Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Winding number of a point outside the curve is 0 I've been looking for the answer to the following question for a little while now:
Let $γ$ be a closed (C1-)curve whose image is contained in ${z: |z| < R}$ for some $R > 0$. Show that for any $z$ with $|z| > R$ we have $\operatorname{Ind}(γ,z) = 0$.
I think I am suppose... | $\mathrm{Ind}(\gamma,z_0) = \frac{1}{2\pi i} \int_{\gamma} \frac{1}{z-z_0} \textrm{d}z$.
Since $\left|z_0\right| > R$, the function $z \mapsto \frac{1}{z-z_0}$ is holomorphic on $\{z : \left|z\right| < R\}$ and the result follows by Cauchy's Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/720418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving Lagrange's Theorem I had no idea how to start the proof so I cheated and looked it up, and the proof that I can understand uses cosets. How do you know that you should start with cosets to perform this proof? I spent about half the exam time trying to think of ideas and ways to try and prove that the order of a... | Proving on demand under time pressure is difficult. It just is. It isn't a realistic reflection of mathematics research, and I hope you are cutting yourself some slack.
I can't speak to your specific question because when I first learned group theory, the relevant section of the book was called "Cosets and the Theorem ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How can I solve the equation $n\log_2n = 10^6$? I was solving some exercises and the most of them I found the algebraic properties on the web.
But in the equation $n\log_2 n = 10^6$ I have no idea, I tried several ways to solve the equation and none of them worked.
Thanks!
| As said in previous answers and comments, equations such as $$n\log_a n = b$$ have no elemental solutions. Only Lambert function provides a solution which is (for the algebraic case)
$$n=\frac{b \log (a)}{W(b \log (a))}$$ For large values of $x$, $$W(x) \simeq \log (x)-\log (\log (x))+\frac{\log (\log (x))}{\log (x)}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Definite integral by u-substitution (1/u^2), u given $$\int_{-3}^0 \frac{-8x}{(2x^2+3)^2}dx; u=2x^2+3$$
I need help solving this integral -- I'm completely bewildered. I've attempted it many times already and I don't know what I am doing wrong in my work, and I seem to be having the same problem with other definite int... | Here specifically is what you did wrong. Because you changed the limits of integration from $(-3, 0)$ to $(21, 3)$, therefore once you found the antiderivative $-{1\over u}$, you should have just plugged in $-{1\over 21}$ and $-{1\over 3}$, not $-{1\over 2(21)^3+3}$ and $-{1\over 2(3)^3+3}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/720695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding the values of $A,B,C,D,E,F,G,H,J$ Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and
$$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$
$$C = B + 1$$
$$H = G + 3$$
find (edit: without a calculator) $A,B,C,D,E,F,G,H,J$
I could only deduce that $D\ge E$, from the ... | We have that $J|A$, and that $\frac{A}{J}|10$. So let's first consider the possible divisors of $10$, which are $1, 2, 5$. Clearly, $\frac{A}{J} \neq 5$ is the most likely option, based on the possible values. So $\frac{A}{J} = 2$. How many ways can we get this? Consider pairs $(A, J)$. We have $(2, 1)$, $(4, 2)$, $(8,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Rationalization of $\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$
Question:
$$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$ equals:
My approach:
I tried to rationalize the denominator by multiplying it by $\frac{\sqrt{2}-\sqrt{3}-\sqrt{5}}{\sqrt{2}-\sqrt{3}-\sqrt{5}}$. And got the result to be (after a long calculat... | What I would do is multiply by the first term plus the conjugate of the last two terms. I have coloured the important parts of the following expression to make it easier to understand.
$$\frac{2\sqrt{6}}{\color{green}{\sqrt{2}+\sqrt{3}+}\color{red}{\sqrt{5}}}\cdot\frac{\color{green}{\sqrt{2}+\sqrt{3}-}\color{red}{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Why do we require radians in calculus? I think this is just something I've grown used to but can't remember any proof.
When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?
|
Radians make it possible to relate a linear measure and an angle
measure. A unit circle is a circle whose radius is one unit. The one
unit radius is the same as one unit along the circumference. Wrap a
number line counter-clockwise around a unit circle starting with zero
at (1, 0). The length of the arc subten... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 11,
"answer_id": 2
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The first odd multiple of a number in a given range As a part of a programming problem I was solving, we were required to find the first offset of a range at which the number is a odd multiple of another number.
For e.g: Take the range $100$ to $120$. Only the odd multiples here are taken for the offsets - so offset 0 ... | Most programming languages have a "mod" operator to calculate the remainder of a division of two integers. Therefore let's assume that $L$ and $n$ are positive integers and let's define the numbers $x$ and $y$ by
$$\begin{align}
x&:=(L+n-1)\mod 2n\\
y&:=L+2n-1-x
\end{align}$$
We see that $L+n-1-x$ is an even multiple o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's. The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order Runge-Kutta method, so I have the ... | Although this answer contains the same content as Amzoti's answer, I think it's worthwhile to see it another way.
In general consider if you had $m$ first-order ODE's (after appropriate decomposition). The system looks like
\begin{align*}
\frac{d y_1}{d x} &= f_1(x, y_1, \ldots, y_m) \\
\frac{d y_2}{d x} &= f_2(x, y_1,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 3,
"answer_id": 1
} |
Question on a modified definition of divergence. I've been looking at this question for a few days but I still can't fully understand it. I know that divergence is defined as flux per unit volume, which corresponds to the same limit you see below but multiplied by $3/(4\pi e^3)$, which is 1 divided by the volume of the... | It all depends on the definition. In my cursus divegence is differential operator $$\mathrm{div} \,\mathbf F=\nabla\cdot \mathbf F.$$ Then we can via Gauss theorem make a connection of flux through surface with integral of divergence in the volume bounded by that surface (mathematically, we should talk about manifolds... | {
"language": "en",
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"question_score": "1",
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Newton's Method for estimating square roots. Sometime ago I wrote a program that used Newtons Method and derivatives to approximate unknown square roots (say $\sqrt 5$) from known square roots like $\sqrt 4$.I have since lost the calculator and the book I got the equation from.
Edit Researched a bit let me see if I h... | To find a square root of $a$ using Newton's Method, we can write:
$$f(x) = x^2 - a$$
This is because the roots would be:
$$f(x) = x^2 - a = 0 \implies x^2 = a \implies x = \pm ~ \sqrt{a}$$
Apply Newton's iteration:
$$x_{n+1} = x_n - \dfrac{f(x)}{f'(x)} = x_n - \dfrac{x^2-a}{2x}$$
Select an $x_0$ and iterate away.
You c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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1-manifold is orientable I am trying to classify all compact 1-manifolds. I believe I can do it once I can show every 1-manifold is orientable. I have tried to show prove this a bunch of ways, but I can't get anywhere.
Please help,
Note, I am NOT assuming that I already know the only such manifolds are [0,1] or $S^1$. ... | If you've already classified orientable $1$-manifolds, then you know that the only connected ones (without boundary) are $\mathbb R$ and $\mathbb S^1$. Now suppose $M$ is a connected, nonorientable $1$-manifold, and let $\pi\colon \widetilde M\to M$ be its universal covering. Then $\widetilde M$ is orientable and simp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 0
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What to do when the second derivative test fails? What do we do when the second derivative test fails?
For example, I'm asked to find all the critical points of the function
$$f(x,y)=x^{2013}−y^{2013}$$
And determine the nature of the critical points.
The critical point that I have found is at $(0,0)$, but I'm unable... | Hint:
*
*Take into consideration higher-order derivatives.
*Note the parity of the first non-zero derivative.
*What are the similarities among $x^3, x^5, x^7,\ldots$ and similarities among $x^2,x^4,x^6,\ldots$ (e.g. how the graphs would look like, and what is the parity of the first non-zero derivative)?
*You ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
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Showing the sequence converges to the square root For any $a > 0$, I have to show the sequence $x_{n+1}$ $=$ $ \frac 12$($x_n+ $ $ \frac {a} {x_n}$)
converges to the square root of $a$ for any $x_1>0$.
If I assume the limit exists ( denoted by $x$) then,
$x$ $=$ $ \frac 12$($x+ $ $ \frac {a} {x}$) can be solved to $x... | As mentioned in the comments, we need to show that the sequence is monotonic and bounded.
First, we observe that
$$
x_n-x_{n+1}=x_n-\frac12\Bigl(x_n+\frac a{x_n}\Bigr)=\frac1{2x_n}(x_n^2-a).
$$
Secondly, we obtain that
\begin{align*}
x_n^2-a
&=\frac14\Bigl(x_{n-1}+\frac a{x_{n-1}}\Bigr)^2-a\\
&=\frac{x_{n-1}^2}4-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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If $E/F$ is algebraic and every $f\in F[X]$ has a root in $E$, why is $E$ algebraically closed? Suppose $E/F$ is an algebraic extension, where every polynomial over $F$ has a root in $E$. It's not clear to me why $E$ is actually algebraically closed.
I attempted the following, but I don't think it's correct:
I let $f$... | Note: feel free to ignore the warzone in the comments; it's not really relevant anymore.
If $F$ is perfect, we can proceed like this. Let $f$ be a polynomial with coefficients in $F$. Let $K/F$ be a splitting field for $f$. Then $K=F(\alpha)$ for some $\alpha \in K$. Let $g$ be the minimal polynomial of $\alpha$ over $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 0
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How to solve quadratic function with degree higher than two? I am struggling to solve the function $z^4 - 6z^2 + 25 = 0$ mostly because it has a degree of $4$. This is my solution so far:
Let $y = z^2 \Longrightarrow y^2 - 6y + 25 = 0$.
Now when we solve for y we get: $y=3 \pm 4i$.
So $z^2 = 3 \pm 4i$. Consequently $z ... | You are very close to the answer. Just put a plus or minus in front of the solution and you have your complete answer. It's always the simple things.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/721725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Absolute value of complex exponential Can somebody explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook says.)
For example:
$$|e^{-2i}|=1, i=\sqrt {-1}$$
| I would like to provide an intuitive understanding, in addition to the previous excellent answers.
Recall that a complex number in Euler’s form can be expressed as $r e^{i \theta}$, in this case, the modulus is 1 and the argument is -2. Graphically, we can visualize complex numbers of modulus 1 as points on a unit circ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 7,
"answer_id": 6
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Vector valued function: What part of the train is always moving backward? So I found out that on a train some part of the wheel will always be moving backward. Thinking about it in terms of a space curve, its the section of the path drawn out that drops below the x axis that corresponds to the part moving backward. Is ... | This does not require a very difficult computation. Every point of the wheel is subject to two motions: the global motion of the train (which is the same for all points) and the rotational relative motion due to the turning of the wheel. For the latter, the horizontal component of the velocity is simply proportional th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
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Eigenvalues and polynomials Hey I'm stuck on this question, I'll be glad to get some help.
$A$ is a matrix, $f(x)$ is a polynomial such that $f(A)=0$.
Show that every eigenvalue of $A$ is a root of $f$.
Well, I thought of something but I got stuck: we know that if $t$ is an eigenvalue of $A$, then $f(t)$ is an eigenval... | Assuming $A$ is diagonalizable, you can write it as $A=PDP^{-1}$ and transcribe your equation into
$f(A)=P( \rm{diag\,}{f(\lambda)}) P^{-1}$
If $f(A)$ is to be zero, and $P$ is nonsingular, then the diagonal matrix on the right must be zero, which explicitly states that all the eigenvalues are roots of $f$.
However, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/722025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question on double dual of C*-algebra Let $A, B$ be the C*-algebra. Assume $A$ is nonunital, $B$ is unital and $\phi: A \rightarrow B$ is a contractive completely positive map.
Then we consider the double adjoint map $\phi^{**}: A^{**}\rightarrow B^{**}$. Identifying double duals with enveloping von Neumann algebras... | The key observation is that the identification between $A^{**}$ and the enveloping von Neumann algebra preserves positivity. So, if $\alpha\in A^{**}_+$, we can find a net $\{a_n\}\in A_+$ with $a_n\to\alpha$ in the $w^*$-topology (every $a\in A''_+$ is a weak-limit of elements in $A_+$).
Now let $f\in B^*_+$. Then
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/722100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing? How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing?
My idea:
*
*$n^2<(n+1)^2 /+1$
*$n^2+1<(n+1)^2+1/ \sqrt{}$
*$\sqrt{n^2+1}<\sqrt{(n+1)^2+1}/+n$
*$\sqrt{n^2+1}+n<\sqrt{(n+1)^2+1}+n$
And now I'm stuck... | Steps:
1) The sequence is decreasing if the denominators are increasing
2) $\sqrt{n^2+1}+n$ is increasing if both $\sqrt{n^2+1}$ and $n$ are increasing
3) $n$ is increasing. $\sqrt{n^2+1}>n$ is also increasing. Q.E.D.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/722171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Want to show that a solution of some ODE is bounded Suppose that $u(t)$ satisfies the differential equation
$$\dot{u}(t)=a(t)[u(t)-\sin(u(t))]+b(t),\;u(0)=u_0$$
for all $t\in\mathbb R$. In addition suppose that $a,b$ are continuous integrable on $\mathbb R$. Now I want to show that $u(t)$ remains bounded on whole $\... | Multiply both sides by $u(t)$ to get
$$\frac12 \frac d{dt} |u|^2 = a u [u-\sin(u)]+ u b \le (|a| + |b|) (|u|+1)^2 \le 2(|a| + |b|)(|u|^2+1).$$
Divide both sides by $2(|u|^2+1)$ to get
$$ \frac14 \frac d{dt} \log(|u|^2+1) \le |a| + |b| .$$
Integrate from $t = 0$ to $t = T$ to get
$$ |u(T)|^2 + 1 \le (|u_0|^2+1) \exp\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/722277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Dominated convergence under weaker hypothesis Let $f_n,\,n\in\mathbb{N}$ be a sequence of real integrable functions, $f_n\to f$ pointwise as $n\to\infty$.
The dominated convergence theorem states that if there exists $g\in L^1$ such that $|f_n(x)|\leq g(x)$ for all $n,x$, then
$$ \int f_n(x)\,dx \to \int f(x)\,dx \quad... | The answer to my question is clearly negative, as Ambram Lipman showed.
Anyway the answer become positive if one improves the hypothesis. Precisely assume:
$\bullet$ the integral is done over a bounded domain (or in general against a finite measure $\mu$);
$\bullet$ the sequence $(f_n)_n$ is uniformly bounded not only ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Characters of a faithful irreducible Module for an element in the centre Basically here is my questions. We have a character $\chi$ which is faithful and irreducible of a group $G$. we have an element $g$ which i needs to show belongs to the centre $Z(G)$, i.e. $gh=hg$ for all $g$ in $G$, If and only If $|\chi(g)|=\chi... | In general, if $\chi \in Irr(G)$, then $Z(G/ker(\chi))=Z(\chi)/ker(\chi)$. Here $Z(\chi)=\{g \in G: |\chi(g)|=\chi(1)\}$ and $ker(\chi)=\{g \in G: \chi(g)=\chi(1)\}$. Faithulness of an irreducible character means $ker(\chi)=1$. See also I.M. Isaacs, Character Theory of Finite Groups Lemma (2.27).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/722467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Solution to the "cubic" Helmholtz equation What is known about the solutions of the differential equation in three-dimensions
$$
\nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3)
$$
Without the cubic term, this gives a linear operator $\mathcal{L} = \nabla^2 + \kappa^2$. In this case I can get a solution via the Green's... | This paper discusses your problem below Eq. 8 and provides the solution in Eq. 9 and 10 for a single plate and Eq. 11 and 12 for two plates.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/722562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Interpretation of Second isomorphism theorem I have a question about the Second Isomorphism Theorem.(Actually my book called it the first), namely, let $G$ be a group, $N$ is a normal subgroup of $G$, and let $H$ be any subgroup of $G$, then $ (HN)/N \cong (H/ (H \cap N))$. So what's the main argument the theorem want ... | I guess it comes from very natural problem.
Let $\phi$ be cononical homomorphism from $G$ to $G/N$. Let $H$ be any subgroup of $G$.
Question is that what is the image of $H$? If $N\leq H$ then answer is simple $\phi(H)=H/N$.
What if $H$ does not contain $N$? We can find answer in two different way and it gives us an e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/722632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
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Maximum of linear combination I have an range like this:
$$x + 2y \leq 40$$
$$4x + 3y \leq 120$$
$$x \geq 0, y \geq 0 $$
I made an plot using wolfram alpha. Now I have a linear combination $$4x+5y$$ and I want to find the maximum of this linear combination, but with restriction to the range which I described. But I wan... | This falls in the domain of Linear Programming which essentially deals with optimizing linear functions subject to linear constraints. In your case, the restrictions you wrote are linear equations and the objective function is also.
Since you wanted a geometric interpretation, here you go:
The first thing you need to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/722716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Osculating plane<=>distance limit Let $r$ be a unit-speed bi-regular curve. (It passes the point $s_0$)
Let $distP(q)$ be the distance between the plane $P$ and the point $q$.
Question.
The plane is equal to the osculating plane of $r$ at $s_0$ if and only if
$P$ contains
$r(s_0)$,$$ \lim_{s \to s_0} \frac{distP(r(s))... | I found the answer. Using the L'Hospital's law, the equation means that the binormal vector of r(s) perpendicular to the normal vector. And it is true because it is bi-regular curve.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/722766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Sign of square root of a real number May we write $\sqrt{x^2}=\pm x$.
Is $\sqrt{x^2}=\sqrt{(\pm x)^2}=\pm{x}$ true ?
But we write $\sqrt{x^2}=|x|=x$
What is the actual logic?
| You are confused because there are two notions at play here which the standard treatment of the square root obfuscates: functions and solutions of equations.
The right way to think about the square root is as a function: given any positive number $a$, the square root function returns the square root $\sqrt a$ (I should... | {
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prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$
show that this equation
$$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$
have no positive integer solution.
This problem is china TST (2014),I remember a famous result? maybe is a Erdos have proof this prob... | This is not an answer, but it's too long for a comment. Perhaps it helps:
Observe that $$\frac{(y+1)\cdots(y+4028)}{4028!}={y+4028\choose y},$$ which is an integer. Then $$(x+1)\cdots(x+2014)=4028!{y+4028\choose y},$$ so the left hand side is an integer multiple of $4028!$.
This yields a large lower bound for $x$ as fo... | {
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"source": "stackexchange",
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If the sum $\sum_{x=1}^{100}x!$ is divided by $36$, how to find the remainder?
If the sum $$\sum_{x=1}^{100}x!$$ is divided by $36$, the remainder is $9$.
But how is it?
THIS said me that problem is $9\mod 36$, but how did we get it?
| One should consider that numbers written in a base are a series of remainders, so for example, 1957 gives 195 remainder 7, and 195 gives 19 remainder 5, and 19 is 1 remainder 9. So 1957 is 1 re 9 re 5 re 7. Adding remainders is then like adding the last n places of a base.
For $n$!, we have all factorials 6! or large... | {
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Continuous decreasing function has a fixed point
Let $f$ be continuous and decreasing everywhere on $\mathbb{R}$. Show that:
1) $f$ has a unique fixed point
2) $f\circ f$ has either an infinite number of fixed points or an odd number of fixed points.
The first part is easy and I am sure it is available on this websit... | Hint: Suppose $h$ only has finitely many fixed points. Call the set of fixed points $A$. Where does $f$ map $A$? (And what does uniqueness of the fixed point of $f$ then tell you?)
Added: Since the problem appears to be solved, here's how to complete the solution: define $$\begin{align}A_<&=\{x\in A\mid x<f(x)\},\\A_=&... | {
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How to calculate volume in R5 Find the volume of $\Omega=\{(x,y,z,u,v):x^2+y^2+z^2+u^2+v^2)<=1\}$.
I have no idea what to do.
| If you know the volume of a sphere, you can integrate like this using polar substitution:
$$\underset{x^2+y^2+z^2+u^2+v^2\leq1}{\int\!\!\!\int\!\!\!\int\!\!\!\int\!\!\!\int}\!\!\!\!\!\!1\ d(x,y,z,u,v)=\iint\limits_{u^2+v^2\leq1}\left(\ \iiint\limits_{x^2+y^2+z^2\leq1-u^2+v^2}\!\!\!\!\!\!\!\!\!\!\!1\ d(x,y,z)\right)d(u,... | {
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Frobenius coin problem induction help I need some help with the actual Induction proof of a Frobenius coin problem. This is the exact problem:
The government of Elbonia has decided to issue currency only in 5 and 9 cent denominations. Prove that there is largest value that Elbonians cannot pay with this denomination
A... | You are all set. From your base cases (32 to 36) you can add 5 to each to get the next stretch of 5 (37 to 41), and from them the next one (42 to 46), and...
To make it formal:
Bases: As you did show, 32 to 36 are all possible.
Induction: Asuming all between $32$ and $n \ge 32$ are representable,
$n + 1$ is representab... | {
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Are there homomorphisms of group algebras that don't come from a group homomorphism? Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces an algebra homomorphism on the gro... | Let $G = \{e, \sigma\}$ be the group of order $2$. Then $\{e, \sigma\}$ is a basis for $\mathbb C[G]$. The linear map given by $e \mapsto e$ and $\sigma \mapsto -\sigma$ is an automorphism of $\mathbb C[G]$ that does not come from a group homomorphism.
| {
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closed ball in euclidean space In general metric spaces the closed ball is not the closure of an open ball.
However, I read that in the Euclidean space with usual metric, closed ball is the closure of an open ball. I'm having trouble rigorously proving this? How can I show this?
| Hint:
Suppose $B_x(r)$ and $B_x[r]$ are the respectively open and closed balls with centre $x$ and radius $r \gt 0$. If $y \in B_x[r]$ then $||y - x|| \le r \implies ||y - x|| \lt r$ or $||y - x|| = r$. For the first case since $y \in B_x(r)$ it is easy to prove that $y$ is also in the closure since $ A \subseteq A^{... | {
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Can someone explain what does this sum mean? I found a solution to my problem in this thread: How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number?
But unfotunately I don't understand the last step.
$$x^n(1-x^{m})^{n}\left(\sum_{k=0}^{\infty} {n+k-1 \choose k} x^k\right)$$
I... | I can answer part of your question:
Not all sums have an upper limit. For example, take $\sum_{i+j=1}i+j$. To figure out what values of $i$ and $j$ to plug in, look at all combinations of 2 integers whose sum equals $1$. So for example, $0$ and $1$, $5$ and $-4$, $-4$ and $5$, etc.
So in your sum, $rm+k=S-n$ is a cond... | {
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Chinese remainder theorem? In the 2014 AIME 1, number 8 says:
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.
I solved this problem using modular arithmetic and a little bit of logic (mainly ... | You could start with Wikipedia A web search will turn up many references. It looks like you were applying it without knowing it. Here you are looking to solve $N \equiv 0 \pmod {2^4}, N\equiv -1 \pmod {5^4}$ or the other way around. Because $2^4,5^4$ are relatively prime, CRT says there will be exactly one solution... | {
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Find the point on a plane $3x + 4y + z = 1$ that is closest to $(1,0,1)$ Is anyone able to help me with regards to this question?
Find the point on a plane $3x + 4y + z = 1$ that is closest to $(1,0,1)$
http://i.imgur.com/ywdsJi7.png
| Let $(x, y, z)$ be the point in question. The distance is given by $\sqrt{(x - 1)^2 + y^2 + (z - 1)^2}$. By Cauchy Schwarz, $\left((x-1)^2 + y^2 + (z-1)^2\right)(3^2 + 4^2 + 1^2) \geq (3x + 4y + z - 4)^2$, so $\left((x-1)^2 + y^2 + (z-1)^2 \right) \geq \frac{9}{26}$
Equality is reached when $\frac{x-1}{3} = \frac{y}{4}... | {
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Question about Normed vector space. Here is the definition of a normed vector space my book uses:
And here is a remark I do not understand:
I do not understand that a sequence can converge to a vector in one norm, and not the other. For instance: Lets say $s_n$ converges to $u$ with the $\|\|_1$-norm. From definition... | You can get in trouble if the convergence is not absolute, i.e. $$\sum_n \|x_n\| = +\infty$$ but
$$\lim_{N\to\infty} \sum_{n\le N} x_n $$
exists.
| {
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Bearings Problem I'm presented with the following bearings problem. I believe I have graphed it correctly, although I don't know where to go from here.
A US Coast Guard patrol boat leaves Port Cleaveland and averaged 35
knots (nautical mph) traveling for 2 hours on a course of 53 degrees
and then 3 hours on a cour... | Won't you calculate the degrees from the east direction?. If you did, it is simple application of pythogorus theorem. It travels 2*35 nm at 53 degress to the east. Then it travels 3*35 nm at 143 degrees to the east. It's position is $\sqrt{70^2 + 105^2} = 126.2$ miles from the port of cleaveland and it is 109.3 deg... | {
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Finding minimum of this function I am helping my brother with his math homework and this problem has stumped me.
I tried (150+4n)(490000/n)+0.75n as the cost function but that doesn't get me anywhere when I take second derivative.
A company needs 490,000 items per year. It costs the company \$150 to prepare a product... | [Edit, sorry, mis-read the numbers. Fixing them below.]
So if x is the number of production runs, then 490000/x is the number of items produced in a production run. Since the cost of producing each item is the same no matter how many production runs we use, it doesn't really matter for the minimization problem. (If ... | {
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If $A=AB-BA$, is $A$ nilpotent? Let matrix $A_{n\times n}$, be such that there exists a matrix $B$ for which
$$AB-BA=A$$
Prove or disprove that there exists $m\in \mathbb N^{+}$such
$$A^m=0,$$
I know
$$tr(A)=tr(AB)-tr(BA)=0$$
then I can't.Thank you
| With the risk of being repetitive, let me just record here the (original?) proof of N. Jacobson in this paper, where you only need to assume that $[A,B]$ and $A$ commute to deduce $[A,B]$ is nilpotent.
Let $[A,B]=A'$, and consider $D(X) = [X,B]$. Then for any polynomial $F$, we have that $D(F(A)) = F'(A)A'$. Now pick ... | {
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Is this differential form closed / exact? Could you check if I calculated the exterior derivative of this differential form $\omega$ correctly?
$\omega \in \Omega_2 ^{\infty} (\mathbb{R}^3 \setminus \{0\})$
$\omega = (x^2 + y^2 + z^2)^{\frac{-3}{2}}(x \mbox{d}y \wedge \mbox{d}z + y \mbox{d}z \wedge \mbox{d}x + z \mbox{... | I did not find any errors in your calculations. So it looks like $\omega$ is not closed.
And how to figure out if you have exact form, given it is closed?
In this case you know that $H^2_{dR}(\mathbb{R}^3 \setminus \{0\}) \simeq \mathbb{R}$ There is just one closed but not exact form(up to scalar multiple). And it is(I... | {
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Representing 2D line as two variables, all cases
Is it possible to represent a line in the general case using only two variables (namely floats or doubles)? By this I mean being able to convert a line into those two variables and successfully converting them back without loss of data or the algorithm failing for speci... | Yes, it is possible to represent any line with only two variables:
p is the length of normal from coordinate origin to line
Theta is angle between OX-axis and direction of normal.
Equation:
x*Cos(Θ) + y*Sin(Θ) - p = 0
To transform canonical equation of line to normal form:
A*x+B*y+C=0
D=Sqrt(A^2+B^2)
Sgn = Sign(C)... | {
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Recommended books/articles for learning set theory What is the recommended reading for thoroughly learning set theory? I'm currently studying Kunen's book [1]. But what then, and in what order? One needs to learn large cardinals, inner models and descriptive set theory.
Is it a good idea to try to read Jech's bible [2]... | Maybe is "too elementary" for your puropses, but I like very much the set-theory parto of the Topology book of Kelley
http://www.zbmath.org/?q=%28%28kelley+topology%29+ai:kelley.john-leroy%29+py:1975
| {
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If $23|3x+5y$ for some $x,y\in \mathbb{Z}$, then $23|5x-7y$? Let $x,y$ be some integers such that $23|3x+5y$. Show that $23|5x-7y$ as well.
This was my exam question and I did not solve it. Could anyone give some insight how to do this? Even after the exam I can not solve it...
| Divisibility questions are often simpler when rephrased as modular arithmetic. You want to show
$$ 3x + 5y \equiv 0 \pmod{23} \implies 5x - 7y \equiv 0 \pmod{23} $$
A simplistic thing to do is to look at the known equation, and simplify it by solving for $x$
$$ x \equiv 3^{-1} \cdot (-5) y = 6y \pmod{23} $$
and using t... | {
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Proof of $\lim_{n \to \infty} a^{1/n} = 1$ and $\lim_{n \to \infty}b^{f(n)} = b^{\lim_{n \to \infty} f(n)}$ where $a>1$ As title says, I am not sure what would be the proof of $$\lim_{n \to \infty} a^{1/n} = 1$$ would be where $a>1$. Also, how do you prove that $$\lim_{n \to \infty}b^{f(n)} = b^{\lim_{n \to \infty} f(n... | 1) If you want to argue more formally. For $a>1$, we have $a^{1/n}>1$. Define $c_n = a^{1/n}-1>0$, then $a=(1+c_n)^n\ge1+nc_n$. Thus
$$0<c_n= a^{1/n}-1\le \frac{a-1}{n}\to0$$
Hence $a^{1/n}-1\to 0$, i.e., $a^{1/n}\to 1$.
2) Try to prove that $b^x$ is continuous if you want I could give a sketch of the proof.
| {
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Are Ideals and Varieties Inclusion Reversing? Let $S_1$, $S_2$ be sets or varieties (I don't think it matters, does it?). Then if $S_1 \subset S_2$, is it always the case that $I(S_2) \subset I(S_1)$ (where I is an ideal)? Also, is it always the case that if $I_1$ $I_2$ are ideals such that $I_1 \subset I_2$, then $V(I... | Yes. This is true and easy to show. The key is to look at the definitions: $$S \subseteq \mathbb A^n : I(S) = \{f \in k[x_1, \ldots, x_n] : f(P) = 0 \text{ for all } P \in S\}$$ and $$I \subseteq k[x_1, \ldots, x_n] : V(I) = \{P \in \mathbb A^n : f(P) = 0 \text{ for all } f \in I\}.$$
Claim: If $S_1 \subseteq S_2$ are... | {
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Question about definition of binary relation Wikipedia says:
Set Theory begins with a fundamental binary relation between and object $o$ and a set $A$. If $o$ is a member of $A$, write $o \in A $.
I thought that a binary relation is a collection of ordered pairs of elements of $A$.
Why is relating one element of a s... | The distinction between a ‘binary relation’ and ‘an ordered pair’ is particularly germane when considering the set membership relation, ‘$\in$’, because it clearly illustrates a case of the chicken vs. the egg.
As the Wikipedia article states, a set theory is some logic (say first order with identity for arguments sake... | {
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Find a unit vector and the rate of change Could anyone help me answer this question? Or point me in the right direction?
Find a unit vector in the direction in which f increases most rapidly at P and find the rate of change of f at p in that direction.
$$f(x,y) = \sqrt{\frac{xy}{x+y}}; \qquad P(1,1)$$
http://i.imgur.co... | A vector in the direction of most change is $(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})$. Divide by its length to find a unit vector.
Add the unit vector to P and evaluate f to find the rate of change.
| {
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General formula for a square in complex numbers I need to find a general formulae for a square, with its interior
included, in terms of complex numbers. Note that your general square
should have (general centre, side-length and orientation.)
I do not know how to deal with the orientation i.e. when the square is 45 degr... | A square with center the origin and sides, of length $2a>0$, parallel to the axes:
$$\max\{|Re(z)|,|Im(z)|\}\leq a.$$
Rotating the square around the origin with angle $\theta$:
$$\max\{|Re(z/e^{i\theta})|,|Im(z/e^{i\theta})|\}\leq a.$$
Translating it:
$$\max\{|Re((z-c)/e^{i\theta})|,|Im((z-c)/e^{i\theta})|\}\leq a.$$
U... | {
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Help with integral $\int^{a\sqrt{1-E/V}}_{-a\sqrt{1-E/V}}\sqrt{2mV(1-x^2/a^2)-E} dx $ Hi I have tried u and trig substitution for this integral and just cant get it can someone offer a pointer or two? Thanks
$\int^{a\sqrt{1-E/V_0}}_{-a\sqrt{1-E/V_0}}\sqrt{2m[V_0(1-x^2/a^2)-E]} dx $
| One can simplify the integral using substitutions.
$$I=\int^{a\sqrt{1-E/V_0}}_{-a\sqrt{1-E/V_0}}\sqrt{2m(V_0(1-x^2/a^2)-E)} \ \mathrm dx=\sqrt{2mV_0}\int^{a\sqrt{1-E/V_0}}_{-a\sqrt{1-E/V_0}}\sqrt{1-\frac{E}{V_0}-\frac{x^2}{a^2}} \ \mathrm dx$$
Let's set $\sqrt{1-E/V_0}=\alpha$, then changing the variable to $t=\frac{x}... | {
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Prove that $\frac{x \log(x)}{x^2-1} \leq \frac{1}{2}$ for positive $x$, $x \neq 1$. I'd like to prove
$$\frac{x \,\log(x)}{x^2-1} \leq \frac{1}{2} $$
for positive $x$, $x \neq 1$.
I showed that the limit of the function $f(x) = \frac{x \,\text{log}(x)}{x^2-1}$ is zero as $x$ tends to infinity. But not sure what to ... | If $x > 1$ we prove equivalent inequality: $2x \ln x \leq x^2 - 1 \iff 2x\ln x - x^2 + 1 \leq 0$.
Look at $f(x) = 2x \ln x - x^2 + 1$ for $x > 1$. We have $f'(x) = 2\ln x + 2 - 2x$, and $f''(x) = \dfrac{2}{x} - 2 < 0$. So $f'(x) < f'(1) = 0$. So $f(x) < f(1) = 0$, and this means $2x\ln x \leq x^2 - 1$.
If $0 < x < 1$ ... | {
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Why is this projective curve in $\mathbf{P}^3_k$ nonsingular?
Consider $C$ in $\mathbf{P}^3_k = \mathrm{Proj}[x_0,...,x_3]$ defined by $$x_0x_3 - x_1^2 = 0$$ and $$x_0^2 + x_2^2 - x_3^2 = 0$$ where $k$ is an algebraically closed field. Why is this curve nonsingular?
I've been trying work this out in scheme-theoretic... | Yours may be a case of excess of technology...
You probably know that to check non-singularity it is enough to do it locally, so you can consider, for example, the standard open covering of $P^3$. Then you are in affine $3$-space, and you can use the Jacobian criterion.
For example, in the set where $x_0\neq0$ we can t... | {
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Find the limit $\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}$ When $\lim_{n\to\infty} x_n = a$, and $\lim_{n\to\infty} y_n = b$, find the limit,
$$\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}.$$
Thank you for your help in advance.
| By the Cesàro mean theorem, if $(x_n)_{n\in\mathbb{N}^*}\to a$ then $\left(\bar{x}_n=\frac{1}{n}\sum_{j=1}^{n}x_j\right)_{n\in\mathbb{N}^*}\to a$.
So, for any $\epsilon>0$ there exists $N\in\mathbb{N}$ such that all the quantities:
$$|x_m-a|,\quad|y_m-b|,\quad|\bar{x}_m-a|,\quad|\bar{y}_m-b|$$
are less than $\epsilon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/725914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 0
} |
Intuition behind chain rule What is the intuition behind chain rule in mathematics in particular why there is a multiplication in between?
| The best way to think about the derivative is: if $f$ is differentiable at $x$, then
\begin{equation*}
f(x + \Delta x) \approx f(x) + f'(x) \Delta x.
\end{equation*}
The approximation is good when $\Delta x$ is small. This is practically the definition of $f'(x)$.
Now suppose $f(x) = g(h(x))$, and $h$ is differentiabl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/725951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
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Knight's metric: ellipse and parabola. Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this new metric?
I apologize if the question is too vague.
| Using Noam D. Elkies's characterization of the knight's distance, here's an animation of $d(x,y)+d(x-a,y)$ as $a$ goes from $0$ to $30$. All cells of the same colour are on the same "ellipse" (except the darkest red ones, which have distance $\ge20$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/726045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Cubic root formula derivation I'm trying to understand the derivation for the cubic root formula. The text I am studying from describes the following steps:
$$x^3 + ax^2 + bx + c = 0$$
Reduce this to a depressed form by substituting $y = x + \frac{a}{3}$. Such that:
$$y^3 = (x + \frac{a}{3})^3 = x^3 + ax^2 + \frac{a^2}... | If you replace $x$ in your original equation by $y-\frac{a}{3}$, you get:
*
*$y^3+p\cdot y+q$
with
*
*$p=b-\frac{a^2}{3}$
*$q=\large\frac{2a^3}{27}\normalsize -\large\frac{ab}{3}\normalsize +c$
And for what? Now you are able to solve the new problem without a quadratic component by Cadano's method (bet... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving two sequences converge to the same limit $a_{n+1}\frac{a_n+b_n}{2} \ , \ b_{n+1}=\frac {2a_nb_n}{a_n+b_n} $
$\text{We have two sequences}$ $(a_n), (b_n)$ where $0<b_1<a_1$ and:
$$a_{n+1}=\frac{a_n+b_n}{2} \ , \ b_{n+1}=\frac {2a_nb_n}{a_n+b_n} $$
Prove both sequences converge to the same limit and try to find... | *
*Just show via an induction that $b_n\le b_{n+1} \le a_{n+1} \le a_n$: this proves that both sequences are convergent.
*Then take the limit in the definition and the previous inequality: you get
$$A = \frac 12 (A+B)
\\A\ge B$$so $A=B.$
details for 1.:
a) The inequality
$$
u<v\implies \frac {u+v}2<v
$$is trivial.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/726201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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invariants of a representation over a local ring from the residual representation Let $(R, \mathfrak m)$ be a local ring (not necessarily an integral domain) and $T$ be a free $R$-module of finite rank $n\geq 2$. Let $\rho: G \to \mathrm{Aut}_{R\text{-linear}}(T)$ be a represenation of a group G. Is it true that if the... | Note that $\mathfrak m^n T/\mathfrak m^{n+1}$ is naturally isomorphic to
$(\mathfrak m^n/\mathfrak m^{n+1})\otimes_k T/\mathfrak m$ as a $G$-representation,
with $G$-acting through the right-hand factor. In particular, if $T/\mathfrak m$
has trivial $G$-invariants, so does $\mathfrak m^n/\mathfrak m^{n+1}$.
From t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Determine value of the integration Determine value of the integration $$I=\iint_{D} \sqrt{\left|y-x^2\right|} \, dx \, dy$$ with $$D=[-1,\: 1]\times [0,\: 2]$$
My tried:
$$I=2 \int_0^1 \left(\int_{x}^{2}\sqrt{y-x^2} \, dy\right) \, dx+I_1$$
Find $I_1$.
| An idea: the wanted integral seems to be
$$2\int\limits_0^1\int\limits_\sqrt x^1\sqrt{x^2-y}\,dydx+2\int\limits_0^1\int\limits_0^\sqrt x\sqrt{y-x^2}\,dydx+2\int\limits_0^1\int\limits_1^2\sqrt{y-x^2}\,dxdy$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trivial zeros of the Riemann Zeta function A question that has been puzzling me for quite some time now:
Why is the value of the Riemann Zeta function equal to $0$ for every even negative number?
I assume that even negative refers to the real part of the number, while its imaginary part is $0$.
So consider $-2$ for exa... | There is a way to prove that $\zeta(-2K) = 0$ :
*
*By definition of the Bernouilli numbers $\frac{z}{e^z-1}= \sum_{k=0}^\infty \frac{B_k}{k!}z^k$ is analytic on $|z| < 2\pi$
*Note that
$ \frac{z}{e^z-1}-\frac{z}{2} = \frac{z}{2}\frac{e^{z/2}+e^{-z/2}}{e^{z/2}-e^{-z/2}}$ is an even function, therefore $\frac{z}{e^z... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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Minimize the minimum - Linear programming Consider an optimization problem with variables $x_1, x_2, \dots, x_n \in \mathbb{R}$ (maybe subject to some linear constraints), and linear functions $\{f_i(x_1, \dots, x_n)\}_{1\leq i\leq m}$. We want to minimize $\min_{1\leq i\leq m} f_i(x_1, \dots, x_n)$.
Is it possible to... | You can maximize a minimum or minimize a maximum with a single LP, but min-min and max-max are both non-convex.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Riemann-integrable functions and pointwise convergence
Hello, I was hoping for some advice on finding a function which will satisfy this. I think I am okay with the actual execution of the answer, but I don't know how I'm supposed to find a suitable function.
Thank you
| Hint: Let $\{r_n\}_{n\in\Bbb N}$ be an enumeration of the rationals in $[0,1]$, that is $$\{r_n\}_{n\in\Bbb N} =\Bbb Q\cap [0,1].$$
Define $g_n:[0,1]\to \Bbb R$ by $$g_n(x)=\begin{cases} 1 &\text{if $x\in \{r_1,\ldots,r_n\}$} \\ 0 &\text{otherwise} \end{cases}$$
Each $g_n$ is Riemann integrable. The function $g$ to whi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing a Mapping Between $\left\langle a,b \mid abab^{-1}\right\rangle$ and $\left\langle c,d \mid c^2 d^2 \right\rangle$ is Surjective Hypothesis:
*
*Let
$$
G \cong \left\langle a,b \mid abab^{-1}\right\rangle
$$
$$
H \cong \left\langle c,d \mid c^2 d^2 \right\rangle
$$
*Let the function $f$ be defined as follo... | A different way to do this problem is to use Tietze transformations. These are specific transformations you can do to group presentations. The key result is that two presentations $\mathcal{P}$ and $\mathcal{Q}$ define isomorphic groups if and only if there exists a sequence of Tietze transformations which takes $\math... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Prove that a power of odd number is always odd by induction. The problem has confused me for like half hour.
An integer is odd if it can be written as d = 2m+1. Use induction to prove that the ${d^n}$ = 1 (mod 2)
by induction, the basecase is pretty simple , let n = 0 then ${d^0}$=1 (mod 2) is correct. But I stucked in... | It's easier to just show that the product of a finite number of odd integers is odd. This can be done inductively if you like. Then your problem is a special case of this.
Base case: a single odd number is odd.
Inductive step: Assume $n_1,n_2,\ldots,n_{k+1}$ are odd and that $p_k = (n_1)(n_2)\cdots(n_k)$ is odd. Then $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find a change in variable that will reduce the quadratic form to a sum of squares Find a change of variable that will reduce the quadratic form
$x_1^2-x_3^2-4x_1x_2+4x_2x_3$
to a sum of squares, and express the quadratic form in terms of the new variable.
| Call the quadratic form, $Q(x)$. Write down the symmetric matrix $A$ such that $Q(x)=x^tAx$; that would be $$A=\pmatrix{1&-2&0\cr-2&0&2\cr0&2&-1\cr}$$ Since $A$ is symmetric, there is an orthogonal matrix $P$ such that $P^tAP=D$ is diagonal. Define new variables $y=(y_1,y_2,y_3)$ by $x=Py$. Then $$Q(x)=Q(Py)=(Py)^tAPy=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The inverse of m with respect to n in modular arithmetics From concrete mathematics problem 4.35.
Let $I(m,n)$ be function that satisfies the relation
$$ I(m,n)m + I(n,m)n = \gcd(m,n),$$
when $m,n \in \mathbb{Z}^+$ with $m ≠ n$. Thus, $I(m,n) = m'$ and $I(n,m) = n'$ in (4.5).
The value of $I(m,n)$ is an inverse of $m$... | If $\gcd(m,n)=1$, so that the equation is $I(m,n)m+I(n,m)n=1$, then $I(m,n)$ is the multiplicative inverse of $m$ modulo $n$, since looking at that equation modulo $n$ yields $I(m,n)m\equiv1\pmod n$.
When $\gcd(m,n)>1$, there is no multiplicative inverse of $m$ modulo $n$, but $I(m,n)$ would be a multiplicative inverse... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Ineffable Cardinals and Critical Point of Elementary Embeddings A cardinal $\kappa$ is a ineffable if and only if for all sequences $\langle A_\alpha : \alpha < \kappa\rangle$ such that $A_\alpha \subseteq \alpha$ for all $\alpha < \kappa$, then there exists $A \subseteq \kappa$ such that $\{\alpha < \kappa : A \cap \a... | Let's let $B = \{ \alpha < \kappa : A_\kappa \cap \alpha = A_\alpha \}$.
Suppose $C \in M$ is a club subset of $\kappa$. We want to show that $C \cap B \neq \varnothing$. It follows that $j(C)$ is a club subset of $j(\kappa)$, and also that $j(B) = \{ \alpha < j(\kappa) : j(A_\kappa) \cap \alpha = A_\alpha \}$.
Now n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How to find the eigenvalue? I have been given a matrix
$$A = \begin{bmatrix}
1&1&0&0\\
1&1&0&0\\
0&0&1&1\\
0&0&1&1
\end{bmatrix}$$
I expanded by row one twice to get the characteristic polynomial:
$(1-\lambda)^2[(1-\lambda)^2 -1] - 1[(1-\lambda)^2 - 1]$
Which I solved lambda and got that $\lambda = 0$. I checked my w... | The characteristic polynomial you calculated is correct, however, it simplifies to $$p(\lambda) = \lambda^2(\lambda - 2)^2$$
There is more than one solution to the equation $p(\lambda)=0$.
In your notes, the mistake you made was at the very end. You got that $$(1-\lambda)^2 = 1$$ and from that, you concluded that $$(1-... | {
"language": "en",
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Linear algebra, polynomial problem Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :(
Let $S$ be the following subset of the vector space $P_3$ of all real polynomials $p$ of degree at most 3:
$$S=\{p\in P_3\mid p(1)=0, p^\prime (1)=0\}$$
where $p^\prime$... | Alright, for part a), you have to look at the definition of a subspace. That is, it must contain the additive identity (the zero-polynomial), which is trivial. It must be closed under multiplication by scalar, which it is, since its degree will not change, regardless of what real number $C$ you multiply a polynomial $P... | {
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Convex function almost surely differentiable. If f: $\mathbb{R}^n \rightarrow \mathbb{R}$ is a convex function, i heard that f is almost everywhere differentiable. Is it true? I can't find a proof (n-dimentional).
Thank you for any help
| For a proof that doesn't rely directly on the Rademacher theorem, try Rockafellar's "Convex analysis", Theorem 25.5.
From the book: Let $f$ be a proper convex function on $\mathbb{R}^n$, and let $D$ be the set of points where $f$ is differentiable. Then $D$ is a dense subset of $(\operatorname{dom} f)^\circ$, and its c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Computing the Legendre symbol of -3, $(\frac{-3}p)$ I'm working on Ireland and Rosen, exercise 6.8.
Let $\omega=e^{2\pi i/3}$ satisfying $\omega^3-1=0$. Show that $(2\omega-1)^2=-3$ and use this result to determine $(\frac{-3}p)$ for $p$ an odd prime.
I've already found that $0=\omega^3-1=(\omega-1)(\omega^2+\omega+1... | answer: use $\omega^3=1$ and $2^p=2\bmod p$.
| {
"language": "en",
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10-hand card is dealt from a well shuffled deck of 52 cards A 10-hand card is dealt from a well shuffled deck of 52 cards. What is the probability that the hand contains at least two cards from each of the four suits?
| You either need the suits distributed $4222$ or $3322$. The chance of $4222$ is $$\frac{{4 \choose 1} (\text{suit with four cards}) {13 \choose 4}{13 \choose 2}^3}{52 \choose 10}$$ The chance of $3322$ is $$\frac{{4 \choose 2} (\text{suits with three cards}) {13 \choose 3}^2{13 \choose 2}^2}{52 \choose 10}$$ for a to... | {
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How to write $x=2\cos(3t) y=3\sin(2t)$ in rectangular coordinates? How would I write the following in terms of $x$ and $y$? I think I use the inverse $\cos$ or $\sin$?
$$x=2\cos(3t)\,, \quad y=3\sin(2t)$$
| This is not an answer (with apologies to Magritte)...
The point is that there is no functional relationship between $x $ and $y$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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$\sin^2(x)+\cos^2(x) = 1$ using power series In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power series:$$\sin(x)=\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}$$ and $$\cos(x)=\sum_{k=0}^\infty\... | You can also prove this identity directly from the power series
$$
\begin{align}
\cos x &= \sum_{n = 0}^\infty \frac{(-1)^n}{(2n)!} x^{2n},\\
\sin x &= \sum_{n = 0}^\infty \frac{(-1)^n}{(2n + 1)!} x^{2n + 1}.
\end{align}
$$
The following is modified from the discussion on Wikipedia's article on the Pythagorean Trig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 2
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Integration Problem with a Trig substitution Okay I am a little stuck on this problem.
$$\int \tan^5(x)\sqrt{\sec(x)} \; dx$$
What should be my first step for a u sub or a trig sub? I have tried to use $u=\sec(x)$ and then $u=\tan(x)$, but I get stuck. A little help?
| Let $u=\cos x$ then $du=\sin x dx$ and then the anti derivative becomes:
$$\int\frac{(1-u^2)^2}{u^5}\frac{du}{\sqrt u}$$
now let $u=t^2$ so we find
$$2\int \frac{(1-t^4)^2}{t^{10}}dt=2\int t^{-10}dt-4\int t^{-6}dt+2\int t^{-2}dt$$
I'm sure that you can take it from here.
| {
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"question_score": "3",
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show lambda is an eigenvalue of matrix A and find one eigenvector x Hello Lovely people of the Overflow :)
I am working on a homework assigntment for my linear algebra class and i am stumped on this pesky question which is as follows:
Show that λ is an eigenvalue of A and find one eigenvector, x, corresponding to this ... | The characteristic polynomial is given by $|A - \lambda I| = 0$, hence:
$$\lambda ^2-3 \lambda -54 = 0 \implies (\lambda +6)(\lambda -9) = 0 \implies\lambda_1 = -6, ~ \lambda_2 = 9$$
The eigenvectors are found by $[A - \lambda I]v_i = 0$. For $\lambda_1 = -6$, we have
$$\begin{bmatrix} 12 &\ 6\\ 6 & 3\\ \end{bmatrix}v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/728461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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A statement for convex sets The following statement is true or false?
Given a convex set $S$ then for any $y \in S$ and $\theta\in[0,1], \theta \in \mathbb R$ there exist $y_1,y_2 \in S, y_1 \ne y, y_2 \ne y$ such that $y=\theta y_1+(1-\theta) y_2.$
| You don't even need $S$ convex: just take $y_1 = y_2 = y$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using U Substitution on 1/(3x) Say I want to find the indefinite integral of 1/(3x).
I can pull out the (1/3) so now I just have 1/x to integrate and I get (1/3)(lnx) as my final answer. This is the correct answer.
But now I'm learning U substitution and I'm wondering why I can't apply this method on this question. So ... | Let's do our u-substitution on the integral
$$\int \frac{1}{3x} \ dx$$
Let $u=3x$, therefore $du=3 \ dx$.
$$\int \frac{1}{3x} \ dx$$
$$=\int \frac{1}{u} \cdot \frac{1}{3} \ du$$
$$=\frac{1}{3} \int \frac{1}{u} \ du$$
$$=\frac{1}{3}\ln|u|+C$$
Now we reverse our substitution:
$$\frac{1}{3}\ln|3x|+C$$
But wait! Remember t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/728633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Evaluating $\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA$ I'm trying to evaluate the double integral
$$
\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA.
$$
Any ideas?
| Hint: use polar coordinates:$$
\frac{1}{2\pi} \int_{\Bbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA
= \lim_{R\to\infty}
\frac{1}{2\pi} \int_{x^2 +y^2 \le R^2}
e^{-\frac{x^2+y^2}{2}} \, dA
\\
= \lim_{R\to\infty}
\frac{1}{2\pi} \int_{0\le r \le R}
e^{-\frac{r^2}{2}} 2\pi r dr
= \lim_{R\to\infty} \left[-e^{-\... | {
"language": "en",
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Nilpotent operator of index $n$ Let $T: \mathbb R^n \to \mathbb R^n$ be a linear operator such that $T^{n-1} \neq 0$ but $T^n = 0$. Prove that $\text{rank}(T)=n-1$ and give an example of such operator.
PS. This was on a homework, I searched a lot but couldn't find the solution/hint. The point is the problem can be solv... | $T^{n-1}\neq 0$ so, there is a $x$ such that $T^{n-1}(x) \neq 0$. It's easy to see that every power $T^1(x), T^2(x), ..., T^{n-2}(x)$ are also different from 0 otherwise the n-1 power would be immediately 0.
Now let's prove that the family $(T^1(x), T^2(x), ... , T^{n-1}(x))$ is linearly independent.
For $\lambda_1, ..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/728825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to show $\lim_{n\to \infty}\sqrt{n}^n (1 - (1 - 1/(\sqrt{n}^n))^{2^n})/2^n = 1$? How can you show the following?
$$\lim_{n\to \infty}\frac{\sqrt{n}^n \left(1 - \left(1 - \frac{1}{\sqrt{n}^n}\right)^{2^n} \right)}{2^n} = 1$$
It certainly seems to be true numerically when I plot it.
| Let $x=\sqrt n^n$, then $2^n=x^2\left(\frac2n\right)^n$. Then we have
$$\displaystyle\begin{align}\lim_{n\to\infty}\frac{x\left(1-\left(1-\frac1x\right)^{x^2\left(\frac2n\right)^n}\right)}{x^2\left(\frac2n\right)^n}=&\lim_{n\to\infty}\frac{1-e^{-x\left(\frac2n\right)^n}}{x\left(\frac2n\right)^n}
=\lim_{n\to\infty}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/728928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
General Solution to a Differential EQ with complex eigenvalues. I need a little explanation here the general solution is $$x(t)=c_1u(t)+c_2v(t)$$ where $u(t)=e^{\lambda t}(\textbf{a} \cos \mu t-\textbf{b} \sin \mu t$ and $v(t)=e^{\lambda t}(\textbf{a} \sin \mu t +\textbf{b} \cos \mu t)$ I am confused on what happened t... | It doesn't really disappear.
Note that $\{u,v\}$ is linearly independent over $\mathbb R$, so if they are solutions of a second degree ordinary differential equation with constant coefficients, they form a basis of solutions.
The $i$ disappears because usually one is interested in real functions.Of course $u+iv$ will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determine whether S is a subspace of P3. Vector space of all real polynomials.
ATTEMPT:
Have given a small attempt just really confused on how to approach.
So I got the general equation of $p(x)= a + bx +cx^2 +dx^3$.
So we find the derivative? and find the values of $a,b,c,d$?
How do I find the polynomials? like the ... | Any element of the vector space $P_3$ is of the form,
$p(x)= a + bx +cx^2 +dx^3$
Substituting x = 1, we get,
$p(1)= a + b.1 + c.1^2 +d.1^3$
$\implies a + b + c + d = 0$
For p'(1), we have after differentiating,
$p'(x)= b + 2cx +3dx^2$
$p'(1)= b + 2c.1 +3d.1^2 = 0 $
$\implies b + 2c +3d = 0$
So, you know every element o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding the Total Curvature of Plane Curves I'm trying to find the total curvature (or equivalently, rotation index, winding number etc.) of a plane curve (closed plane curves) given by
$$\gamma(t)=(\cos(t),\sin(nt)), 0\le t\le 2\pi$$for each positive integer $n$.
Looking at the image of thess curves makes me believe t... | I am assuming that the index is the rotation index.
This said, there are much easier ways to compute the index. One way is to compute all zeros of the two components of the derivative of the curve (trivial in the above example); each zero represents a time at which the tangent points towards a cardinal direction; in be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
An identity in Ring of characteristic $p$ prime Is it true that in a ring of prime characteristic $p$ results that
$(x-1)^{p-1}=1+x+x^2+...+x^{p-1}$ ?
If this is not true in general, the assumption that $x$ is a nilpotent element (let's say $x^{p^n}=0$) make it works?
| The identity does indeed hold in general. One way to see this is from the binomial coefficient identity
$$\binom{p-1}{n}\equiv (-1)^n\pmod p$$
To see that this identity holds, notice that
$$\binom{p-1}{n}=\frac{(p-1)(p-2)\ldots(p-n)}{1\cdot2\cdot\ldots\cdot n}\equiv\frac{-1\cdot-2\cdot\ldots\cdot-n}{1\cdot2\cdot\ldots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find constants of function I have this equality :
$$f(x)=\frac{9}{(x-1)(x+2)^2}$$
I am required to find the constants A, B and C so that,
$$f(x) = \frac{A}{(x-1)} + \frac{B}{(x+2)} + \frac{C}{(x+2)^{2}} $$
How do we go about solving such a question?
I am not sure on how to solve such questions. Approach and Hints to ... | Add the three fractions on the right hand side of your equation
$$\frac{A}{(x-1)}+\frac{B}{(x+2)}+\frac{C}{(x+2)^2}=\frac{A(x+2)^2+B(x-1)(x+2)+C(x-1)}{(x-1)(x+2)^2}$$
I'll leave it to you to simplify the numerator and solve for $A$,$B$ and $C$, such that.
$$A(x+2)^2+B(x-1)(x+2)+C(x-1)=9$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/729350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 0
} |
Can I compute $\lim_{n \to \infty}(\frac{n-1}{3n})^n$ this way? Can I do the following to compute the limit: $\lim_{n \to \infty}(\frac{n-1}{3n})^n = \lim_{n \to \infty}(\frac{n}{3n}-\frac{1}{3n})^n = \lim_{n \to \infty}(\frac{1}{3}-\frac{1}{3n})^n = (\frac{1}{3}-0)^\infty = (\dfrac{1}{3})^\infty = 0$
| Basically, no.
When you go from the third step to the fourth one, you make a false assumption. The same one that confuses people with $e$: $$\lim_{n\to\infty}\left(1+\frac1n\right)^n=e\neq \lim_{n\to\infty}\left(1+\frac1\infty\right)^\infty=1
$$
In simple terms, the reason why this is so is that if you expand the expre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
A proof for a (non-constant) polynomial can't take only primes as value I know a proof of this statement, see How to demonstrate that there is no all-prime generating polynomial with rational cofficents?
My question is that, in the book
Introduction to Modern Number Theory - Fundamental Problems, Ideas and Theories
b... | suppose the polynomial P(x) takes prime value for each integer k.assume r is a root of P(x).so P(r)=0
*
*now k -r|P(k) -P(r)$ \Rightarrow $k -r|P(k)=m=a prime. so clearly k -r=m or 1
*so clearly k-n=a prime or 1 for all roots n of P(x)
*consider P(x)=a(x-r)(x-c)(x-d)....................(x-v)[r,c,d,.........,v are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why $\cos^3 x - 2 \cos (x) \sin^2(x) = {1\over4}(\cos(x) + 3\cos(3x))$? Wolfram Alpha says so, but step-by-step shown skips that step, and I couldn't find the relation that was used.
| Hint: Taking the real part of De Moivre's formula
$$
\cos(3x)+i\sin(3x)=(\cos(x)+i\sin(x))^3
$$
and applying $\cos^2(x)+\sin^2(x)=1$, we have
$$
\begin{align}
\cos(3x)
&=\cos^3(x)-3\sin^2(x)\cos(x)\\
&=4\cos^3(x)-3\cos(x)
\end{align}
$$
Furthermore, the left side is
$$
\cos^3(x)-2\cos(x)\sin^2(x)=3\cos^3(x)-2\cos(x)
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Harmonic numbers probability similar to coupon collector We're ordering beer with uniform probability and with replacement. I calculated the expected value of receiving $n$ different brands of beer from some company is $E(X) = n\cdot H_n$. I've defined a random variable as:
$X = $ total number of distinct brand receive... | Your question is less than clear, but I suppose you have $n$ distinct and equally probably brands and you take a sample of size $m$ with replacement, receiving $X$ distinct brands.
You can work out the probablity of not getting a particular brand as $\left(1-\frac1n\right)^m$ and so the expected number of distinct bran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/729684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
what are the spherical coordinates What are the spherical coordinates of the point whose rectangular coordinates are
(3 , 1 , 4 ) ?
I got that =sqrt26 but I could not find the values for the others
| In terms of Cartesian:
Cylindrical coordinates:
\begin{eqnarray}
\rho&=&\sqrt{x^2+y^2}\\
\theta&=&\tan^{-1}{\frac{y}{x}}\\
z&=&z
\end{eqnarray}
Spherical coordinates:
\begin{eqnarray}
r&=&\sqrt{x^2+y^2+z^2}\\
\theta&=&\cos^{-1}{\frac{z}{\sqrt{x^2+y^2+z^2}}}\\
\phi&=&\tan^{-1}{\frac{y}{x}}
\end{eqnarray}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/729811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability of Head in coin flip when coin is flipped two times Probability of getting a head in coin flip is $1/2$.
If the coin is flipped two times what is the probability of getting a head in either of those attempts?
I think both the coin flips are mutually exclusive events, so the probability would be getting hea... | Let $A$ be the event of getting a tail in both tosses, then $A'$ be the event of getting a head in either tosses. So $P(A') = 1 - P(A) = 1 - 0.5*0.5 = 0.75$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/729920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 0
} |
If one of the hypotheses holds, then one of the conclusions holds. (looking for a proof) Using a huge truth table, I proved the theorem below.
I cannot find a more elegant proof. I tried to rewrite expressions; e.g. using the distributive laws and the laws of absorption - to no avail. Is there another proof - or any hi... | So you want to prove $$((A \Rightarrow A') \wedge (B \Rightarrow B')) \Rightarrow (A \vee B \Rightarrow A' \vee B')$$
You can rewrite it as:
$$\begin{align}
& \equiv (\neg (A \Rightarrow A') \vee \neg (B \Rightarrow B')) \vee (A \vee B \Rightarrow A' \vee B') \\
& \equiv ((A \wedge \neg A') \vee (B \wedge \neg B')) \v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and vice versa Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and $\overline{\text{int}(A)}\not\... | Consider $\Bbb R$ with the usual metric. Let $A$ be the set of rationals from $0$ to $1$ together with the set of reals from $2$ to $3$, all four endpoints excluded. Then
$${\rm int}(\overline A)=(0,1)\cup(2,3)$$
and
$$\overline{{\rm int}(A)}=[2,3]\ .$$
Neither is a subset of the other.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/730158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Strict convexity of $c_0$ Let $c_0$ be a spaces of sequences converging to $0$ with the following norm
$$
\|x\|=\sup\{|x_i|: i\in \mathbb{N}\}+\left(\sum_{i=1}^{\infty}\left(\frac{x_i}{i}\right)^2\right)^{\frac{1}{2}}
$$
Prove that $(c_0,\|.\|)$ is strictly convex but not uniformly convex.
Thank you for your kind help... | Following the guidance of Robert Israel on the link
Strict convexity of a norm on $C[0,1]$
we can prove that $(c_0, \|.\|)$ is strictly convex. Indeed, suppose that $x, y\in c_0\setminus\{0\}$ such that $\|x+y\|=\|x\|+\|y\|$.
Consider the inner product and the norm on $c_0$ given by
$$
\langle x,y\rangle=\sum_{i=1}^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What does it mean for ultrafilter to be $\kappa$-complete? What does it mean when ultrafilter is said to be $\kappa$-complete? I cannot find suitable Internet resource, so I am asking here.
| If $\cal U$ is a filter, we say that it is $\kappa$-complete if whenever $\gamma<\kappa$, and $\{A_\alpha\mid\alpha<\gamma\}\subseteq\cal U$, then $\bigcap_{\alpha<\gamma} A_\alpha\in\cal U$. (In this context, let an intersection over an empty family to be the set $X$ over which $\cal U$ is taken.)
If $\cal U$ is a $\k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
What is the distribution of the dot product of a Dirichlet vector with a fixed vector? I am trying to get the distribution of a weighted sum when the weights are uncertain:
$S = \sum\limits_{i=1}^N w_iC_i = \mathbf{w}\cdot \mathbf{C}$ where vector $\mathbf{w}$ is random with components having an N-dimensional Dirichlet... | I'm very interested in this distribution as well, unfortunately I believe TenaliRaman's answer is only an approximation. Perhaps it only holds when the distribution is sufficiently concentrated to be approximately Gaussian..?
I tried the following in R:
require(gtools)
library(MASS)
n <- 7000
C <- c(0,1,0.2,0.5)
alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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