qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
697,458 | <p>Using a trivial example to illustrate the question -</p>
<p>$$\int_0^\infty 2 dx$$</p>
<p>$=2x \mid_0^\infty = 2(\infty) - 2(0)$</p>
<p>Can we actually say $2(\infty)$? It doesn't seem valid me.</p>
| Community | -1 | <p>The expression $2 \cdot (+\infty) - 2 \cdot 0$ does indeed make sense. It is rather useful to do arithmetic/analysis with $\pm \infty$ alongside the other real numbers; e.g $\lim_{x \to +\infty}$ has all of the same properties that $\lim_{x \to a}$ does, and $\lim_{x \to a} f(x) = +\infty$ behaves much more like a c... |
1,531,050 | <p>a. Say $X_n$ and $Y_n$ are two sequences such that $|X_n-Y_n| < \frac{1}{n}$ and $\lim X_n$ = $z$. Prove $\lim Y_n$ = $z$.</p>
<p>b. Say $X_n$ takes values in closed interval $[a,b]$ and $\lim X_n$ = $z$. Prove $z$ takes values in closed interval $[a,b]$.</p>
<p>c. Does b. hold when the interval is open?</p>
| G H | 282,682 | <p>a) Let $N$: $\forall n > N$ $\ \ \ |X_n - z| < \frac{1}{n}$.</p>
<p>$|z - Y_n| < |z - X_n| + |X_n - Y_n| < \frac{2}{n}$ for $n > N$.</p>
<p>b) If not than $\exists$ $\epsilon $ : $z \in (z - \epsilon, z + \epsilon)$ because $\mathbb{R} / [a,b]$ is open.</p>
<p>But $\exists N$: $\forall n > N$ $... |
2,507,565 | <p>I'm failing to find any reasonable solution to this given problem.</p>
<p>Find explicit formula of $n$-th element from given sequence:
$$\begin{cases}
a_1 = 1\\
a_2 = 2\\
a_{n+2} = 4 a_{n+1} + 4 a_n + 2^n
\end{cases}$$</p>
<p>I have tried finding some clues whilst typing out few first expressions to no avail. I al... | Benedict W. J. Irwin | 330,257 | <p>Solving recurrences can be tricky. A systematic approach is to try to solve for the <a href="https://en.wikipedia.org/wiki/Generating_function" rel="nofollow noreferrer">generating function</a> of the sequence and then work from there, but to get the general sequence term from the generating function can also be har... |
538,870 | <p>Evaluate</p>
<p>$$\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right).$$</p>
<p>This sequence looks extremely horrible and it makes me crazy. How can we evaluate this?</p>
| robjohn | 13,854 | <p>Stirling's asymptotic formula (without any series enhancement)
$$
n!\sim\frac{n^n}{e^n}\sqrt{2\pi n}\tag{1}
$$
put into Landau notation is
$$
n!=\frac{n^n}{e^n}\sqrt{2\pi n}\,(1+o(1))\tag{2}
$$
which gives
$$
\begin{align}
(n!)^{1/n}
&=\frac ne(2\pi n)^{\frac1{2n}}(1+o(1/n))\\
&=\frac ne\left(1+\frac{\log(2\... |
1,299,552 | <p>Can you tell me why my answer is wrong?</p>
<p>$$\frac {x+y} {x-y} + \frac 1 {x+y} - \frac {x^2+y^2} {y^2-x^2} = \frac {x^2 + y^2} {x^2-y^2} + \frac {x-y} {x^2-y^2} + \frac {x^2+y^2} {x^2-y^2} = 2x^2 + 2y^2 + x-y$$</p>
| jejeje | 243,585 | <p>$(x+y)(x+y)$ does not equal $x^2+y^2$. It equals $(x+y)^2$ or $x^2+2xy+y^2$.</p>
|
57,642 | <p>I'm looking for the shortest and the clearest proof for this following theorem:</p>
<p>For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , I need to show $V= \ker T^n \oplus $ Im $T^n$.</p>
<p>Any hints? I don't know where to start from.</p>
<p>Thank you.</p>
| user89987 | 89,987 | <p>This appears in Axler Sheldon's LADR in the exercises to chapter 8. The exercise right before it reads as follows:</p>
<p>Suppose $P \in \mathcal L (V)$ and $P^2 =P$. Prove that $V=\mathrm{null}P \oplus \mathrm{range}P$.</p>
<p>In the same chapter it is shown that if $n=\mathrm{dim}V$ then $\mathrm{null}T^n =\math... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | BlueRaja - Danny Pflughoeft | 136 | <p>Most of us know that, being deterministic, computers cannot generate true <a href="http://en.wikipedia.org/wiki/Pseudorandom_number_generator" rel="noreferrer">random numbers</a>.</p>
<p>However, let's say you have a box which generates truly random binary numbers, but is biased: it's more likely to generate eithe... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | BlueRaja - Danny Pflughoeft | 136 | <p>Assuming you have unlimited time and cash, is there a strategy that's guaranteed to win at roulette?</p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Sev | 1,630 | <p>Relatively simple, but still fun...</p>
<p>Unknown source, other than my friend telling me yesterday</p>
<p>There is a room with 50 doors in it, each door leading to a cell that is completely sound-proof and light-proof, and each cell can hold a maximum of 1 person. There are 50 prisoners. There also exists a li... |
3,387,024 | <p>We denote by <span class="math-container">$H$</span> the subspace of <span class="math-container">$C([0,1])$</span> of functions <span class="math-container">$h$</span> such that <span class="math-container">$h(0)=0$</span>, <span class="math-container">$h$</span> is absolutely continuous and its derivative <span cl... | N. S. | 9,176 | <p><strong>Hint</strong> For each odd prime <span class="math-container">$p$</span>, if <span class="math-container">$p^k |m \leq n$</span> then
<span class="math-container">$$p^k |2^{\frac{p-1}{p}m}-1$$</span> and hence
<span class="math-container">$$p^k| 2^{n}-2^{n-\frac{p-1}{p}m}$$</span> </p>
|
3,387,024 | <p>We denote by <span class="math-container">$H$</span> the subspace of <span class="math-container">$C([0,1])$</span> of functions <span class="math-container">$h$</span> such that <span class="math-container">$h(0)=0$</span>, <span class="math-container">$h$</span> is absolutely continuous and its derivative <span cl... | WhatsUp | 256,378 | <p>A nice solution is given by the natural embedding of <span class="math-container">$S_n$</span> into <span class="math-container">$\mathrm{GL}_n(\mathbb{F}_2)$</span>.</p>
<p>Here <span class="math-container">$S_n$</span> is the symmetric group of <span class="math-container">$n$</span> elements. Every element of <s... |
655,591 | <p>$S=(v_1, \cdots v_n)$ of vectors in $\mathbb{F^n}$ is a basis iff the matrix obtained by forming a matrix (call it A) of the co-ordinate vectors of $v_i$ is invertible</p>
<p>My Idea: I was able to prove the reverse direction wherein we can show that $AX =0$ has trivial solutions so linearly independent and also sp... | Blah | 6,721 | <p>You can rewrite $A$ as a row of vectors $A_1,\dots,A_n$. Then
$$
Ax = x_1 A_1 + \dots + x_n A_n
$$
Think about</p>
<p>$Ax=b$ always solvable means $A_i$ generate $V$</p>
<p>$Ax=b$ uniquely solvable (if at all) means $A_i$ independent</p>
<p>If these two conditions are met, you can find $n$ solutions for $Ax_i=e_i... |
724,302 | <p>I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is.</p>
<p>Is category theory a content in ZFC-set theory? (Just like measure theory, group theory etc.) If not, is it just another formal logic system independent from the standard ZFC-set theory?</p>
<p>Fol... | Geremia | 128,568 | <p>See:</p>
<ul>
<li>Bell, J. L., “<a href="https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/IY3V5UCD/Bell,%20J.%20L.%20-%201981%20-%20Category%20Theory%20and%20the%20Foundations%20of%20Mathematics.pdf" rel="nofollow noreferrer">Category Theory and the Foundations of Mathematics</a>,” <em>British ... |
387,519 | <p>The domain of the following function $$y=2$$ is just 2? And the image of it?</p>
<p>I don't think I quiet understand what the image of a function means. The domain is all values that it can assume, correct?</p>
<p>Could you please try to define the image of this equation too: $$y = 2x - 6$$ so I can try to underst... | fgp | 42,986 | <p>The <em>image</em> of a function is simply the set of all <em>possible</em> values the function can take. If you have a function $f(x)$, and evaluate it for all possible values of $x$ (i.e., for all $x$ in the <em>domain</em> of f), then the resulting set of values is called the <em>image</em> of $f$.</p>
<p>You ca... |
506,767 | <p>What is the largest number on multiplying with itself gives the same number as last digits of the product?</p>
<p>i.e., $(376 \times 376) = 141376$</p>
<p>i.e., $(25\times 25) = 625$</p>
<p>If the largest number cant be found out can you prove that there is always a number greater than any given number? (only in ... | robjohn | 13,854 | <p>Looking for $k$ so that $k^2\equiv k\pmod{10^n}$, we get
$$
k(k-1)\equiv0\pmod{10^n}
$$
Since $\gcd(k,k-1)=1$, we must have one of
$$
\begin{align}
k&\equiv0\pmod{10^n}\tag{1}\\
k&\equiv1\pmod{10^n}\tag{2}\\
k&\equiv0\pmod{2^n}\quad\text{and}\quad k\equiv1\pmod{5^n}\tag{3}\\
k&\equiv1\pmod{2^n}\quad\... |
2,050,939 | <p>My daughter got the following optional question for 8th grade homework mixed into her simultaneous equations questions. All of the other questions were simple for me to solve using substitution or elimination which is what she is being taught at the moment but this one seems to me to be at a much higher grade.</p>
... | Community | -1 | <p>Let us consider $m$ red balls and $n$ blue balls. Let $u_{mn}(r)$ and $u_{mn}(b)$ be the probability that the sequence of balls being discarded should begin and end with a red ball and a blue ball respectively. We require $u_{mn} = u_{mn}(r) + u_{mn}(b)$.</p>
<hr>
<p>Consider $u_{mn}(r)$. The probability of choosi... |
2,084,671 | <p>I've tried simplifying the term by <strong>substituting</strong> $y := x^2$ and also to use a certain <strong>algorithm</strong> that outputs the f(a), but it gets too big for the input r, that is a variable and not a number to begin with (added little example):</p>
<p><a href="https://i.stack.imgur.com/8C9sV.png" ... | Arnaldo | 391,612 | <p><strong>Hint</strong></p>
<p>Completing square (it is equivalent to use Baskara's formula):</p>
<p>$x^4-6qx^2+q^2=(x^2-3q)^2-8q^2=0 \rightarrow (x^2-3q)^2=8q^2 \rightarrow x^2=(3 \pm2\sqrt{2})q=(1\pm \sqrt{2})^2q$</p>
<p>$$x^2=(1\pm \sqrt{2})^2q$$</p>
<p>Can you finish?</p>
|
42,258 | <p>I have a Table of values e.g. </p>
<pre><code>{{x,y,z},{x,y,z},{x,y,z}…}
</code></pre>
<p>How do I replace the the "z" column with a List of values?</p>
| bill s | 1,783 | <p>When I read this question, it seems to be asking for set-delayed. For example, say we want to calculate <code>D[f[t],t]</code> over and over but want to give it a name like <code>q</code>. Then</p>
<pre><code>q := D[f[t], t]
</code></pre>
<p>does this. If you don't have f defined then</p>
<pre><code>q
Derivative[... |
549,585 | <p>Let $a_1=1$ and $a_{n+1}=ma_n+10$ for $n\geq 2$. Determine for which value of m, the sequence $(a_n)$ is convergent and find the limit.</p>
<p>I know that if if we simplify we will get $0=L(M-1)+10$, which is equal to $10/(1-M)=L$ but I have no clue what to do from there</p>
| detnvvp | 85,818 | <p>If $m=1$, then $a_n=10(n-1)+1$ for all $n\in\mathbb N$, so the sequence does not converge. Suppose that $m\neq 1$. Then, we can prove that $a_{n+1}=m^n+10\frac{m^n-1}{m-1}$ for $n\geq 1$ by induction: this holds for $n=2$, and, if it holds for some $n\geq 2$, then $$a_{n+2}=ma_n+10=m\left(m^n+10\frac{m^n-1}{m-1}\rig... |
1,143,889 | <p>I am interested in counting the number of hyperedges in the complete, $t$-uniform hypergraph on $n$ vertices which intersect a specified set of $k$ vertices. This is trivial, the answer is:</p>
<p>$$\sum_{i=1}^t {k \choose i}{n-k \choose t-i}.$$</p>
<p>My questions is whether there is a nice simplification of thi... | Alijah Ahmed | 124,032 | <p>We can use the <a href="http://en.wikipedia.org/wiki/Binomial_coefficient">Chu-Vandermonde identity</a> (see Equation 7 in linked page):-
$$\sum_{i=0}^t {k \choose i}{n-k \choose t-i} = {n \choose t}$$
so that the sum can be simplified to
$$\sum_{i=1}^t {k \choose i}{n-k \choose t-i} = {n \choose t}-{n-k \choose t}$... |
173,745 | <p>I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is separable.' It seems pretty straightforward, but I would really appreciate it if someone could verify that my proof works.<... | Kevin Arlin | 31,228 | <p>Your proof goes through. For countable products, make the analogous argument using the metric
$$d(x,y)=\sum_{i=1}^\infty \frac{1}{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}$$ The summands are now bounded by $\frac{1}{2^i}$, so the sum always converges, and defines a metric on the product space.</p>
<p>Incidentally, if... |
173,745 | <p>I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is separable.' It seems pretty straightforward, but I would really appreciate it if someone could verify that my proof works.<... | Jonathan Hole | 661,524 | <p>The answer to your question whether the countable product of seperable metric spaces is seperableis affirmative. However I thnik it's useful to prove a slight generalization of this, namely that the countable product of second countable topological spaces is countable. Indeed, suppose <span class="math-container">$\... |
3,699,303 | <p>Let <span class="math-container">$R$</span> be a commutative ring with <span class="math-container">$1$</span>. This is a simple question, but I can't see whether it is true or not.</p>
<p>Suppose <span class="math-container">$R$</span> does not have a "unit" of (additive) order <span class="math-container">$2$</sp... | Ender Wiggins | 424,790 | <p>No, consider the example <span class="math-container">$R = \mathbb{C}[X]/\langle X^2-1\rangle$</span>. If you denote <span class="math-container">$x = X + \langle X^2-1\rangle$</span> then <span class="math-container">$x^2=1$</span> and hence it has order <span class="math-container">$2$</span> in the multiplicative... |
2,812,518 | <p>I am self learning calculus and have a problem that may be simple here but I cant find an answer on the web so here it is:</p>
<p>If we have the equation:</p>
<p>$$y = y^2x$$</p>
<p>and differentiate it (implicitly and using product rule) we get:</p>
<p>$$\frac{dy}{dx} = \frac{y^2}{1-2xy}.$$</p>
<p>However, $y... | Eric Wofsey | 86,856 | <p>Remember that you found that $y=\frac{1}{x}$. So, $$\frac{y^2}{1-2xy}=\frac{1/x^2}{1-2}=-\frac{1}{x^2}$$ and the two answers you have found actually are the same.</p>
<p>However, there are some issues with division by $0$ in your second approach. It is <em>not</em> correct to simplify $y=y^2x$ to $1=xy$; that is ... |
1,144,540 | <p>I'm looking to determine the convergence of $\{e^{in}\}_{n=1}^\infty$ in $\mathbb{C}$, where $n \in \mathbb{N}$. Using the definition of convergence that iff $\exists \ \epsilon > 0: \forall \ n_0 \in \mathbb{N}, \exists \ n \ge n_0, |z_n - z| \le \epsilon$. </p>
<p>Using $z_n = e^{in} = \cos{n} + i\sin{n}$, a... | BCLC | 140,308 | <ol>
<li><p>$$\infty > \sum_{n} \lambda_n = \sum_{n} E[X_n] = E[\sum_{n} X_n]$$</p></li>
</ol>
<p>$$\to \infty > \sum_{n} X_n$$</p>
<ol start="2">
<li>$$E[X_n] = Var[X_n] = \lambda_n$$</li>
</ol>
<p>Use contrapositive of <a href="https://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem" rel="nofollow"... |
1,785,044 | <p>Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side (say $AB$) of the polygon is shorter than the circular arc with the same endpoints ($\stackrel{\frown}{AB}$). Summin... | Oscar Lanzi | 248,217 | <p>Pick a point $F$ on one of the arcs of the circle, let us say it's on arc that faces $C$ in your circumscribed quadrilateral. Construct a line segment $GFH$ with $G$ on $BC$ and $H$ on $CD$, tangent to the circle at $F$. $GFH$,being a straight segment, is shorter than $GC+CH$, so the circumscribed pentagon $ABGHD$... |
1,785,044 | <p>Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side (say $AB$) of the polygon is shorter than the circular arc with the same endpoints ($\stackrel{\frown}{AB}$). Summin... | Richard Hevener | 17,411 | <p>This proof assumes the fact (sometimes used as a definition) that the circumference of a circle is the least upper bound of the perimeters of all polygons inscribed within it.</p>
<p>Def.: If $A$ is a polygon, $P(A)$ is its perimeter. If $C$ is a circle, $P(C)$ is its circumference.</p>
<p>Lemma: If $A$ and $B$ ar... |
3,176,482 | <p>In my Econometrics class yesterday, our teacher discussed a sample dataset that measured the amount of money spent per patient on doctor's visits in a year. This excluded hospital visits and the cost of drugs. The context was a discussion of generalized linear models, and for the purposes of Stata, he found that a g... | r.e.s. | 16,397 | <p>(Expanding on a comment by @Michael, this might help some readers understand the R computations in the accepted answer by @Henry.)</p>
<p>Let <span class="math-container">$$Y=
\begin{cases}
0 &\text{ if }K=0\\
X_1 + \ldots + X_K&\text{ if }K>0
\end{cases}$$</span>
where the <span class="math-container">$... |
988,566 | <p>For x ∈ ℝ, define by:
⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋).</p>
<p>Use this definition to prove or disprove the following with a structured proof technique:
∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋.</p>
<p>I understand I need to start with assuming the domain to be true aswell as the antecedent, then equati... | Ben Grossmann | 81,360 | <p>It is clear that the matrix is symmetric. It suffices, therefore, to show that its eigenvalues are non-negative.</p>
<p>Note that the eigenvalues of $11^T$ are $0$ and $n$. Thus, the eigenvalues of $H$ must be $1-\rho$, and $(1 - \rho) + \rho n = 1 + (n-1) \rho$ respectively. Note that both of these must be non-... |
1,728,910 | <p>Whenever I get this question, I have a hard time with it. </p>
<p>An example of a problem:</p>
<p>In the fall, the weather in the evening is <em>dry</em> on 40% of the days, <em>rainy</em>
on 58% of days and <em>snowy</em> 2% of the days. </p>
<p>At noon you notice clouds in the sky. </p>
<p>Clouds appear
at noo... | DeepSea | 101,504 | <p>Hint: $P\left(S \mid C \right) = \dfrac{P\left(C \mid S \right)P\left(S\right)}{P\left(C\right)}$.</p>
|
226,534 | <p>I have the following function for which I want to know the range</p>
<pre><code>FunctionRange[{1/
32 (8 - Sqrt[(-8 - 36 Abs[y] (1 + 2 Sqrt[Abs[z]]) -
27 (Abs[y] + 2 Abs[y] Sqrt[Abs[z]])^2)^2 -
64 (1 + Abs[y] + 2 Abs[y] Sqrt[Abs[z]])] +
36 Abs[y] (1 + 2 Sqrt[Abs[z]]) +
27 (Abs[y] + 2 Ab... | Bob Hanlon | 9,362 | <pre><code>expr = 1/32 (8 -
Sqrt[(-8 - 36 Abs[y] (1 + 2 Sqrt[Abs[z]]) -
27 (Abs[y] + 2 Abs[y] Sqrt[Abs[z]])^2)^2 -
64 (1 + Abs[y] + 2 Abs[y] Sqrt[Abs[z]])] +
36 Abs[y] (1 + 2 Sqrt[Abs[z]]) +
27 (Abs[y] + 2 Abs[y] Sqrt[Abs[z]])^2);
</code></pre>
<p>Looking at the plot,</p>
<pre><code... |
259,751 | <p>I'm playing this video game where people can get kills, deaths, and assists , and all this is recorded on a stats website. The stats website gives you a rating by directly manipulating these numbers.</p>
<p>In the first entry, I have 26 kills, 5 deaths, and 19 assists. The KDA ratio the website gave me was 29.8. At... | Dan Brumleve | 1,284 | <p><a href="http://terrytao.files.wordpress.com/2011/02/polymath.pdf" rel="nofollow">No prime-generating function is known to be computable in polynomial time</a>. However, maybe there is a solution to the "almost-all" case.</p>
|
167,891 | <p>I'm working through the James Stewart Calculus text to prep for school. I'm stuck at this particular point. </p>
<p>How would you sketch the graph for the parametric equations:
$x = \cos t$, $y = \sin t$, and $z = \sin 5t$? I understand that if it were the case that $z=t$, I'd merely get a helix around the $z$-axis... | Zev Chonoles | 264 | <p>Here is an animation I made that might help. The left is a plot of $(\cos(t),\sin(t),\sin(5t))$ and the right is a plot of $\sin(5t)$.</p>
<p><img src="https://i.stack.imgur.com/y0gen.gif" alt="enter image description here"></p>
<p>For the case of $(\cos(t),\sin(t),\ln(t))$, here is the corresponding animation:</p... |
1,687,500 | <p>Prove that, any group of order $15$ is abelian (without help of Sylow's theorem or its application).</p>
<p>What I have done so far is, </p>
<p>by class equation we know that $|G|=|Z|+\sum\frac{|G|}{C(a_i)}$. Now if I can show that $|G|=|Z|$ then the theorem is proved. Now order of $|Z|$ can not be $3$ or $5$, bec... | Shahab | 10,575 | <p>If there is a non basic variable $x_r$ such that $z_r−c_r=0$ (and all others are $\ge 0$) and point 3 (the one about the min ratio) is satisfied one can still force $x_q$ to leave and $x_r$ to enter. This will not improve the optimal value but give an alternative solution</p>
|
1,687,500 | <p>Prove that, any group of order $15$ is abelian (without help of Sylow's theorem or its application).</p>
<p>What I have done so far is, </p>
<p>by class equation we know that $|G|=|Z|+\sum\frac{|G|}{C(a_i)}$. Now if I can show that $|G|=|Z|$ then the theorem is proved. Now order of $|Z|$ can not be $3$ or $5$, bec... | fredq | 297,080 | <h1>1. Lets show $x_q$ can't leave the basis</h1>
<h2>Intuitively</h2>
<p>If the row $s$ contains all zeros except in the column of $x_q$, it means the problem contains a constraint of the type</p>
<p>$$x_q = b_s$$</p>
<p>In that case, only solutions with $x_q = b_s$ are feasible, so the variable $x_q$ never leaves... |
89,638 | <p>I'm trying to solve an exercise as it follows:</p>
<p>$\alpha, \beta \in \mathbb{C}$ such that $a^{37}=2=\beta^{17}$. Note that both are prime.</p>
<p>a) Find the minimum polynomials of $\alpha$ and $\beta$ over $\mathbb{Q}$.</p>
<p>Well, I guess that $(x^{37}-2)$ and $(x^{17}-2)$ are irreducible over $\mathbb{Q}... | Community | -1 | <p>If $\varphi$ is the characteristic function of $X$, then $\Re(\varphi)$ is the characteristic function of a symmetrized version $X_{\rm sym}$ of $X$. </p>
<p>That is, $X_{\rm sym}=\varepsilon X$, where $\varepsilon$ is independent of $X$ and $\mathbb{P}(\varepsilon=+1)=\mathbb{P}(\varepsilon=-1)=1/2$. </p>
|
2,431,052 | <p>Let $M$ be a an oriented riemannian manifold.
I have seen the following definition for the Hodge-star operator acting on a differential form. Starting with $\beta\in \Omega^p(M)$ we have
$$\alpha \wedge \star \beta = \left<\alpha,\beta\right>\text{vol} ~~\forall \alpha \in \Omega^p(M)$$</p>
<p>where $ \left&l... | Ted Shifrin | 71,348 | <p>The Hodge star is not any different. But, depending on the notation you're using, if you have Lie-algebra valued forms, you use wedge on the forms and Lie bracket on the Lie-algebra part. To be explicit, if you have
$F= \sum\limits_j \omega_j\otimes v_j$ and $G = \sum\limits_k\eta_k\otimes w_k$, with $\omega_j,\eta_... |
2,389,581 | <p>A town has 2017 houses. Of these 2017 houses, 1820 have a dog, 1651 have a cat, and 1182 have a turtle. If x is the largest possible number of houses that have a dog, a cat, and a turtle, and y is the smallest possible number of houses that have a dog, a cat, and a turtle, then what is the value of x−y.</p>
| Math Lover | 348,257 | <p>You already figured out $x=1182$. Now for computing $y$, we take the following strategy.</p>
<p>Without loss of generality, we presume each of the houses 1 to 1820 has a dog. To minimize the 'overlap', each of the remaining houses (1821 to 2017 i.e. 197 houses) contains both a cat and a turtle. So we are left with ... |
1,887,395 | <p>Let $D=\{Z\in\mathbb{C}:|z|<1\}$. Suppose $f:\overline D\to\mathbb{C}$ is continuous, holomorphic in $D$, and $f(\partial D)\subset \mathbb{R}$. Define $h:\mathbb{C}\to\mathbb{C}$ by
$$
h(z)=\begin{cases}
f(z) &\text{if}\ z\in \overline D\\
\overline{f(1/\overline z)} &\te... | Jack D'Aurizio | 44,121 | <p>We have:
$$ -\log(1-x) = \sum_{n\geq 1}\frac{x^n}{n}\tag{1} $$
hence:
$$ \frac{-\log(1-x)}{x(1-x)} = \sum_{n\geq 1} H_n x^{n-1} \tag{2} $$
and:
$$ \sum_{n\geq 1}H_n\left(\frac{1}{n}-\frac{1}{n+k}\right) = \int_{0}^{1}\frac{-\log(1-x)}{x(1-x)}(1-x^k)\,dx \tag{3}$$
can be easily computed by expanding $\frac{1-x^k}{1-x... |
2,415,957 | <p>I would like to prove the Second Isomorphism Theorem in an arbitrary abelian category without appealing to Mitchell's Embedding Theorem.</p>
<p>Given two subobjects $M', M''$ of an object $M$ in an abelian category $\mathcal A$, we define the intersection $M'\cap M'' := \ker(M\rightarrow M/M' \oplus M/M'')$ and the... | emma | 406,460 | <p>An important identity in trigonometry is that $\sin^2 \theta + \cos^2 \theta = 1$. One way to see that this is true, at least for $0^{\circ} \leq \theta \leq 90^{\circ}$, is to draw a right triangle. Choose one of the acute angles as $\theta$, and label the leg opposite that angle as $o$, the side next to that angle... |
2,415,957 | <p>I would like to prove the Second Isomorphism Theorem in an arbitrary abelian category without appealing to Mitchell's Embedding Theorem.</p>
<p>Given two subobjects $M', M''$ of an object $M$ in an abelian category $\mathcal A$, we define the intersection $M'\cap M'' := \ker(M\rightarrow M/M' \oplus M/M'')$ and the... | user577215664 | 475,762 | <p>$$ sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac 1 9 = \frac 8 9 $$
$$ \sin \theta = \sqrt \frac 8 9 = \frac { 2 \sqrt 2} {3} $$</p>
<p>Once you have one of the trigonometric functions you get all of them, $\tan \theta $ or $ \cos \theta $</p>
|
186,395 | <p>i want write a module to find the integer combination for a multi variable fomula. For example</p>
<p>$8x + 9y \le 124$</p>
<p>The module will return all possible positive integer for $x$ and $y$.Eg. $x=2$, $y=12$.
It does not necessary be exactly $124$, could be any number less or equal to $124$. Must be as close... | Gerry Myerson | 8,269 | <p>Given a finite set $U$ of positive integers, and positive integers $B$ and $K$, the problem of determining whether there are non-negative integers $a(u)$ such that $K\le\sum_{u{\rm\ in\ }U}a(u)u\le B$ is NP-complete (which implies that it is computationally infeasible now and, perhaps, forever). This is discussed un... |
301,318 | <p>As the title stated , what is the meaning of infinitely many ? When we say a set contains infinitely many elements, does this mean we cannot finish counting all the elements in the set ? Does infinitely many same as $\forall$ ? </p>
| Asaf Karagila | 622 | <p>No, $\forall$ means that <em>for all</em> elements. Infinitely many just means there are infinitely many elements with a certain property.</p>
<p>For example $A=\{p\in\mathbb N\mid p\text{ is a prime and } p\equiv 1\pmod 4\}$ contains <em>infinitely many</em> prime numbers, but it is not true that all the prime num... |
301,318 | <p>As the title stated , what is the meaning of infinitely many ? When we say a set contains infinitely many elements, does this mean we cannot finish counting all the elements in the set ? Does infinitely many same as $\forall$ ? </p>
| Gautam Shenoy | 35,983 | <p>You have to understand this from basic definitions.</p>
<p>Firstly, a set A is said to be infinite if it is "not finite" OR equivalently, there exists a bijection from A to a <strong>strict</strong> subset of itself. Additionally, we don't say a set contains infinitely many elements, instead we say it is infinite. ... |
531,895 | <p>Assume K is a field that is not algebraically closed, then how to prove that for any m>0, there is a polynomial with m variables over K that possess a unique zero point?</p>
| Hagen von Eitzen | 39,174 | <p>As $K$ is not algebraically closed, there exists $p\in K [X]$ with no root in $K$ and $\deg p\ge1$. Let $f(X,Y)=Y^{\deg p}p\left(\frac XY\right)$, that is, if $\deg p=d$ and
$$p(X)=a_0+a_1X+\ldots +a_dX^d $$
we let
$$ f(X,Y)=a_0Y^d+a_1XY^{d-1}+\ldots +a_dX^d. $$</p>
<p>Then clearly $f(0,0)=0$. If $v\ne 0$ then $f(u... |
1,283,541 | <p>Find the least value of $f(x)=3^{-x+1} + e^{-x-1}$.</p>
<p>I tried to use the maxima/minima concept but it was of no use. Please help.</p>
| WimC | 25,313 | <p>On some suitable interval around $x$ $$1\geq \cos(x) \geq \cos(x) - x \sin(x)\geq 0.$$ Now integrate on this interval to find for $x\geq 0$ $$x \geq \sin(x) \geq x \cos(x)$$ and the opposite inequalities for $x\leq 0$.</p>
|
89,622 | <p>I'm having a hard time trying to understand a proof of the Principle of Local Reflexivity. I'm following the proofs from </p>
<p>1) Topics in Banach Space Theory (by Fernando Albiac, Nigel J. Kalton)
2) <a href="http://www.math.tamu.edu/~schlump/sofar.pdf" rel="nofollow">http://www.math.tamu.edu/~schlump/sofar.pdf<... | Bill Johnson | 2,554 | <p>Enlarge $G$, if necessary, so that the restriction mapping from $F$ to $G^*$ is one to one. Then what you want is completely obvious from the equality 4 lines from the bottom of p. 274 of [AK]. (The proof is correct without this step, but enlarging $G$ makes it clear without thinking.)</p>
|
3,004,767 | <p>If <span class="math-container">$\displaystyle\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$</span> for <span class="math-container">$n \in \mathbb{N}$</span>. Then prove that <span class="math-container">$\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$</span></p>
<p><span class="math-container">$$\lambda_n=\int_{0}^{1} ... | Yiorgos S. Smyrlis | 57,021 | <p>Actually,
<span class="math-container">$$
\int_0^1 \frac{dt}{(1+t)^n}=\left.\frac{1}{1-n}\frac{1}{(1+t)^{n-1}}\,\right|_0^1=\frac{1}{n-1}-\frac{2^{-n+1}}{n-1}
$$</span>
and hence, for all <span class="math-container">$n>1$</span>
<span class="math-container">$$
\frac{1}{2(n-1)}<\int_0^1 \frac{dt}{(1+t)^n}<\... |
1,080,572 | <p>I am having problem with this question , kindly please help me with this ,</p>
<p>Let $$S = \{x : x^{6} -x^{5} \leq 100\}$$</p>
<p>And $$T =\{x^{2} - 2x : x \in ( 0, \infty)\}$$</p>
<p>Then I have to show that set $S \cap T$ is closed and bounded. $S$ appears to be closed and bounded as it less than or equal to $... | agha | 118,032 | <ol>
<li>Set $S \cap T$ is bounded. Let $x \in S \cap T$. Then $x \in S$, so $x \geq -1$ (it's because $(t-1)^=t^2-2t+1 \geq 0$ for all $t \in \mathbb{R}$) and $x \leq 101$ (because for $t>101$ we have $t^6-t^5=t^5(t-1)>100$). So $x \in [-1,101]$, thus $S \cap T \in [-1,101]$.</li>
<li>Set $S \cap T$ is closed. N... |
121,631 | <p>How to prove most simply that if a polyonmial $f$, has only real coefficients and $f(c)=0$, and $k$ is the complex conjugate of $c$, then $f(k)=0$?</p>
| Martin Argerami | 22,857 | <p>You use the fact that the coefficients of $f$ are real to show that
$$
f(\overline c)=\overline{f(c)}.
$$</p>
|
1,317,974 | <p>Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. This is problem 1.1.8 from Cox, Little, and Schenck, but my question regards a more general notion: How exactly does one... | Viktor Vaughn | 22,912 | <p>If you haven't already, you should consult Cox, Little, and O'Shea's <a href="http://www.cs.amherst.edu/~dac/iva.html" rel="nofollow">Ideals, Varieties, and Algorithms</a>: they treat this exact problem in Chapter 3 Elimination, particularly in $\S3$ Implicitization. The main result is the following theorem.</p>
<... |
921,868 | <p>I'm not quite sure whether this question belongs here, because it has no definite answer. But I'll give it a shot. If any of the mods objects, then I will, of course, respectfully delete this contribution.</p>
<p>A friend of mine, who knows a little arithmetic and only the bare rudiments of algebra, has asked me to... | user164587 | 164,587 | <p>A useful introduction to proof by induction is to take a $2 \times 2$ square with one corner removed as in the diagram below.</p>
<p><img src="https://i.stack.imgur.com/D2tji.png" alt="The basic building block"></p>
<p>The problem is to prove that using these shapes, one can cover any $2^n \times 2^n$ square with ... |
2,696,097 | <p>$ \lim_{n\to \infty} ( \lim_{x\to0} (1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)))^{\frac{1}{n^3x^2}} $</p>
<p>The answer should be $ {e}^\frac{1}{3} $</p>
<p>I haven't encountered problems like this before and I'm pretty confused, thank you.</p>
<p>I guess we must use the remarkable limit of $ \frac{\tan(x)}{x} ... | G Tony Jacobs | 92,129 | <p>Suppose $h\in H$ and $k\in K$. Where is $hk$? If $H$ and $K$ are both subgroups, with $H\cup K=G$ and $H\cap K=\{e\}$, then we have a problem. Do you see why?</p>
|
2,385,188 | <p>$c_R,R_f$ are known (constant part of return and risk free rate). Let $R=R_f(e^r-1)=c_R+\epsilon_R$, $r\sim N(\mu_r,\sigma_r)$, how to specify the distribution of $\epsilon_R\sim Lognormal(?,?)$ by mean and variance of a normal distribution?</p>
| Graham Kemp | 135,106 | <p>The implication/conditional: "$P\to Q$" means "If $P$, then $Q$" .</p>
<p>It is a guarantee that $Q$ is true whenever $P$ is true.</p>
<p>When $P$ is false, we might say the guarantee is <em>counterfactual</em>, but it is not <em>falsified</em>. If the guarantee is not false, what is it?</p>
<p>The guarant... |
2,104,382 | <p>I was trying to understand the proof of a theorem from Munkres's book.
He wants to prove that a compact subset $Y$ of an Hausdorff space $X$ is closed. He simply proves that, for all $x\in X-Y$, there is a neighborhood which is disjoint from $Y$. Why is this sufficient to conclude that $X-Y$ is open?</p>
| Kendall | 400,051 | <p>You can also prove this by showing that the closure of Y in X is open. </p>
|
3,346,543 | <p>I am aware this is a pretty big topic, but the attempts at layman's explanations I have seen either barely provide commentary on the formal proofs, or fail to provide an explanation (e.g "it gets too complex" does not really say anything)</p>
<p>Is there a good intuitive explanation as to why we fail to obtain a ge... | IamKnull | 610,697 | <p>There is intutive explanation given <a href="https://math.stackexchange.com/a/1733136/610697">here</a> through Galois Theory</p>
<p>you have to take Galois group of the equation, the group of automorphisms of the <em>splitting field</em> of the polynomial. For polynomial of any degree <span class="math-container">... |
225,730 | <p>I am reading a paper "2-vector spaces and groupoid" by <a href="http://arxiv.org/abs/0810.2361" rel="noreferrer">Jeffrey Morton </a> and I need a help to understand the following.</p>
<p>Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor category from $X$ to $\mathbb{Vect}$. Suppose we have... | Qiaochu Yuan | 232 | <p>You should learn something about the representation theory of finite groups. It suffices to understand the case that $X, Y$ have one object (note that the definition is as a direct sum of things that happen at one object); call those objects $x, y$ and let $G = \text{Aut}(x), H = \text{Aut}(y)$. Then $f$ may be rega... |
542,397 | <p>Consider the sequence $\{a_n\}_1^\infty$ such that $ a_n = $ ${n^m + 1}\over {n^{m+1} + 1}$ and $m \in \mathbb{R}$</p>
<p>EDIT: This is incorrect for $m < -1$, Then add the condition $m \geq -1$</p>
<p>I want to show this sequence is monotonically decreasing.<br>
This is not homework, I've seen many specific e... | Prahlad Vaidyanathan | 89,789 | <p>Take
$$
f(x) = \frac{x^m + 1}{x^{m+1} + 1}
$$</p>
<p>When you differentiate, your numerator should be
$$
(x^{m+1} +1)mx^{m-1} - (x^m+1)(m+1)x^m = -x^{m-1}(x^{m+1} + (m+1)x - m)
$$
Now, $x^m + (m+1)x - m > 0$ for large values of $x$, so $a_n$ is <em>eventually</em> decreasing. However, a little checking with sma... |
3,150,645 | <p>Suppose <span class="math-container">$\dim V = \dim W$</span>. Given <span class="math-container">$\phi:V\rightarrow V$</span> and <span class="math-container">$\psi:W\rightarrow W$</span> when do we have bijective linear maps <span class="math-container">$a,b:V\rightarrow W$</span> such that
<span class="math-cont... | egreg | 62,967 | <p>This is essentially the change of bases.</p>
<p>Suppose that <span class="math-container">$\phi$</span> and <span class="math-container">$\psi$</span> have the same rank <span class="math-container">$k$</span>. Take a basis <span class="math-container">$\{v_1,\dots,v_n\}$</span> of <span class="math-container">$V$<... |
286,932 | <p>For convenience we work with commutative rings instead of commutative algebras.</p>
<hr>
<p>Fix a commutative ring $R$. Consider the functor $\mathsf{Mod}\longrightarrow \mathsf{CRing}$ defined by taking an $R$-module $M$ to $R\ltimes M$ (with dual number multiplication). Following the <a href="https://ncatlab.org... | Nick Mertes | 178,527 | <p>Your question can be rephrased as follows: I know something about split square-zero extensions. How can I (categorically) conclude something about all square-zero extensions? The key observation is that square-zero extensions are the <em>torsors</em> for the split square-zero extensions. I first learned about this g... |
1,529,731 | <p>I'm currently studying for the GRE and a specific type of question has me stumped. I have a two part question, but first here is one problem and my work:</p>
<p><strong>"If x is the remainder when a multiple of 4 is divided by 6, and y is the remainder when a multiple of 2 is divided by 3, what is the greatest poss... | Anurag A | 68,092 | <p>When any integer $a$ is divided by $6$ then the possible remainders
are $0,1,2,3,4,5$. So if $4a$ is divided by $6$, then the possible remainders will be $0,4,2,0,4,2$ respectively (if you are familiar with modular arithmetic then this can be written more elegantly). So the maximum possible remainder when a multipl... |
1,529,731 | <p>I'm currently studying for the GRE and a specific type of question has me stumped. I have a two part question, but first here is one problem and my work:</p>
<p><strong>"If x is the remainder when a multiple of 4 is divided by 6, and y is the remainder when a multiple of 2 is divided by 3, what is the greatest poss... | Archis Welankar | 275,884 | <p>The pattern which is followed when divided by 6 a multiple of 4 is 0,4,2 starting from 0 and pattern followed by remainder when a multiple of 2 is divided is 0,2,1 when started from 0 so greatest value if $x+y=4+2=6$. Hope it helped you.</p>
|
82,865 | <p>I've looked through the other posts on rotating a function around the x or y axis and haven't had any luck.</p>
<p>my question is this how do i rotate the area between the functions $f(x)=x^2+1$, and $f(x)=2x^2-2$, whereby $x>0$ and $y>0$ and the region is bounded by the y and x axis.</p>
<p>This is a plot o... | Dr. belisarius | 193 | <pre><code>Show[RevolutionPlot3D[#, {x, 0, Sqrt[3]}, MeshStyle -> None,
PlotStyle -> Opacity[.5]] & /@ {2 x^2 - 2, x^2 + 1}]
</code></pre>
<p><img src="https://i.stack.imgur.com/Jq6nx.png" alt="Mathematica graphics"></p>
|
82,865 | <p>I've looked through the other posts on rotating a function around the x or y axis and haven't had any luck.</p>
<p>my question is this how do i rotate the area between the functions $f(x)=x^2+1$, and $f(x)=2x^2-2$, whereby $x>0$ and $y>0$ and the region is bounded by the y and x axis.</p>
<p>This is a plot o... | ubpdqn | 1,997 | <p>Just for something different:</p>
<pre><code>pp = ParametricPlot3D[{Sqrt[3] Cos[t], Sqrt[3] Sin[t], 4}, {t, 0,
2 Pi}, PlotStyle -> {Red, Thickness[0.04]}, Boxed -> False];
rp = RegionPlot3D[
0 < x^2 + y^2 < 3 &&
Max[0, 2 x^2 + 2 y^2 - 2] < z < x^2 + y^2 + 1, {x, -2, 2}, {y, -2... |
1,160,454 | <p>I missed two classes in calculus and we're on a subject that I do not understand at all. If someone could just walk me through this problem I could probably begin to comprehend the rest.</p>
<p>The base of a solid elliptical cylinder is given by $ (x/5)^2 + (y/3)^2 = 1.$ A solid is formed by cutting off or removin... | Narasimham | 95,860 | <p>$$ y = 3 \sqrt{ 1 -(x/5)^2 } $$</p>
<p>$$ \int dA = 4 \int_{-5} ^5 9 ( 1 -(x/5)^2 ) dx $$ </p>
<p>and you can take it from there.</p>
<p>EDIT1:</p>
<p>To 3D visualize sections perpendicular to x-axis are squares:
<img src="https://i.stack.imgur.com/stAiS.png" alt="SquareXnSolid"></p>
|
1,160,454 | <p>I missed two classes in calculus and we're on a subject that I do not understand at all. If someone could just walk me through this problem I could probably begin to comprehend the rest.</p>
<p>The base of a solid elliptical cylinder is given by $ (x/5)^2 + (y/3)^2 = 1.$ A solid is formed by cutting off or removin... | user84413 | 84,413 | <p>You can use the fact that the volume can be found by integrating the areas of the cross-sections.</p>
<p>In this case, the base of the solid is the region bounded by an ellipse, and the cross-sections are squares with sides $s=2\left(\frac{3}{5}\right)\sqrt{25-x^2}$, so</p>
<p>$\displaystyle V=\int_{-5}^{5}A(x)\;d... |
3,567,245 | <p>I am trying to compute <span class="math-container">$\int_{C}Log(z+3)$</span>, where C is a circle centered at the origin, with radius of 2, oriented once counterclockwise. </p>
<p>I kind of get the idea of how to compute <span class="math-container">$\int Log(z)$</span> on the unit circle. I know <span class="math... | Paweł Czyż | 551,592 | <p>Let's apply <a href="https://regularize.wordpress.com/2011/11/29/substitution-and-integration-by-parts-for-functions-of-a-complex-variable/" rel="nofollow noreferrer">Proposition 2</a> to:</p>
<ul>
<li>the circle <span class="math-container">$\gamma\colon [0,2\pi]\to \mathbb C, t\mapsto 2\exp it$</span>,</li>
<li>f... |
2,226,150 | <blockquote>
<p>The following diophantine equation came up in the past paper of a Mathematics competition that I am doing soon: $$ 2(x+y)=xy+9.$$ </p>
</blockquote>
<p>Although I know that the solution is $(1,7)$, I am unsure as of how to reach this result. Clearly, the product $xy$ must be odd since $2(x+y)$ must b... | Michael Rozenberg | 190,319 | <p>We need to solve $$xy-2(x+y)+4=-5$$ or
$$(x-2)(y-2)=-5$$
and it remains to solve the following systems.</p>
<p>$x-2=1$ and $y-2=-5$;</p>
<p>$x-2=-1$ and $y-2=5$;</p>
<p>$x-2=5$ and $y-2=-1$ and</p>
<p>$x-2=-5$ and $y-2=1$,</p>
<p>which gives the answer: $\{(1,7),(7,1), (3,-3),(-3,3)\}$.</p>
|
693,550 | <p>Consider the function $F(x) = x^2-2x+2$.</p>
<p>Find an interval in which the function is contractive and find the fixed point in this interval.</p>
<p>What is the convergence rate of the fixed point iteration: $x_{n+1} = F(x_n)$ in that interval?</p>
<p>I'm lost on fixed point iteration. I've watched a few youtu... | Ian Coley | 60,524 | <p>You need an interval $[a,b]$ so that there's a $0\leq c<1$ such that $|f(x)-f(y)|\leq c|x-y|$ for all $x,y\in[a,b]$. In our case, we have
$$
|f(x)-f(y)|=|x^2-y^2-(2x-2y)|\leq c|x-y|.
$$
Notice that, in fact, $f(1)=1$, so you're going to want to choose an interval somewhere around $1$.</p>
|
504,997 | <p>I have a dynamic equation,
$$ \frac{\dot{k}}{k} = s k^{\alpha - 1} + \delta + n$$
Where $\dot{k}/k$ is the capital growth rate as a function of savings $s$, capital $k$, capital depreciation rate $\delta$, and population growth rate $n$.</p>
<p>I have been asked to find the change in the growth rate as $k$ increase... | Patrick | 109,962 | <p>The more formal/modern/natural way to answer this is via a change of variables.</p>
<p>Let <span class="math-container">$z \equiv \log(k)$</span> on <span class="math-container">$\mathbb R_{>0}$</span>, so that pairs <span class="math-container">$(k, z) $</span> belong the graph of the function <span class="math-... |
1,635,682 | <p>I have the functions
$$ f_n(x) = x + x^n(1 - x)^n $$</p>
<p>that $\to x$ as $n \to \infty $ (pointwise convergence).</p>
<p>Now I have to look whether the sequence converges uniformly, so I used the theorem and arrived at:
$$ \sup\limits_{x\in[0,1]}|f_n(x)-x| = |x+x^n(1-x)^n - x| = |x^n(1-x)^n| = 0, n \to \infty $... | Andrew | 307,012 | <p>You are correct in assuming the formula is invalid because of the presence of a quantifier inside a predicate. The following formula captures the essence of your English sentence: $$\forall x_1 (\forall x_2 (\mathop{\mathrm{Love}}(x_1,x_2)) \rightarrow \mathop{\mathrm{Good}}(x_1)).$$</p>
|
930,458 | <p>would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry:
I have to find smallest algebraic variety (irreducible algebraic set) in $\mathbb{C^{2}}$ that contains all the points with integer coordinates. I don't really know how to start, so at least an idea, h... | Jyrki Lahtonen | 11,619 | <p>Extended hint: Let
$$
F(X,Y)=\sum_{i=0}^n F_i(Y)X^i
$$
be a non-zero polynomial from $\Bbb{C}[X,Y]$. Assume that it vanishes at all the points where $(X,Y)\in\Bbb{Z}^2$. The polynomial $F_n(Y)$ is non-zero as the leading coefficient. It can only have finitely many zeros. Therefore there exists an integer $m$ such t... |
289,367 | <p>Given a positive definite matrix $Q\in\mathbb{R}^{n \times n}$, I want to find a diagnonal matrix $D$ such that $rank(Q-D) \leq k < n$.</p>
<p>I think this can be regarded as a generalization of eigenvalue problem, which is basically problem of finding a diagonal matrix $\lambda I$ such that $rank(Q-\lambda I) &... | Aaron Meyerowitz | 8,008 | <p>You can limit the question to real symmetric matrices $A$ (with diagonal $0$ if desired). For large enough $k,$ the matrix $A+kI$ will be positive definite and you are then allowed to change all the diagonal entries.</p>
<p>As noted in a comment (and overlooked in my earlier answer) this $5 \times 5$ matrix (where ... |
50,023 | <p>two problems from Dugundji's book page $156$. (I don't know why the system deletes the word hi in this sentence)</p>
<p>$1$. Let $X$ be a Hausdorff space. Show that:</p>
<p>a) $\bigcap \{F: x \in F , F \ \textrm{closed}\} = \{x\}$.</p>
<p>b) $\bigcap \{U : x \in U, U \ \textrm{open}\} = \{x\}$.</p>
<p>c) The ab... | Mark | 4,460 | <p>a) Hausdorff implies $T_1$ which implies singletons are closed. Note that closure of $\{x\}$ is defined: $\bigcap\{F:x\in F\}$, and this must equal $\{x\}$ by the aforementioned property of $T_1$ spaces.</p>
<p>b) Choose a point $y \not= x$. We note that there is an open set containing $x$, that does not contain $y... |
1,259,961 | <p>I was looking at $$\int_0^1\left(\frac{1}{\sqrt{s}}\right)^2\ ds.$$</p>
<p>So in calculus, I would evaluate $\ln(1) - \ln(0)$ as the answer. What I don't get and I don't remember why is the answer $\infty$?</p>
<p>I know $\ln(1)=0$ and $\ln(0)=-\infty$</p>
<p>Wouldn't the answer be $-\infty$? Any explanation woul... | HeeHaw | 235,885 | <p>The answer would be $0 - (- \infty) = + \infty$ (modulo technicalities, the wholesome way to write this is to say that the integral diverges to $\infty$).</p>
|
711,549 | <p>Consider the successor of the largest finite ordinal that will ever be considered alone. But then it wasn't the largest finite ordinal that will ever be considered alone. How do we get around this paradox? The largest finite ordinal that will ever be considered alone does exist, and yet we can consider its successor... | Charles | 1,778 | <p>This is essentially <a href="https://en.wikipedia.org/wiki/Richard%27s_paradox">Jules Richard's paradox</a>. You define an ordinal in a language and then talk about its consequences in the metalanguage. But it's not problematic in the original language and doesn't have the desired property in the metalanguage.</p>
... |
1,157,877 | <p>We have $n$ bags of sand, with volume $$v_1,...,v_n, \forall i: \space 0 < v_i < 1$$ but not essentially sorted. we want to place all bag to boxes with volumes 1. We propose one algorithm:</p>
<blockquote>
<p>At first we place all bags in the original order. Then we select one
box and place on it, bag $1,... | DeepSea | 101,504 | <p><strong>hint</strong>: Solve for $\displaystyle \int_{0}^x f(\alpha)d\alpha$, and then taking derivative both sides to proceed. </p>
|
2,526,865 | <p>I came across this question while preparing for an interview.</p>
<p>You draw cards from a $52$-card deck until you get first Ace. After each card drawn, you discard three cards from the deck. What's the expected sum of cards until you get the first Ace? </p>
<p>Note</p>
<ol>
<li><p>J, Q, K have point value 11, 1... | Charlotte | 499,360 | <p>Although an exact solution is complicated and worth more thoughts, I did a quick simulation which hopefully can provide some insights.</p>
<p><a href="https://i.stack.imgur.com/tC8TB.png" rel="nofollow noreferrer">Expected sum v.s. number of cards thrown</a></p>
<p>Essentially the plot above demonstrates how the e... |
1,461,484 | <p>For example, look at this sentence from Perko's text on dynamical system</p>
<p>"It follows from Cauchy Schwarz inequality that if $T \in L(R^n)$ is represented by the matrix $A$ with respect to the standard basis for $R^n$ $_\cdots$" pg 11</p>
<p>What does it mean for a $T: R^n \to R^n$ to be represented by a mat... | Brian M. Scott | 12,042 | <p>The operator $T$ is a function from $\Bbb R^n$ to $\Bbb R^n$; an $n\times n$ matrix of real numbers is not a function. If one chooses the matrix $A$ properly, one can say that the function $T(x)=Ax$ for each $x\in\Bbb R^n$, but that is far from saying that the function $T$ <strong>is</strong> the matrix $A$.</p>
<p... |
766,694 | <p>I am stuck trying to get the values for x, y, and z. I keep moving variables around but I end up getting answers like x = x or z = z and I do not think that is what I want. It's really just algebra but I seem to have forgotten that. </p>
<p>(x/2) + (2z/3) = x</p>
<p>(x/2) + (y/3) = y</p>
<p>(2y/3) + (z/3) = z</p>... | DomJack | 145,285 | <p>Step 1: Move all unknowns to the left, in the same order, i.e. a*x + b*y + c*z = d</p>
<p>-1/2 * x + 0 * y + 2/3 * z = 0 (1)</p>
<p>1/2 * x - 2/3 * y + 0 * z = 0 (2)</p>
<p>0 * x + 2/3 * y - 2/3 * z = 0 (3)</p>
<p>This is a homogeneous linear system, and is normally expressed as the product of a matrix... |
2,282,359 | <p>I am trying to calculate this limit:
$$\lim_{x \to \infty} x^2(\ln x-\ln (x-1))-x$$
The answer is $1/2$ but I am trying to verify this through proper means. I have tried L'Hospital's Rule by factoring out an $x$ and putting that as $\frac{1}{x}$ in the denominator (indeterminate form) but it becomes hopeless afterwa... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Set $1/x=h$ to get $$-\lim_{h\to0^+}\dfrac{\ln(1-h)+h}{h^2}$$</p>
<p>Now either use Taylor Expansion or L'Hospital</p>
<p>or See the third Question of <a href="https://math.stackexchange.com/questions/387333/are-all-limits-solvable-without-lh%c3%b4pital-rule-or-series-expansion">Are all limits solvab... |
11,315 | <p>I just answered this question:</p>
<p><a href="https://math.stackexchange.com/questions/528880/boolean-formula">Boolean formula over 64 Boolean variables X</a></p>
<p>By the time I had posted my answer, another user had edited the question so as to remove all the mathematical interest.</p>
<p>What is going on her... | user642796 | 8,348 | <p>This isn't going to directly address the question (which has been amply answered by others), but I do want to highlight <a href="https://math.meta.stackexchange.com/questions/11315/it-looks-like-anyone-can-edit-anyones-questions-is-that-right#comment42866_11317">a comment made by Lord_Farin</a>:</p>
<blockquote>
... |
1,149,907 | <p>What are some different methods to evaluate</p>
<p>$$ \int_{-\infty}^{\infty} x^4 e^{-ax^2} dx$$</p>
<p>for $a > 0$.</p>
<p>This integral arises in a number of contexts in Physics and was the original motivation for my asking. It also arises naturally in statistics as a higher moment of the normal distributio... | Simon S | 21,495 | <p><strong>1</strong> Here's a relatively elegant method.</p>
<p>Notice that $\frac{\partial \ }{\partial a} e^{-a x^2} = - x^2 e^{-a x^2}$ and hence $\frac{\partial^2 \ }{\partial a^2} e^{-a x^2} = + x^4 e^{-a x^2}$ </p>
<p>Thus, as the integrand is bounded and $C^\infty$ in both variables,</p>
<p>$$I = \int_{-\in... |
4,048,335 | <p>Question:</p>
<blockquote>
<p>Assume that a recurrence relation is given as below:</p>
<p><span class="math-container">$T(n)=2T(n/2)+\log(n)$</span></p>
<p>and we know that <span class="math-container">$T(1) = 1$</span></p>
<p>We want to solve the relation (find an explicit definition of <span class="math-container"... | jjagmath | 571,433 | <p>The recurrence relation is not well defined. You can't calculate, for example, <span class="math-container">$T(3)$</span>.</p>
|
4,048,335 | <p>Question:</p>
<blockquote>
<p>Assume that a recurrence relation is given as below:</p>
<p><span class="math-container">$T(n)=2T(n/2)+\log(n)$</span></p>
<p>and we know that <span class="math-container">$T(1) = 1$</span></p>
<p>We want to solve the relation (find an explicit definition of <span class="math-container"... | Cesareo | 397,348 | <p>Assuming <span class="math-container">$\log(n) = \log_2 n$</span> we have</p>
<p><span class="math-container">$$
T\left(2^{\log_2 n}\right) = 2T\left(2^{\log_2 \frac n2}\right)+\log_2 n
$$</span></p>
<p>now calling <span class="math-container">$\mathcal{T}(\cdot) = T\left(2^{(\cdot)}\right)$</span> and <span class="... |
591,207 | <p>Show a proof that $Z(G),$ the center of $G,$ is a normal subgroup of $G$ and every subgroup of the center is normal. I know that $Z(G)$ is a normal subgroup and it is abelian, but how would I show that every subgroup of $Z(G)$ is also normal?</p>
| Igor Rivin | 109,865 | <p>Since if $H<Z(G),$ and $h \in H,$ for every $g \in G$ $g^{-1}hg = h,$ (since $h$ is central), it follows that $g^{-1}H g = H$ (it is true elementwise).</p>
|
873,183 | <p>Are variables logical or non-logical symbols in a logic system?
I understand constants are 0-ary logical operation symbols. I think variables are non-logical symbols.</p>
<p>But here are two contrary examples:</p>
<p>It seems that variables are logical symbols in a propositional logic system, according to <a href=... | hmakholm left over Monica | 14,366 | <p>As Thomas Andrews pointed out in a comment, this terminology is not something where one can expect consistency from book to book.</p>
<p>However, <strong>as a practical matter</strong>, when one specifies that the non-logical language of a particular theory is such-and-such, it is highly unusual to have to say, "oh... |
2,869,305 | <p>I need some help evaluating this limit... Wolfram says it blows up to infinity but I don't think so. I just can't prove it yet.</p>
<p>$$ \lim_{x\to\infty}(x!)^{1/x}-\frac{x}{e} $$</p>
| gammatester | 61,216 | <p>I used a simple one-liner and asked Maple 7 to <code>simplify(series(GAMMA(x+1)^(1/x)-x/exp(1), x=infinity,3));</code> and the answer is <code>1/2*(ln(2)+ln(Pi)+ln(x)+2*O(1/x)*exp(1))*exp(-1)</code>, this can be converted to mathematical notation as:
$$\Gamma\left(x+1\right)^{1/x}-\frac{x}{e} = \frac{1}{2e}\ln(2\... |
2,707,744 | <h2>The Problem</h2>
<p>I am really struggling to wrap my head around the definition of the integral in measure theory. According to <a href="http://measure.axler.net/" rel="nofollow noreferrer">Axler</a>, the integral of a nonnegative function is defined as follows:</p>
<blockquote>
<p>Suppose $(X, \mathcal S,\mu)... | Emilio Novati | 187,568 | <p>It seems that you know homogeneous coordinates. Using these coordinates, a point $X$ in an affine space that corresponds to the vector $x=[x_1,x_2]^T$ is represented by the triple $X=[x_1,x_2,1]^T$ and the matrix that represents a translation of vector $a=[2,5]^T$ is :
$$
T_a=\begin{bmatrix} 1 & 0 & 2\\0 &am... |
4,580,484 | <p>It is often said that "differential forms are used for integration".</p>
<p>Typically people like to talk about the integral <span class="math-container">$\int_M \omega$</span> of a differential form <span class="math-container">$\omega$</span>, and exterior derivative, one of the most important operation ... | F. Conrad | 403,916 | <p>While I love the answer of Quaere Verum and cant give enough +1s on it, here is a few more "simple" facts about the symplectic form <span class="math-container">$\omega$</span> on a symplectic manifold <span class="math-container">$(M,\omega)$</span> of dimension <span class="math-container">$2n$</span>:</... |
2,623,544 | <p>I have to diagonolize a matrix A
\begin{bmatrix}0&-3&-1&1\\2&5&1&-1\\-2&-3&1&1\\2&3&1&1\end{bmatrix}</p>
<hr>
<p>I do $det(A-λ)=0$ and I get $λ_{1}=1$, $λ_{2}=2$, $λ_{3}=2$, $λ_{4}=2$</p>
<p>So possible diagonal matrix looks like:
\begin{bmatrix}1&0&0&0\... | Radek Suchánek | 300,610 | <p>It seems you have not made any errors when computing the eigenvectors. I checked your results on the following matrix calculator: <a href="https://matrixcalc.org/en/#%7B%7B1,1,0,0%7D,%7B-1,0,1,0%7D,%7B1,0,0,1%7D,%7B-1,2,3,1%7D%7D%5E%28-1%29%2A%7B%7B1,0,0,0%7D,%7B0,2,0,0%7D,%7B0,0,2,0%7D,%7B0,0,0,2%7D%7D%2A%7B%7B1,1,... |
2,249,929 | <p>Is there a more general form for the answer to this <a href="https://math.stackexchange.com/q/1314460/438622">question</a> where a random number within any range can be generated from a source with any range, while preserving uniform distribution? </p>
<p><a href="https://stackoverflow.com/q/137783/866502">This</a>... | Ian | 83,396 | <p>The simplest way to proceed is rejection sampling. For this we need to first assume $M \geq N$. If this isn't the case (maybe you're playing D&D with a d6, so $N=20,M=6$), then you should roll at least $k=\lceil \log_M(N) \rceil$ times, and then label the $k$-tuples of rolls. For the discussion of the rejection ... |
1,546,599 | <p>What method should I use for this limit?
$$
\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}
$$</p>
<p>I tried ratio test but I ended with the ugly answer
$$\lim_{n\to \infty}\frac{(n+1)^{n-1}}{n^{n-1}}
$$
which would go to 1? Which means we cannot use ratio test. I do not know how else I could find this limit.</p>
| André Nicolas | 6,312 | <p>Note that for $n\ge 2$ the top is equal to $n\cdot n^{n-2}$.</p>
<p>The bottom is $(1)(2)\left[(3)(4)\cdots(n)\right]$. The product $[(3)(4)\cdots(n)]$ consists of $n-2$ terms, all $\le n$. So the bottom is $\le (1)(2)n^{n-2}$.
It follows that for $n\ge 2$ we have
$$\frac{n^{n-1}}{n!}\ge \frac{n\cdot n^{n-2}}{(1)(2... |
98,932 | <p>Suppose I would like to use a method for data prediction, and that I have some empirical data (i.e., sequence of samples of the form [time, value]). Would it be possible to know in advance, based on the data only, if it makes sense to use a model for prediction (based on the samples, used as a training set). </p>
... | Andrew | 23,236 | <p>I lack a theoretical proof, however I strongly suspect that no such test exists.</p>
<p>My reasoning is that it is possible to design pseudo random number generators that meet strong statistical measures of randomness ( For example, the BSI tests. See wikipedia for details ). Nonetheless, such generators are determ... |
2,576,801 | <p>$$\frac {|A-B|}{15} = \frac {|B-A|}{10} = \frac {|A\cap B|}{6}$$</p>
<ul>
<li>What is the minimal value of $A \cup B$?</li>
</ul>
<p>This question is from my exam. I've tried to solve it by giving values and added them up. </p>
<p>Regards</p>
| random | 513,275 | <p>Multiplying with $30$ gives $2|A-B|=3|B-A|=5|A\cap B|$, which implies that |A-B| must be divisible by 3 and by 5 and therefore by $15$.</p>
|
4,007,450 | <p>Let <span class="math-container">$y=f(x)$</span> the graph of a real-valued function. We define its curvature by : <span class="math-container">$$curv(f) = \frac{|f''|}{(1+(f')^2)^{3/2}}$$</span></p>
<p>I would like to know if there is any function (apart from the trivial anwser <span class="math-container">$f(x)=0$... | Robert Lewis | 67,071 | <p>Choose</p>
<p><span class="math-container">$a = b; \tag 1$</span></p>
<p>then</p>
<p><span class="math-container">$aba = b \tag 2$</span></p>
<p>becomes</p>
<p><span class="math-container">$a^3 = a, \tag 3$</span></p>
<p>which implies</p>
<p><span class="math-container">$a^2 = e, \tag 4$</span></p>
<p><span class="m... |
2,336 | <p>I have just started teaching mathematics up to secondary level. I don't have much idea as to how to handle the class. </p>
<p>In order to make students learn well, how can we divide the time in order to put a mix of lecture and specially the group work?</p>
| Dan Goldner | 1,525 | <p>In my 75-min algebra 2 class, I use this schedule daily: 5 min for warmup, 35 min for individual presentations to the class of homework solutions, up to 10 min for direct instruction if needed from me, 20 min for classwork in groups, and 5 for some quick exit exercise that I collect as a check on understanding. It's... |
2,336 | <p>I have just started teaching mathematics up to secondary level. I don't have much idea as to how to handle the class. </p>
<p>In order to make students learn well, how can we divide the time in order to put a mix of lecture and specially the group work?</p>
| Mike Pierce | 5,220 | <ol>
<li><p>Identify what you want you'd like to teach the students for the class. Maybe more importantly, identify the <em>goals</em> of the course. What do you want the students to <em>know</em> and what do you want them to be able to <em>do</em> once the course is over? </p></li>
<li><p>Write up a reasonable calenda... |
3,373,576 | <p>What are the non-trivial solutions of
<span class="math-container">$$\tan x = \arctan x$$</span></p>
<p>Can these solutions be expressed e.g. in terms of <span class="math-container">$\pi$</span> or in radicals?
I mean are they some "nice" numbers?</p>
<p>E.g. do we know if these solutions are irrational, ... | user | 505,767 | <p>We have by <span class="math-container">$y=x-1 \to 0$</span></p>
<p><span class="math-container">$$\lim\limits_{x\to1}\frac{\tan(x^2-1)}{\sin(x^2-4x+3)}=\lim\limits_{y\to0}\frac{\tan(y(y+2))}{\sin(y(y-2))}$$</span></p>
<p>and</p>
<p><span class="math-container">$$\frac{\tan(y(y+2))}{\sin(y(y-2))}=\frac{\tan(y(y+2... |
1,361,948 | <p>$\frac{df}{dx} = 2xe^{y^2-x^2}(1-x^2-y^2) = 0.$</p>
<p>$\frac{df}{dy} = 2ye^{y^2-x^2}(1+x^2+y^2) = 0.$</p>
<p>So, $2xe^{y^2-x^2}(1-x^2-y^2) = 2ye^{y^2-x^2}(1+x^2+y^2)$.</p>
<p>$x(1-x^2-y^2) = y(1+x^2+y^2)$</p>
<p>$x-x^3-xy^2 = y + x^2y + y^3$</p>
<p>Is the guessing the values of the variables the only way of so... | Jan Eerland | 226,665 | <p>I've checked my answer with Mathematica and it gives me the same, it's much more complicated as you thought, so your answer isn't good!</p>
<hr>
<p>$$z = \frac{1 + i}{\sigma \delta \left(1 - e^{-(1 + i)t/\delta} \right)}=
\frac{1 + i}{\sigma \delta \left(1 - e^{-\frac{(1+i)t}{\delta}} \right)}$$</p>
<p>Assuming $... |
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