idea. You should have written
$$f(x; \alpha, \theta) = \alpha \theta^\alpha x^{-(\alpha+1)}, \quad x \ge \color{red}{\theta},$$ and $$\ell(\theta) = \log \mathcal L(\theta) = n \log \alpha + \alpha n \log \theta - (\alpha +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3174150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to simplify the the binomial coefficient in the binomial series? Use binomial series to expand the function $\frac{5}{(6+x)^3}$ as a power series.
I understand the process to get the following summation:
$\frac{5}{6^3}\sum_{n=0}^{\infty} {-3 \choose n} (\frac{x}{6})^n $
However, I am stuck on seeing what's going on... | $$\frac {(3)(4)(5)\cdots(n+2)}{n!} = \frac {(3)(4)(5)\cdots(n)(n+1)(n+2)}{(1)(2)(3)(4)(5)\cdots(n)} = \frac {(n+1)(n+2)}{(1)(2)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3174277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Opposite real number identities(why $\cos(-x)=\cos x$)
Ive been studying about opposite real number identities and Ive been stuck on this question on why $\cos(-x)=\cos x$.
Okay so if we consider that the given circle is a unit circle and triangle $pom$ and triangle $qom$ are congruent then how $\cos(-x)=\cos x$?
Acco... | A non-geometry approach would be to consider the series definition for cosine. With this, for all $x\in\mathbb R$ the series $\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)!}$ converges and
$$\cos(x)=\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)!}.$$
For every $x\in\mathbb R$, you can now trivially see that
$$\cos(-x)=\sum_{n=0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the easiest way to get: $2+ \sqrt{-121} = (2+ \sqrt{-1})^3$ I was reading the book Seventeen equations have changed the world.
At some point, while the book was talking about complex numbers, I see this equation:
$2+ \sqrt{-121} = (2+ \sqrt{-1})^3$
Even if it's easy to proof the truth of this equivalence (it is... | If it is to prove:
$$2+11i=2+i+10i=2+i+(2+i)(2+4i)=(2+i)(3+4i)=(2+i)(2+i+1+3i)=(2+i)(2+i+(2+i)(1+i))=(2+i)(2+i)(2+i).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3174607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving that $ \tan 2x \cdot (1 + \tan x) \cdot \cot x = \frac{2}{1 - \tan(x)} $
Given the following expression,
$$ \tan(2x) \cdot (1 + \tan(x)) \cdot \cot(x) $$
the exercise asks to simplify the expression and
$$ \frac{2}{1 - \tan(x)} $$
should be the simplified expression.
I have tried everything I possibly... | $$\tan(2x) (1+\tan(x)) \cot(x) = \frac{2\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}\left(\frac{\sin(x)+\cos(x)}{\cos(x)} \right)\frac{\cos(x)}{\sin(x)}$$
Simplifying you get
$$\tan(2x) (1+\tan(x)) \cot(x) = \frac{2(\sin(x)+\cos(x))\cos(x)}{(\cos(x)-\sin(x))(\cos(x)+\sin(x))} = \frac{2\cos(x)}{\cos(x)-\sin(x)}$$
i.e.
$$\tan(2x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Continuous function which is always rational Let $f:\mathbb{R} \to \mathbb{R} $ be a continuous function such that $f(x) \in \mathbb{Q} $, $\forall x\in \mathbb{R} $. Is it true that the only functions with this property are the constant functions? Intuitively, I believe it is, but I am not sure.
EDIT: I had a typo, th... | No, $f(x)$ could be any polynomial function with rational coefficients. Polynomial functions are continuous, and a polynomial with rational coefficients evaluated at a rational number is rational.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3174853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Symmetrizability of shallow water equations Consider the shallow water equation
\begin{equation}h_t+(hu)_x=0\\
(hu)_t+\left(hu^2+\frac{g}{2}h^2 \right)_x=0
\end{equation}
I want to know the entropy of this system?
I understood that if their exists a change of variable which symmetrizes the system, then system admits s... | Follow the steps in Sec 3.2 of (1). Let's subtract $u$ times the first equation to the second one. After division by $h$, we get the following conservation law for $u$:
$$
u_t + (\tfrac12 u^2 + gh)_x = 0 \, .
$$
Now, multiply the conservation law for $h$ by $\frac12 u ^2 + gh$, multiply the conservation law for $u$ by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3175027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
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Can natural deduction prove it's own rules, as my logic book says? Is there a level confusion there? I'm currently studying John Nolt's Outline of Logic ( Schaum's series).
According to the author, one can use natural deduction to prove some rules of natural deduction itself, for example the absorption rule ( chap. 4,... | Natural Deduction consists of a set of fundamental rules, which are each independent, and justified by the semantics of the connectives. The fundamental rules can be used to prove sentences which may be used to justify derived rules. Sometimes these sentences may be called Tautological Consequences (TautCon).
Here ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Magnitude of Complex Numbers Let $\alpha \neq 1$ be a complex number such that the distance from $\alpha^2$ to 1 is twice the distance from $\alpha$ to 1, while the distance from $\alpha^4$ to 1 is four times the distance from $\alpha$ to 1. Enter all possible values of $\alpha,$ separated by commas.
I have no idea how... | An alternative method:
Use the first condition to find$$|\alpha^2-1|=2|\alpha-1|\\|\alpha+1|=2\\\alpha=-1+2e^{i\theta}$$
Use the second condition to get $$|\alpha^4-1|=4|\alpha-1|\\|\alpha^3+\alpha^2+\alpha+1|=4\\|(2e^{i\theta}-1)^3+(2e^{i\theta}-1)^2+2e^{i\theta}|=4\\\left|8e^{3i\theta}-12e^{2i\theta}+6e^{i\theta}-1+4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3175366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Addition formula for elliptic integral of second kind Let $k\in(0,1)$ and the incomplete elliptic integral integral $E(u, k) $ be defined by $$E(u, k) =\int_{0}^{u}\operatorname {dn} ^2(t,k)\,dt\tag{1}$$ where $\operatorname {dn} (u, k) $ represents one of the Jacobian elliptic functions. When the value of $k$ is evide... | On searching further in Fundamenta Nova I found the key to the problem. Not only Jacobi found the Fourier series for elliptic functions, but he found such series for their integer powers also using purely algebraic approach. Here one needs the following Fourier series for $\operatorname {dn} ^2(u,k)$ $$\left(\frac{2K}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3175504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can a Cauchy sequence converge for one metric while not converging for another? Is there an easy example of one and the same space $X$ with two different metrics $d$ and $e$ such that one and the same sequence $\{x_n\}$ is a Cauchy sequence for both metrics, but converges only for one of them?
| You can always have some artificial example where you just "move the limit elsewhere". For example, let $X=\mathbb{R_{\ge 0}}$, $d_1$ be the Euclidean metric and $d_2(x, y)=|\hat{x}-\hat{y}|$, where $ \hat{x}=-1$ if $x=0$ and $ \hat{x}=x$ otherwise. Then the sequence $x_n=\frac 1n$ converges in $(X, d_1)$ but not in $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3175604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Boundary condition inserted in the local PDE Let us consider the following problem:
$$
\begin{align}
-&u_{xx}=0&&\forall x\in(0,L)&&\tag{1}\\
&u(0)=0\tag{2}\\
&u_x(L)=\alpha\tag{3}
\end{align}
$$
It is possible to insert (3) in (1) as follows:
$$
\begin{align}
-&u_{xx}=\alpha\delta (x-L)&&\forall x\in(0,L]\tag{4}\\
&u(... | I am suggesting a solution below (but I am not convinced). From (1), (2) and (3), it is clear that the sought solution is $u(x)=\alpha x$. Let us try to solve (4), (5) and (6) in the sense of distributions. Integrating (4) twice yields:
$$-u(x)=ax+b+\alpha (x-L)H(x-L)$$
Condition (5) implies $b=0$ and condition (6) imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3175747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Determine the real numbers $a$, $b$, $c$ such that $1$, $\frac1{1+\omega}$ and $\frac1{1+\omega^*}$ are zeroes of the polynomial $p(z)=z^3+az^2+bz+c$ I am stuck on this question:
Let $1$, $\omega$ and $\omega^*$ be the cube root of unity.
a. Show that $\dfrac1{1+\omega}=-\omega$ and $\dfrac1{1+\omega^*}=-\omega^*$.
b... | For those three numbers to be roots of the cubic equation means that if you set $z$ equal to any one of them, then the cubic is zero. Therefore you can write $$z^3+az^2+bz+c\equiv(z-1)\left(z-\frac1{1+\omega}\right)\left(z-\frac1{1+\omega^*}\right)$$ To determine $a,b,c$, simply multiply this out and equate the coeffic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3175872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the symmetrical matrix $A$ so that $Q(\vec x) = \vec x^TA \vec x$ $Q(\vec x) = x_1^2+x_1x_2+x_2^2$
The matrix $A=\begin{bmatrix}1 & 0.5 \\ 0.5 & 1\end{bmatrix}$ seems to do the job. But what's the general procedure for finding a solution?
I can just think of setting it up like this for more clarity:
$\begin{bmatri... | $A$ is called the matrix associated with the quadratic form $Q$. The general procedure is rather simple: put the coefficients of $x_i^2$ in the diagonal $a_{ii}$ and divide the coefficient of $x_{ij}$ in $2$, writing it twice in $A$: once in $a_{ij}$ and once in $a_{ji}$.
In your example, the coefficient of $x_1^2$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3176052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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how to check a propriety using r studio I have to check that this propriety
$Z \sim N(0,1)$ and $U\sim \chi ^{2}(10)$ then $ Z/\sqrt{U/10} \sim T(10)$
is true using r studio if anyone can help , much appreciate
| One approach could be simulation of thousands of values:
*
*Simulate $Z$ using rnorm
*Simulate $U$ using rchisq
*Do the division $Y = Z / \sqrt{U / 10}$
*Simulate the same number of $T$ from the hypothesised $t$-distribution using rt
*Sort $Y$ and $T$ and plot them against each other - you want to see a diag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3176151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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What does $*$ mean in equivalence relations? The notation of "$*$" started being used in my proof textbook in the section of equivalence relations and partitions yet it never once said what it means.
An example from the textbook:
Let $\mathbb{Z}^* = \mathbb{Z} - \{0\}.$ Define the relation on $\mathbb{Z} \times \math... | In an algebraic context, many authors use $A^*$ to denote the set $A$ without the zero element. In your specific example, the author uses $\mathbb Z^*$ to denote the set $\mathbb Z$ without $0$, i.e. $\mathbb Z-\{0\}$. So, $*$ is just used in a context of notation and does not denote any particular operation.
In a sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3176322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f$ is continuous at $a$ and $f' < 0$ on $(b, a)$, then $a$ is a minimum on $(b, a)$. Could you please verify my proof? I feel like it's terribly overwrought and I'm missing a much simpler explanation. (It may even be invalid, see the bold).
If $f$ is continuous at $a$ and $f' < 0$ on $(b, a)$, then $a$ is a minimu... | You have $f(a)<f(x)+\epsilon$ and $f(x)<f(y).$ This does NOT imply $f(a)<f(y). $
(1). Suppose $c\in (b,a)$ and $f(c)< f(a).$
Let $e= (f(a)-f(c))/2.$ Let $d\in (c,a)$ such that $f(d)-f(a)>-e.$ We know that $d$ exists because $f$ is continuous at $a.$ By the MVT there exists $d'\in (c,d)$ such that $$f'(d')=\frac {f(d)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3176430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $B$ satisfies $A B = B A^{-1}$, show that $B^2$ is diagonalisable
Let $A$ be a $n \times n$ non-singular matrix having distinct eigenvalues. If $B$ is a matrix satisfying $A B = B A^{-1}$, show that $B^2$ is diagonalisable.
Answer:
Let $\lambda_i, \ i=1,2,3, \cdots, n$ be the $n$ distinct eigenvalues.
Now, $AB=... | We can show that $AB^2=ABABA=B^2A$. So $B^2$ and $A$ commute. Then the claim follows from these duplicates (replacing $B$ by $B^2$):
$AB=BA$. Prove $B$ is diagonalizable.
If $AB=BA$, show that $B$ is diagonalizable.
Indeed, we have $AB^2=(AB)B=ABABA$ and $B^2A=B(BA)=ABABA$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3176751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Integral as infimum of integrals I am trying to understand if the following formula holds. I cannot prove it but cannot find a counterexample either.
For $\mu$ a probability measure on $\mathbb{R}^d$ and $p \geqslant 1$ does it hold that
$\int |x|^p d\mu(x) = \inf \limits_{y \in \mathbb{R}^d } \int |x+y|^p d\mu(x)$ ?... | Not always. For example, if $\mu$ is a Dirac measure at a point $x \neq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3176876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the last root of $p(x) = x^5 + a_3 x^3 + a_2 x^2 + a_1x + a_0$ given that...
The polynomial $p(x) = x^5 + a_3 x^3 + a_2 x^2 + a_1x + a_0$ has real coefficients and has 2
roots of $x = -3$, and two roots of $x=4$. What is the last root, and
how many times does it occur?
At first I expanded $(x+3)^2(x-4)^2$... | If $(x-r)$ is a repeated root of $p(x)$ then $p(r) =p'(r) =0$.
So $p(-3)=p(4)=p'(-3)=p'(4)=0$, allowing you to form a system of four linear equations in terms of four unknowns. Solve for the coefficients and find the last root (which has to be a single root as the polynomial is quintic).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Prove atleast one of the equations $x^2+b_1x+c_1=0$ and $x^2+b_2x+c_2=0$ has real roots if $b_1b_2=2(c_1+c_2)$
If $b_1b_2=2(c_1+c_2)$, then prove that atleast one of the equations $x^2+b_1x+c_1=0$ and $x^2+b_2x+c_2=0$ has real roots.
$$
\Delta_1.\Delta_2=(b_1^2-4c_1)(b_2^2-4c_2)=b_1^2b_2^2-4b_2^2c_1-4b_1^2c_2+16c_1c_... | We can assume both $c_1,c_2$ greater than zero.
$b_1b_2=2(c_1+c_2) = 4\dfrac{(c_1+c_2)}{2}$
$b^2_1b^2_2=16\dfrac{(c_1+c_2)^2}{4}\geq 16c_1c_2$
$b^2_1b^2_2 \geq 4c_14c_2$
So if $b^2_1 \lt 4c_1$ , then $b^2_2 \gt 4c_2$ and vice versa.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Evaluate $\lim_{(x,y) \to (0,0), x+y \neq 0}{\frac{\ln(1-x-y)}{x+y} } $ This was the question of a test. My question is if my attempt to solve it is correct, and if it is, why is it correct.
$$\lim_{(x,y) \to (0,0), x+y \neq 0}{\frac{\ln(1-x-y)}{x+y} } $$
My attempt:
Let $\xi = -x-y $. Then $\xi \to 0$ whenever $(x,y)... | We want to show that $\forall \epsilon \gt 0 : \exists \delta \gt 0 : ||(x,y)||< \delta \implies |\frac{\ln(1-x-y)}{x+y}+1| \lt \epsilon$.
Fix $\epsilon \gt 0$. We know that $\lim_{\phi \to 0} \frac{ln(1+\phi)}{-\phi} = -1$. So there's $\delta_1 \gt 0$ such that $|\phi| \lt\delta_1 \implies |\frac{ln(1+\phi)}{-\phi}+1|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Combinatorics problems that can be solved via infinite descent I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a problem(s) from combinatorics and/or any other field of mathema... | What about this one :
Let $(a,b,c)$ in $\mathbb{N}$ such that $(a^2+b^2)/(1+ab) =c $
Prove that $c = p^2$ with $p \in \mathbb{N}$
I don't have a proof though...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 8,
"answer_id": 5
} |
Algebra. Solving for a given gamma function: $\ln L = n \ln(\Gamma(a+1)) - n \ln (\Gamma(a)) + (a-1) \sum_{i=1}^{n} \ln x_i$ $\ln L = n \ln(\Gamma(a+1)) - n \ln (\Gamma(a)) + (a-1) \sum_{i=1}^{n} \ln x_i$
so given this I want to solve the derivative for $a$ then solve for $a$, $\ln L = 0$
$0 = \frac{n(\Gamma(a+1)')}{\G... | It's simply that
$$ \frac{\Gamma(a + 1)}{\Gamma(a)} = a,$$
by a standard property of the Gamma function.
So
$$ \ln L = n \ln a + (a - 1) \sum_{i=1}^n \ln x_i.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
union of non-regular language and finite language Is the union of a non-regular language and a finite language necessarily non-regular?
My suspicion is that it is, and I am yet to think of a counterexample, but am not sure how one might set out a proof.
| Say $L_1$ is nonregular and $L_2$ is finite and so regular. Note that $L_1\cap L_2$ is also regular.
If $L_1\cup L_2$ were regular then we would have $$L_1=((L_1\cup L_2)-L_2)\cup (L_1\cap L_2)$$ so that $L_1$ is also regular. Therefore $L_1\cup L_2$ must be irregular.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does the degree of a polynomial give the number of roots? I am aware of the fundamental theorem of algebra, i.e., the degree of a polynomial is the number of roots of the polynomial. For example, $x^2 - 9 = 0$ would have two solutions: $x=3$ and $x=-3$. However, sometimes I come across quadratic polynomials that only h... | Two main things overlooked as far as I can tell:
*
*Multiplicities, the number of times a root occurs.
*Complex roots, $x^2+1=0$ has no real roots, but two complex roots i, and -i.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why is the total differential divided by the norm of h bounded? I'm trying to proof the product rule for functions $f: R^{n} \to R$. There is already a good thread on this - total differential of $f+g$, $fg$ and $\frac fg$.
However, I am not sure about the last step, i.e. showing that
$$\lim \limits_{h \to 0} \frac{(f... | I have given this another thought and have come up with a solution.
Since $dg_{x}(h)$ is a linear map on a finite-dimensional vector space, it is a bounded linear operator which means that $\lvert dg_{x}(h)\rvert<C \|h\|, C \in \mathbb{R}$ for all $h$ (see https://en.wikipedia.org/wiki/Discontinuous_linear_map).
Theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$8^n-3^n$ Divisible by 5 - Proof Verification. Statement: $\frac{8^k-3^k}{5}=M, M\in\mathbb{N}$
Base case: $P(1): \frac{8-3}{5}=1\in\mathbb{N}$
Assume $P(n): \frac{8^n-3^n}{5}=N$
Then, $P(n+1)=8^{n+1}-3^{n+1}=5K$, where $K$ is in terms of $M$
Writing LHS in terms of $N$:
$8^n-3^n=5N \to 8\cdot8^n-8*3^n=40N$
$8^{n+1}-... | Yes, your proof is correct. Below I explain how to view the arithmetical essence of the matter more conceptually as the result of a product rule, first using congruences, and later using bare divisibility (in case you don't know congruences).
Conceptually the induction follows very simply by multiplying the first two ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3177988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Extending $\|H^{\frac{1}{2}}XK^{\frac{1}{2}}\|\leq\frac{1}{2}\|HX+XK\|$ from matrices to operators I saw in some literature that many author works in finite dimensional (matrix) is because it can be extended into infinite dimensional (operator). The case is as follows:
If the following inequality
$$\|H^{\frac{1}{2}}XK... | I dont'think you are on the right track. The limit $\lim_nP_n=I$ occurs in the strong operator topology (and other weak topologies on $B(H)$) but not in the norm topology.
If you are looking for the particular inequality mentioned in your question, the usual proof works the same for operators, you gain nothing by goin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3178129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Partition of space I have a problem with the following exercise:
Let $\left(A_{k}\right)_{k=1...n}$ be a sequence of subsets of space $\Omega$. Introduce the notation $A^{0} = \Omega \setminus A$ and $A^{1}=A$.
For $\epsilon \in \{0,1\}^{n}$, we put $$A_{\epsilon} = \bigcap^{n}_{k=1} A_{k}^{\epsilon_{k}}.$$
1. Show tha... | An attempt with a direct computation was a step in the right direction. I suspect, however, you may have tried to do the computation for the general case. Generally, what is recommended is first to try the computation on elementary special cases.
So, first, let
$$n = 2, \quad \Omega = \{ \omega_1, \omega_2, \omega_3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3178234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$. I believe that this may require the use of Fermat's Little Theorem. I rewrote it as $(a+1)^p - a^p \equiv 1 \pmod{p}$ because the right-hand side looks similar to Fermat's Little Theorem, but I was unable to figure out how I can get the left-han... | Hint 1:
$$(a+1)^p=\sum_{i=0}^{p}a^i\binom{p}{i}$$
Hint 2:
What is $\binom{p}{i}~\text{mod}~p$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3178374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
How is the relation "the smallest element is the same" reflexive? Let $\mathcal{X}$ be the set of all nonempty subsets of the set $\{1,2,3,...,10\}$. Define the relation $\mathcal{R}$ on $\mathcal{X}$ by: $\forall A, B \in \mathcal{X}, A \mathcal{R} B$ iff the smallest element of $A$ is equal to the smallest element of... | Why are you testing reflexivity by looking at two different elements of $\mathcal{X}$? The definition of reflexivity says that a relation is reflexive iff each element of $\mathcal X$ is in relation with itself.
To check whether $\mathcal R$ is reflexive, just take one element of $\mathcal X$, let's call it $x$. Then c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3178532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Finding the equivalence classes of a relation Define a relation on the set of all real numbers $x, y \in \mathbb{R}$:
$x ≃ y$ if and only if $x − y \in \mathbb{Z}$
Prove that this is an equivalence relation, and find the equivalence class of the number $1/3$.
I proved that the relation is:
reflexive
$$x-x = 0 \in \m... | Opps, carefull: $$x-y\ne y-x$$
You should write: if $x\sim y$ then $x-y\in Z$ so $-(x-y) = y-x\in Z$ so $y\sim x$.
Also, if $x\sim y$ and $y\sim z$ then $x-y,y-z\in Z$, so $(x-y)+(y-z) \in Z$ so $x-z\in Z$ so $x\sim z$.
And equivance class is $$Z+{1\over 3} = \{...-{5\over 3},-{2\over 3},{1\over 3}, {4\over 3},{7\over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3178651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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