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 |
|---|---|---|---|---|
627,258 | <p>Helly everybody,<br>
I'm trying to find another approach to topology in order to justify the axiomatization of topology. My idea was as follows:</p>
<p>Given an <strong>arbitrary</strong> collection of subsets of some space: $\mathcal{C}\in\mathcal{P}^2(\Omega)$<br>
Define a closure operator by: $\overline{A}:=\big... | GEdgar | 442 | <p>This can be done. But it is not very interesting.
Any of these equivalent versions of "topology" could be used its starting point:</p>
<p>(1) In axioms for open sets, do not require the whole space to be open</p>
<p>(2) In axioms for closed sets, do not require the empty set to be closed</p>
<p>(3) In axioms for... |
132,003 | <p>I have a time consuming function that is going to be iterated in a <code>Nest</code> or <code>NestList</code> and I would like to know if there is a good way to monitor the progress. I have found a partial work-around, but it requires an extra global variable (n). </p>
<pre><code>fun[x_] := Module[{}, n++; Pause[1]... | MarcoB | 27,951 | <p>Inspired by <a href="https://mathematica.stackexchange.com/a/227/27951">this older post</a> by Andrew Moylan, I tried combining <code>Monitor</code> with your <code>ProgressIndicator</code>, then wrapping the whole thing in a <code>DynamicModule</code> to limit the scope the <code>n</code> variable:</p>
<pre><code>... |
73,785 | <p>I am new to Mathematica, and I have read this <a href="https://mathematica.stackexchange.com/questions/29203/determine-the-2d-fourier-transform-of-an-image">post</a> to understand how to perform Fourier transform on an image. My mission is to extract information on the typical distance between the black patches in t... | Anton Antonov | 34,008 | <p>This is not an answer more of an extended comment...</p>
<blockquote>
<p>My mission is to extract information on the typical distance between
the black patches in the image I have attached here.</p>
</blockquote>
<p>Do we have to use Fourier transform for this?</p>
<p>For example we can get the required estim... |
313,030 | <p>I often find myself writing a definition which requires a proof. You are defining a term and, contextually, need to prove that the definition makes sense. </p>
<p>How can you express that? What about a definition with a proof?</p>
<p>Sometime one can write the definition and then the theorem. But often happens tha... | Francois Ziegler | 19,276 | <p>Dixmier always solves this as follows, e.g. in <em>C*-algebras</em> — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book from which to learn about <em>C*</em>-algebras”):</p>
<blockquote>
<p><strong>16.1. The com... |
356,353 | <p>I learned in my Intro Algebraic Number Theory class that there exist infinitely many integer pairs $(x,y)$ that satisfy the hyperbola $x^2-ny^2=1$; just consider that there are infinitely many units in $\mathcal{O}_{\mathbb{Q}(\sqrt{n})}$, and their norms satisfy the desired equation. Although this is a nice connect... | Zander | 25,711 | <p>See Wikipedia on <a href="http://en.wikipedia.org/wiki/Pell%27s_equation" rel="nofollow">Pell's equation</a>, also <a href="http://mathworld.wolfram.com/PellEquation.html" rel="nofollow">MathWorld</a>.</p>
<p>If there's one solution with $x,y\ge1$ then see Brahmagupta's method or the section "Additional solutions f... |
61,798 | <p>Are there any generalisations of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$ ?</p>
<p>For example can $\sum {k^m} = \left(\sum k\right)^n$ be valid for anything other than $m=3 , n=2$ ?</p>
<p>If not, is there a deeper reason for this identity to be true only for the case $m=3 , ... | Davide Giraudo | 9,849 | <p>We can't have a relationship of the form $$\forall n\in\mathbb N^*, \sum_{k=1}^nk^a=\left(\sum_{k=1}^nk^b\right)^c$$ for $a,b,c\in\mathbb N$, except in the case $c=1$ and $a=b$ or $a=3$, $b=1$ and $c=2$. Indeed, we can write $$\sum_{k=1}^nk^a =n^{a+1}\frac 1n\sum_{k=1}^n\left(\dfrac kn\right)^a$$ hence $$\sum_{k=1}^... |
888,101 | <p>Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?</p>
| Giuseppe Negro | 8,157 | <p>If a topology is metrizable, then the "diagonal sequence trick" is available. This means that if you have a sequence
$$
x_{(n)} \to x,\qquad n \to \infty
$$
and each term of the sequence is the limit of another sequence, belonging to a "good" set $G$:
$$
G\ni x^{k}_{(n)}\to x_{(n)}, \qquad k\to \infty
$$
then you... |
1,285,213 | <p>Let $f\in P_2(\mathbb R)$, the space of second-order polynomials with real coefficients, and let the linear operator $T$ be defined as $T[f(x)] = f(0)+f(1)(x+x^2)$.</p>
<p>Is $T$ diagonalizable? If so, find a basis $\beta$ of $P_2(\mathbb R)$ in which $[T]_\beta$ is a diagonal matrix.</p>
| Brian Fitzpatrick | 56,960 | <p>Let
\begin{align*}
e_1(t) &= 1 & e_2(t) &= t & e_3(t) &= t^2
\end{align*}
and note that $\alpha=\{e_1,e_2,e_3\}$ is a basis for $P_2(\Bbb R)$. Furthermore,
note t... |
1,285,213 | <p>Let $f\in P_2(\mathbb R)$, the space of second-order polynomials with real coefficients, and let the linear operator $T$ be defined as $T[f(x)] = f(0)+f(1)(x+x^2)$.</p>
<p>Is $T$ diagonalizable? If so, find a basis $\beta$ of $P_2(\mathbb R)$ in which $[T]_\beta$ is a diagonal matrix.</p>
| Hagen von Eitzen | 39,174 | <p>We need to find eigenvectors.
Since the image of $T$ is inly twodimensional, it is clear that one eigenvalue must be $0$. Indeed, $f(x)=x\cdot(x-1)=x^2-x$ is obviously in the kernel of $T$ (and the kernel consists precicely of the multiple of this - why?)</p>
<p>It should be clear from the definition of $T$ that a... |
1,673,452 | <p>Let $\{a_j\}_{j=1}^N$ be a finite set of positive real numbers. Suppose </p>
<p>$$\sum_{j=1}^{N} a_j = A,$$ prove</p>
<p>$$\sum_{j=1}^{N} \frac{1}{a_j} \geq \frac{N^2}{A}.$$ </p>
<p>Hints on how to proceed?</p>
| Lutz Lehmann | 115,115 | <p>Try the Cauchy-Schwarz inequality. This would be a 3 line proof.
$$
\left(\sum_{i=1}^Nx_iy_i\right)^2\le\sum_{i=1}^Nx_i^2·\sum_{i=1}^Ny_i^2
$$
Now chose $x_i,y_i$ so that one recognizes the sums in the task and that $x_iy_i=1$.</p>
|
1,673,452 | <p>Let $\{a_j\}_{j=1}^N$ be a finite set of positive real numbers. Suppose </p>
<p>$$\sum_{j=1}^{N} a_j = A,$$ prove</p>
<p>$$\sum_{j=1}^{N} \frac{1}{a_j} \geq \frac{N^2}{A}.$$ </p>
<p>Hints on how to proceed?</p>
| Svetoslav | 254,733 | <p>$$N=\sqrt{a_1}.\frac{1}{\sqrt{a_1}}+...+\sqrt{a_n}.\frac{1}{\sqrt{a_n}}\leq \sqrt{a_1+...+a_n}\sqrt{\frac{1}{a_1}+...+\frac{1}{a_n}}$$
Now square both sides of the inequality.</p>
|
2,603,799 | <p>Good morning, i need help with this exercise.</p>
<blockquote>
<p>Prove all tangent plane to the cone $x^2+y^2=z^2$ goes through the origin</p>
</blockquote>
<p><strong>My work:</strong></p>
<p>Let $f:\mathbb{R}^3\rightarrow\mathbb{R}$ defined by $f(x,y,z)=x^2+y^2-z^2$</p>
<p>Then,</p>
<p>$\nabla f(x,y,z)=(2x... | user | 505,767 | <p>The equation of the tangent plane is</p>
<p>$$z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$</p>
<p>For the cone we have</p>
<p>$$f(x,y)=\sqrt{x^2+y^2}\implies f_x=\frac{x}{\sqrt{x^2+y^2}} \quad f_y=\frac{y}{\sqrt{x^2+y^2}}$$</p>
<p>Thus the tangent plane at $(x_0,y_0,z_0)$ is
$$z-z_0=\frac{x_0}{z_0} (x-x_0)+\f... |
3,319,629 | <p>The question is from a practice exam I am currently trying to do:
<a href="https://i.stack.imgur.com/VWQhs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VWQhs.png" alt="enter image description here"></a></p>
<p>I am really not sure how to go about this one. In essence, I'd imagine that the idea... | Toby Mak | 285,313 | <p>You can just calculate the area of the <span class="math-container">$\text{outer curve} - \text{inner curve}$</span>:</p>
<p><span class="math-container">$$\frac{1}{2} \int_0^{2\pi} \big(4 - \cos(3\theta) \big)^2 - \big(2 + \sin(3 \theta) \big)^2$$</span></p>
|
2,981,063 | <p>I have seen this statement in a quiz:</p>
<blockquote>
<p>Let <span class="math-container">$X_i$</span> denote state <span class="math-container">$i$</span> in a Markov chain. It is necessarily true
that <span class="math-container">$X_{i+1}$</span> and <span class="math-container">$X_{i-1}$</span> are uncorrel... | grand_chat | 215,011 | <p>The key phrase is "given the present". If past and future are independent given the present, it doesn't follow that past and future are unconditionally independent.</p>
<p>For example, consider the simple random walk that takes a step either left or right with equal probability. If you know where I am today, then t... |
16,725 | <p>I was directed just now to a post with the following abbreviated time-line:</p>
<ul>
<li>Question was posted <strong>21</strong> hours ago</li>
<li>Question was closed as "unclear what you are asking" <strong>19</strong> hours ago</li>
<li>Question was deleted by the votes of three 10K users <strong>4</strong> hour... | Community | -1 | <p>Concrete suggestion: let's avoid voting to delete questions that display [on hold] unless they are of the spam/offensive/trolling/joke kinds. </p>
<p>Also, given that the system automatically deletes nonpositively scored closed posts that got no positively scored answer, I see no reason to spend <a href="http://met... |
3,489,086 | <p>I was given this integral:</p>
<p><span class="math-container">$$\int^{\infty}_{0}\frac{\arctan(x)}{x}dx$$</span></p>
<p>As the title says, I have to find out whether it is convergent or not.
So far, I have tried integrating by parts and substituting <span class="math-container">${\arctan(x)}$</span>, and neither ... | Community | -1 | <p><span class="math-container">$$\int_0^\infty\frac{\arctan(x)}{x}dx=\int_{-\infty}^\infty\arctan(e^t)\,dt$$</span> but this integrand does not vanish.</p>
|
30,141 | <p>I need to define a function <code>fun</code>, and then re-define this function iteratively. The code is given at the end.</p>
<p>First, a function <code>fun[x_, y_, d_]</code> is defined, which is a polynomial in $x$ and $y$, and $d$ is the degree of this polynomial.</p>
<p>My goal is to modify <code>fun</code> ac... | Hector | 8,803 | <p>Try the following code:</p>
<pre><code>fun[n_Integer?Positive][x_, y_, d_Integer] :=
fun[n - 1][x, y, d] + Coefficient[fun[n - 1][x, y, d], x, n - 1 ];
fun[0][x_, y_, d_Integer] :=
Sum[(i + j) x^i y^j Subscript[a, i, j], {i, d}, {j, d}];
Table[fun[n][x, y, 1], {n, 0, 2}] // TableForm
</code></pre>
|
30,141 | <p>I need to define a function <code>fun</code>, and then re-define this function iteratively. The code is given at the end.</p>
<p>First, a function <code>fun[x_, y_, d_]</code> is defined, which is a polynomial in $x$ and $y$, and $d$ is the degree of this polynomial.</p>
<p>My goal is to modify <code>fun</code> ac... | ubpdqn | 1,997 | <p>Another way (using same definition of f[x,y,d]:</p>
<pre><code> nestpol[n_Integer, d_Integer] :=
NestList[{#[[1]] + 1, #[[2]] + Coefficient[#[[2]], x^(#[[1]])]} &,
{1, f[x, y, d]}, n][[;; , 2]]
</code></pre>
<p>note
(i) if just want the last value just change NestList to Nest
(ii) in Hector's answer (b... |
55,482 | <p>I write a code that creates a compiled function, and then call that function over and over to generate a list. I run this code on a remote server via a batch job, and will run several instances of it. Sometimes when I make changes to the code, I make a mistake, and inside the compiled function is an undefined vari... | Jason B. | 9,490 | <p>Adding this option for <code>Compile</code></p>
<pre><code>"RuntimeOptions" -> {"RuntimeErrorHandler" ->Function[Throw[$Failed]]}
</code></pre>
<p>will cause it to abort evaluation if any error messages come up. To more directly control the memory usage, and stay on the sysadmin's good side, wrap the call t... |
3,853,351 | <p>Given an n-dimensional ellipsoid in <span class="math-container">$\mathbb{R}^n$</span>, is any orthogonal projection of it to a subspace also an ellipsoid? Here, an ellipsoid is defined as</p>
<p><span class="math-container">$$\Delta_{A, c}=\{x\in \Bbb R^n\,:\, x^TAx\le c\}$$</span></p>
<p>where <span class="math-co... | Narasimham | 95,860 | <p>Indeed, ellipsoids cast ellipse shape shadows on the ground.</p>
<p>The intersection of any conicoid and a <em>first</em> degree equation <em>plane</em> illumination terminator between two tangential points is a conic section. It can be proved by elimination to the conic <em>second</em> degree equation.</p>
<p><a hr... |
2,929,203 | <p>Suppose we define the relation <span class="math-container">$∼$</span> by <span class="math-container">$v∼w$</span> (where <span class="math-container">$v$</span> and <span class="math-container">$w$</span> are arbitrary elements in <span class="math-container">$R^n$</span>) if there exists a matrix <span class="mat... | Acccumulation | 476,070 | <p>Given any unit vectors, there is a nonsingular rotation matrix that takes the first vector to the other. Given any non-zero, non-unit vector, there is some non-singular scaling matrix that takes the vector to a unit vector. So given two arbitrary non-zero vectors <span class="math-container">$v$</span> and <span cla... |
3,582,585 | <p>Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the
die again and if any other number comes, toss a coin. Find the conditional probability of the event <strong>‘the coin shows a tail’</strong>, given that <em>‘at least one throw of die shows a 3’</em>.</p>
<p>I don't know how to deal w... | John Omielan | 602,049 | <p>You have</p>
<p><span class="math-container">$$3^{15a} = 5^{5b} = 15^{3c} = 3^{3c}5^{3c} \tag{1}\label{eq1A}$$</span></p>
<p>Taking natural logarithms (although any other logarithm base, e.g., common (i.e., base <span class="math-container">$10$</span>), will also work) gives</p>
<p><span class="math-container">$... |
1,419,209 | <p>How do I evaluate this (find the sum)? It's been a while since I did this kind of calculus.</p>
<p>$$\sum_{i=0}^\infty \frac{i}{4^i}$$</p>
| GAVD | 255,061 | <p>Let me try. </p>
<p>Set $$S = \sum_{i\geq 0}\frac{i}{4^i}.$$</p>
<p>Then we have $$4S = \sum_{i \geq 0} \frac{i+1}{4^i} = \sum_{i\geq 0}\frac{i}{4^i} + \sum_{i\geq 0}\frac{1}{4^i} = S + \frac{1}{1-\frac{1}{4}}$$</p>
<p>So, $3S = \frac{4}{3}$. It implies that $$S = \frac{4}{9}.$$</p>
|
2,077,664 | <p>I can show that $3^{3^{3^n}}\equiv7\pmod{10}$ since</p>
<p>$3^1\equiv3\pmod{10}$</p>
<p>$3^2\equiv9\pmod{10}$</p>
<p>$3^3\equiv7\pmod{10}$</p>
<p>$3^4\equiv1\pmod{10}$</p>
<p>Thus, it reduces to $3^{(3^{3^n}\mod4)}$. I can then notice that</p>
<p>$3^1\equiv3\pmod4$</p>
<p>$3^2\equiv1\pmod4$</p>
<p>Reducing ... | N. S. | 9,176 | <p>We know that
$$3^2 \equiv 1 \pmod{4} \\
3^{20} \equiv 1 \pmod{25}$$
with the last following from Euler Theorem. Therefore
$$3^{20} \equiv 1 \pmod{100}$$</p>
<p>The problem then reduces to finding the powers of $3 \pmod{20}$. </p>
<p>Again
$$3^2 \equiv 1 \pmod{4} \\
3^4 \equiv 1 \pmod{5} \\$$</p>
<p>Therefore $3^... |
2,077,664 | <p>I can show that $3^{3^{3^n}}\equiv7\pmod{10}$ since</p>
<p>$3^1\equiv3\pmod{10}$</p>
<p>$3^2\equiv9\pmod{10}$</p>
<p>$3^3\equiv7\pmod{10}$</p>
<p>$3^4\equiv1\pmod{10}$</p>
<p>Thus, it reduces to $3^{(3^{3^n}\mod4)}$. I can then notice that</p>
<p>$3^1\equiv3\pmod4$</p>
<p>$3^2\equiv1\pmod4$</p>
<p>Reducing ... | lab bhattacharjee | 33,337 | <p>As $3\equiv-1\pmod4,3^{2n+1}\equiv-1\equiv3$ for any integer $n\ge0$</p>
<p>So, $3^{3^{3^{\cdots}}}$ can be written as $\displaystyle3^{4k+3}$ or $\displaystyle3^{3^{(4b+3)}}$</p>
<p>Now,$\displaystyle3^{4k+3}=27(10-1)^{2k}=27(1-10)^{2k}$</p>
<p>and $\displaystyle(1-10)^{2k}\equiv1-\binom{2k}110\pmod{10^2}\equiv1... |
251,182 | <p>Is 13 a quadratic residue of 257? Note that 257 is prime.</p>
<p>I have tried doing it. My study guide says it is true. But I keep getting false. </p>
| Bill Dubuque | 242 | <p>$\rm mod\ 257\!:\ 13 \,\equiv\, 13\!-\!257 \,\equiv\, -61\cdot 4 \,\equiv\, 196\cdot 4\,\equiv\,49\cdot 4\cdot 4 \,\equiv\, 28^2\ \ $ (took $\,< 10$ secs mentally)</p>
<p><strong>Remark</strong> $\ $ Because of the <em>law of small numbers</em>, such negative twiddling and pulling out small square factors often ... |
939,237 | <p>Prove $n^2 < n!$.</p>
<p>This is what I have gotten so far</p>
<p>basis step: $p(4)$ is true
Inductive Hypothesis assume $p(k)$ true for $k \ge 4$</p>
<p>Inductive Step $p(k+1)$ : $(k+1)^2 < (k+1)!$</p>
<p>$$(k+1)^2 =k^2 + 2k + 1 < k! + 2k +1$$</p>
<p>Can someone please explain the last step this is fr... | user164515 | 166,497 | <p>$$n^n \geq n!$$</p>
<p>Proof: Let $n\in\mathbb{N}$. Then
$$ n^n = n\cdot n\cdot n\cdot...\cdot n$$
where as
$$ n! = n\cdot(n-1)\cdot (n-2)\cdot...\cdot1$$</p>
<p>For each term in the product you can compare
$$n = n $$
$$n > n-1 $$
$$n > n-2 $$
and so on.
Thus $n^n \geq n!$ </p>
|
1,006,562 | <p>So I am trying to figure out the limit</p>
<p>$$\lim_{x\to 0} \tan x \csc (2x)$$</p>
<p>I am not sure what action needs to be done to solve this and would appreciate any help to solving this. </p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> We have $\csc(2x)=\frac{1}{\sin(2x)}$ and $\sin(2x)=2\sin x\cos x$.</p>
|
844,832 | <p>How to find the derivative of this function $$ 7\sinh(\ln t)?$$</p>
<p>I don't know from where to start, so i looked at it in wolfram alpha and it was saying that the $$ 7((-1 + t^2) / 2t) $$ I did not get that. How did they jump from $$ 7\sinh(\ln t) $$ to this step? Is there an equation for it that I am missing?<... | Hayden | 27,496 | <p>Assuming you know the derivatives of $\ln t$ and $\mathrm{sinh} t$, then you can use the Chain Rule, which states that $(f\circ g)'(t)=g'(t)f'(g(t))$.</p>
|
2,373,073 | <p>Let $a, b, c$ be distinct integers, and let $P$ be a polynomial with integer coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$ at the same time. </p>
| problembuster | 356,262 | <p>Assume otherwise.</p>
<p>By the remainder theorem, $a-b$ divides $b-c$, $b-c$ divides $c-a$, and $c-a$ divides $a-b$.</p>
<p>Then, $b-c$ also divides $a-b$, therefore, $2a = b+c$.</p>
<p>$c-a$ also divides $b-c$, therefore $2c = a+b$</p>
<p>Substracting both equations:</p>
<p>$2(a-c) = -(a-c)$</p>
<p>$a-c=0$</... |
208,008 | <p>Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ in $C$. Also, let $\{C_{11},C_{12}\}$ and $\{C_{21},C_{22}\}$ be the sets of two arbitrary tangent circles with radiu... | Ilya Bogdanov | 17,581 | <p>Do I misunderstand something, or such an easy argument works?</p>
<p><b>1.</b> Assume that such set <span class="math-container">$S$</span> of <span class="math-container">$k$</span> points exists. Choose an arbitrary position of <span class="math-container">$C_1$</span> such that its center <span class="math-conta... |
2,428,243 | <p>How can I evalute this product??</p>
<p>$$\prod_{i=1}^{\infty} {(n^{-i})}^{n^{-i}}$$</p>
<p>Unfortunately, I have no idea.</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>Taking the logarithm,</p>
<p>$$\log p(n)=-\log n\sum_{i=1}^\infty i\,n^{-i}.$$</p>
<p>This summation, which is a modified geometric series, has a closed-form formula.</p>
|
2,335,831 | <p>I am trying to implement an Extended Kalman Filter (EKF) and it is becoming harder than I thought.</p>
<p>I have one question. I noticed that the covariance matrix which should get updated over each iteration is not symmetric. I am debugging through MATLAB.
I know that P should be symmetric and stay symmetric.</p>... | Forrest Voight | 36,097 | <p>A quick ad-hoc fix that (in my experience) works great is to simply "symmetrize" the $P$ matrix every time you calculate a new potentially asymmetric value for it by doing:</p>
<p>$P'=\dfrac{P+P^T}{2}$</p>
|
3,265,403 | <p>While trying to compute the line integral along a path K on a function, I need to parametrize my path K in terms of a single variable, let's say this single variable will be <span class="math-container">$t$</span>.
My path is defined by the following ensemble: <span class="math-container">$$K=\{(x,y)\in(0,\infty)\ti... | Pjotr5 | 157,405 | <p>I think that using trigonometric function is overcomplicating it in this case. You can let <span class="math-container">$y$</span> correspond to a parameter <span class="math-container">$t$</span>, then, since <span class="math-container">$x$</span> is given to be positive, we can say that <span class="math-containe... |
666,217 | <p>If $a^2+b^2 \le 2$ then show that $a+b \le2$</p>
<p>I tried to transform the first inequality to $(a+b)^2\le 2+2ab$ then $\frac{a+b}{2} \le \sqrt{1+ab}$ and I thought about applying $AM-GM$ here but without result</p>
| SomeStrangeUser | 68,387 | <p>First note that if $a=0$ or $b=0$, then the question is easy. So assume that both are non-zero. Consider the value $ab$. We can show that we must have $|ab|\le1$.
Assume by contrary that we have: $|ab| > 1$. This means $$\frac{1}{|b|}<|a|$$
Expanding: $$0\le(b^2-1)^2$$
we get:$$2\le \frac{1}{b^2}+b^2,$$
Thus... |
1,748,751 | <p>By K values, I mean the values described here:</p>
<p><a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Explicit_Runge.E2.80.93Kutta_methods" rel="nofollow">https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Explicit_Runge.E2.80.93Kutta_methods</a></p>
<p>I know how the K values in the Rung... | William Oliver | 89,122 | <p>I figured out the answer to my question, with the help of the Peter in the comments of this question. I decided to post what I've found here in case it might help other people (because I can't seem to find a good explanation of this anywhere else online). </p>
<p>First of all, for those who do not know, the <a href... |
1,748,751 | <p>By K values, I mean the values described here:</p>
<p><a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Explicit_Runge.E2.80.93Kutta_methods" rel="nofollow">https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Explicit_Runge.E2.80.93Kutta_methods</a></p>
<p>I know how the K values in the Rung... | Community | -1 | <p>From the original article ("Beitrag zur näherungweisen Integration totaler Differentialgleichungen"), Kutta used the method of <em>indeterminate coefficients</em>, expressing the increment of the function when moving from $f(x,y)$ to $f(x+\delta,y+\Delta)$, where $\Delta$ is estimated in terms of several intermediat... |
1,637,879 | <p>can you help me identify the mistake I'm making while integrating?</p>
<p>Question:</p>
<p>$$\int{\frac{2dx}{x\sqrt{4x^2-1}}}, x>\frac{1}{2}$$</p>
<p>my solution</p>
<p>$$\int{\frac{2dx}{x\sqrt{4x^2-1}}}=2\int{\frac{dx}{x\sqrt{(2x)^2-1}}}$$</p>
<p>let $$u=2x, x=1/2u, du=2dx, 1/2du=dx$$</p>
<p>$$=\frac{2}{2}... | John B | 301,742 | <blockquote>
<p>The equality of the <strong>time average</strong> and the <strong>space average</strong> essentially means that each trajectory travels through the space so randomly that all happens as if it reaches everywhere and even more spends a time on each region proportional to the size of that region.</p>
</b... |
1,530,702 | <p>Can anybody help me out with getting an expression of the values of $\lambda$ for a matrix $A$ for which $det(A-\lambda I)$ equals the determinant of a matrix with on the main diagonal $-\lambda$, on the diagonal above the main diagonal $\dfrac{1}{2}$ and on the diagonal under the main diagonal $\frac{1}{2} \lambda$... | mvw | 86,776 | <p>The determinant of the tridiagonal matrix can be expressed by the recurrence (<a href="https://en.wikipedia.org/wiki/Tridiagonal_matrix#Determinant" rel="nofollow">link</a>):
$$
f_n = -\lambda f_{n-1} -\frac{1}{4}\lambda f_{n-2} \quad (*)
$$
and the initial values $f_0=1$, $f_{-1}= 0$.</p>
<p>For $\lambda = 0$ the ... |
1,517,456 | <blockquote>
<p>Rudin Chp. 5 q. 13:</p>
<p>Suppose <span class="math-container">$a$</span> and <span class="math-container">$c$</span> are real numbers, <span class="math-container">$c > 0$</span>, and <span class="math-container">$f$</span> is defined on <span class="math-container">$[-1, 1]$</span> by</p>
<p><span... | Thomas Andrews | 7,933 | <p>This is a "between" argument, skipping field theory, but using linear algebra.</p>
<p>If $\sqrt[3]{2}\in\mathbb Q(\sqrt[4]{5})$, then multiplication by $\sqrt[3]{2}$ is a linear transformation, $T,$ of $\mathbb Q(\sqrt[4]{5})$ as a vector space over $\mathbb Q$. That linear transformation has a minimal polynomial, ... |
364,800 | <p>Let <span class="math-container">$V$</span> be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve <span class="math-container">$U$</span> such that there is a non-constant holomorphic map <span class="math-container">$U\to V$</span... | Francesco Polizzi | 7,460 | <p>The title and the body of the question seem to ask two different things. Let me give an answer to the second one. Since you are not asking that the compactification of <span class="math-container">$U$</span> is smooth, we can build up an example as follows.</p>
<p>Let <span class="math-container">$\bar{U}$</span> be... |
3,400,766 | <p>I know that by considering projection <span class="math-container">$q : \mathbb{R}^2 \to \mathbb{R}$</span>, <span class="math-container">$(x, y) \to x$</span>, and the closed subset </p>
<p><span class="math-container">$$G = \left\{(x, y) : y \ge \frac 1 x, x > 0\right\}$$</span></p>
<p>will prove that <span c... | DanielWainfleet | 254,665 | <p>The set of equivalence classes of <span class="math-container">$\sim,$</span> where <span class="math-container">$(x,y)\sim (x',y')\iff x=x',$</span> is <span class="math-container">$\Bbb R^2/\Bbb M=\{\{x\}\times \Bbb R: x\in \Bbb R\}.$</span></p>
<p>The quotient topology on <span class="math-container">$\Bbb R^2/\... |
172,080 | <p>Here is a fun integral I am trying to evaluate:</p>
<p>$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$</p>
<p>I thought about integrating by parts $2n$ times and then using the binomial theorem for $\sin(x)$, that is, using $\dfrac{e^{ix}-e^{-ix}}{2i}$ form in the binomial se... | robjohn | 13,854 | <p>Since $\dfrac{\sin^{2n+1}(x)}{x}$ is an even function, we can integrate over the whole real line and divide by $2$.</p>
<p>Write $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$. Since there are no singularities and the integrand vanishes as $|x|\to\infty$, we can move the path of integration in the direction of $-i$. Expand u... |
1,896,024 | <p><span class="math-container">$f(n) = 2n^2 + n$</span></p>
<p><span class="math-container">$g(n) = O(n^2)$</span></p>
<p>The question is to find the mistake in the following process:</p>
<blockquote>
<p><span class="math-container">$f(n) = O(n^2) + O(n)$</span></p>
<p><span class="math-container">$f(n) - g(n) = O(n^... | naslundx | 130,817 | <p><strong>Hint</strong></p>
<p>What if $g(n) = n^2$?</p>
<p>What does $f(n) - g(n)$ simplify to, and is it $O(n)$ or $O(n^2)$?</p>
<p><strong>Clarification</strong></p>
<p>With the above, we get $f(n) - g(n) = 2n^2 + n - n^2 = n^2 + n = O(n^2)$.</p>
<p>Remember that, by definition, $f(n) = O(n^p)$ means $f(n) \le... |
1,077,594 | <p>Let $C[a,b]$ be the space of continuous functions on $[a,b]$ with the norm
$$
\left\Vert{f}\right\Vert=\max_{a \leq t \leq b}\left| f(t)\right|
$$</p>
<p>Then $C[a,b]$ is a Banach space. </p>
<p>Let's view $C^1[a,b]$ as a subspace of it. My question is, is this $C^1[a,b]$ a Banach space?</p>
<p>I think it is, sin... | Ron Gordon | 53,268 | <p>The cosine function does not vanish on the semicircle as $R \to \infty$; in fact, it does the opposite. You need to either 1) take the real part of $e^{i x}$ in the upper half plane, or 2) use $\cos{x} = (e^{i x}+e^{-i x})/2$ and use both the upper and lower half planes, respectively.</p>
|
2,523,112 | <p>Let $f\left(x\right)$ be differentiable on interval $\left(a,b\right)$ and $f'\left(x\right)>0$ on that interval. If $\underset{x\rightarrow a+}{\lim}f\left(x\right)=0$, $f\left(x\right)>0$ on that interval?</p>
<p>I think this proposition is true by my intuitive, but I wonder whether intuitive is mathematica... | Paramanand Singh | 72,031 | <p>This is a consequence of mean value theorem. Redefine $f$ at $a$ by $f(a) =0$ so that $f$ is continuous on $[a, b) $. If $x\in(a, b) $ then we have via mean value theorem $$f(x) =f(a) +(x-a) f'(c_{x} )>0$$ where $c_{x} $ is some point in $(a, x) $ depending on $x$. </p>
|
4,612 | <p>I would like to make a slope field. Here is the code</p>
<pre><code>slopefield =
VectorPlot[{1, .005 * p*(10 - p) }, {t, -1.5, 20}, {p, -10, 16},
Ticks -> None, AxesLabel -> {t, p}, Axes -> True,
VectorScale -> {Tiny, Automatic, None}, VectorPoints -> 15]
</code></pre>
<p>I solved the diffe... | Heike | 46 | <p>I'm assuming here that the curves you mentioned are streamlines of the vector field. You can plot those automatically without having to solve any differential equations by using the options <code>StreamPoints</code>, for example to plot the stream lines going through the points</p>
<pre><code>points = Transpose@Ar... |
2,913,017 | <p>Imagine I have a real random variable $X$ with some distribution (continuous, discrete or continuous with atoms)</p>
<p>Now Imagine I have i.i.d. copies $X_1,...,X_n$, all independently and equally distributed as $X$</p>
<p>My claim is:</p>
<p>$$\mathbb{P}(X_2>X_1)=\mathbb{P}(X_2<X_1)$$
My secondy claim is ... | leonbloy | 312 | <p>Let $Y_n =\min(X_1,X_2 \cdots X_n)$, and let $y_n=\sum_{i=1}^n[X_i=Y_n]$ count the number of elements that attain that minimum.
Analogously, let $Z_n$ and $z_n$ be the maximum and maximum-count.</p>
<p>Then, by symmetry $P( X_{n} = Y_n \wedge y_n=1)=P(X_n=Y_n) P(y_n=1 \mid X_n=Y_n)=\frac{1}{n} P(y_n=1)$</p>
<p>The... |
3,262,714 | <p>So, I need an exponential function on the form <span class="math-container">$e^{-ax}$</span> that is 1 at <span class="math-container">$x=0$</span> and approaches <span class="math-container">$0.3$</span> as <span class="math-container">$x \rightarrow \infty$</span>. I tried doing <span class="math-container">$e^{-a... | Peter | 82,961 | <p>Choose the function <span class="math-container">$$f(x)=0.3+0.7e^{-x}$$</span></p>
|
1,134,854 | <blockquote>
<p>In complex analysis, let $a, b>0$ in $\mathbb R$, $f(s)=\int^{b}_{a}1/t^s dt$, then $f$ is holomorphic for $Re(s)>0$.</p>
</blockquote>
<p>If $s\neq 1$, then $f(s)=\frac{a^{1-s}}{(1-s)}-\frac{b^{1-s}}{(1-s)}$, but if $s=1$, then $f(s)=\ln\big(\frac{b}{a}\big)$, they seems quite different in th... | Christian Blatter | 1,303 | <p>Assume $0<a<b$ and write
$$f(s):=\int_a^b {1\over {\mathstrut t}^s}\>dt=\int_a^b e^{-s\,\log t}\>dt=\int_{\log a}^{\log b} e^{(1-s)u}\>du\ .$$
Now its obvious that $f$ is an entire function: We can differentiate under the integral sign.</p>
|
184,564 | <p>If $\frac{a}{c} > \frac{b}{d}$, then the mediant of these two fractions is defined as $\frac{a+b}{c+d}$ and can be shown to lie striclty between the two fractions. </p>
<p>My question is can you prove the following property of mediants: if $|\frac{a}{c} - x| > |x - \frac{b}{d}|$ then $|b/d - mediant| < |me... | bartgol | 33,868 | <p>In my opinion, trying to learn CFD before having at least a basic knowledge of Numerical Analysis is like trying to learn multiplication before addition: it's not impossible, but not the best idea.</p>
<p>For Numerical Analysis, I studied on <a href="http://books.google.com/books?id=31m4ahn_KfkC&printsec=frontc... |
2,056,309 | <p>$\textbf{Question}$: Let $f$ be absolutely continuous on the interval $[\epsilon, 1]$ for $0<\epsilon<1$. Does the continuity of $f$ at 0 imply that $f$ is absolutely continuous on $[0,1]$? What if f is also of bounded variation on $[0,1]$?</p>
<p>$\textbf{Attempt}$:</p>
<p>My thoughts are that $f$ is NOT ab... | user251257 | 251,257 | <p>Let $h = \mathbb 1_{(0,1]}$. For $x\in \mathbb R$ defines
$$ g(x) = \sum_{n=1}^\infty 2^n h(x \cdot 2^n - 1) \frac{(-1)^{n+1}}{n}$$
and
$$ f(x) = \int_0^x g(t) dt. $$</p>
<ol>
<li><p>Then, $g$ is improper Riemann integrable and thus $f$ is continuous. </p></li>
<li><p>Further, on every compact interval without $0$,... |
54,506 | <p><a href="http://www.hardocp.com/news/2011/07/29/batman_equation/" rel="noreferrer">HardOCP</a> has an image with an equation which apparently draws the Batman logo. Is this for real?</p>
<p><img src="https://i.stack.imgur.com/VYKfg.jpg" alt="Batman logo"></p>
<p><strong>Batman Equation in text form:</strong>
\beg... | J. M. ain't a mathematician | 498 | <p>In fact, the five linear pieces that consist the "head" (corresponding to the third, fourth, and fifth pieces in Shreevatsa's answer) can be expressed in a less complicated manner, like so:</p>
<p>$$y=\frac{\sqrt{\mathrm{sign}(1-|x|)}}{2}\left(3\left(\left|x-\frac12\right|+\left|x+\frac12\right|+6\right)-11\left(\l... |
54,506 | <p><a href="http://www.hardocp.com/news/2011/07/29/batman_equation/" rel="noreferrer">HardOCP</a> has an image with an equation which apparently draws the Batman logo. Is this for real?</p>
<p><img src="https://i.stack.imgur.com/VYKfg.jpg" alt="Batman logo"></p>
<p><strong>Batman Equation in text form:</strong>
\beg... | Community | -1 | <p>The following is what I got from the equations using MATLAB:
<img src="https://i.stack.imgur.com/vHI8K.jpg" alt="enter image description here"></p>
<hr>
<p>Here is the M-File (thanks to this <a href="https://gist.github.com/1119139">link</a>):</p>
<pre><code>clf; clc; clear all;
syms x y
eq1 = ((x/7)^2*sqrt(abs... |
54,506 | <p><a href="http://www.hardocp.com/news/2011/07/29/batman_equation/" rel="noreferrer">HardOCP</a> has an image with an equation which apparently draws the Batman logo. Is this for real?</p>
<p><img src="https://i.stack.imgur.com/VYKfg.jpg" alt="Batman logo"></p>
<p><strong>Batman Equation in text form:</strong>
\beg... | copper.hat | 27,978 | <p>The 'Batman equation' above relies on an artifact of the plotting software used which blithely ignores the fact that the value $\sqrt{\frac{|x|}{x}}$ is undefined when $x=0$. Indeed, since we’re dealing with real numbers, this value is really only defined when $x>0$. It seems a little ‘sneaky’ to rely on the solv... |
2,461,615 | <p>I am still at college. I need to solve this problem.</p>
<p>The total amount to receive in 1 year is 17500 CAD.
And the university pays its students each 2 weeks (26 payments per year). </p>
<p>How much does a student have to receive for 4 months?
I have calculated this in 2 ways (both seem ok) but results are di... | mlc | 360,141 | <p>a) assumes 4 weeks per 12 months, which is equivalent to a 48-week year.</p>
<p>b) assumes a 52-week year.</p>
<p>Given that the question explicitly relates payments to "weeks", the second convention seems preferable. The correct answer (in my opinion) is b). </p>
|
914,936 | <p>Does anyone know where I can find the posthumously published (I think) chapter 8 of Gauss's Disquisitiones Arithmaticae?</p>
| Beans on Toast | 257,517 | <p>Go to archive.org and look up Gauss' Werke, Band 1. This is in German and it includes the unfinished notes that would have become part of Section 8. But I would strongly recommend reading Mathews book on number theory first because it attempts to go over the content of Gauss' DA in a more up-to-date and accessible... |
3,395,044 | <p>Wikipedia states:</p>
<blockquote>
<p>In mathematics, a <strong>formal power series</strong> is a generalization of a <strong>polynomial</strong>, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the <strong>polynomial</strong> with an arbitr... | Eric Towers | 123,905 | <p>The Wikipedia definitions are not perfect. Let's go at this a little differently.</p>
<p>A formal polynomial is a finite ordered list of coefficients. We use powers of an indeterminant to mark where in the list each coefficient is. For instance (where I write out all the numbers that are normally elided) <span c... |
2,216,601 | <p>Alright so I have this Transformation that I know isn't one to one transformation, but I'm not sure why. </p>
<p>A Transformation is defined as $f(x,y)=(x+y, 2x+2y)$.</p>
<p>Now my knowledge is that you need to fulfill the 2 conditions: Additivity and the scalar multiplication one. I tried both of them and somehow... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Note you can write
$$
f(x,y) = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix},
$$
and since $f$ can be represented by matrix multiplication, it must be linear, since matrix multiplication is linear...</p>
<p>As for it being one-to-one... |
2,216,601 | <p>Alright so I have this Transformation that I know isn't one to one transformation, but I'm not sure why. </p>
<p>A Transformation is defined as $f(x,y)=(x+y, 2x+2y)$.</p>
<p>Now my knowledge is that you need to fulfill the 2 conditions: Additivity and the scalar multiplication one. I tried both of them and somehow... | Swistack | 251,704 | <p>The transformation $f$ is linear indeed, but it is not one-to-one transformation. <em>One-to-one</em> means that the system
\begin{equation*}
x + y = u \\
2(x + y) = v
\end{equation*}
has a unique solution for variables $(x, y)$, i.e. they can be expressed trough the variables $u$ and $v$. But the determinant of thi... |
2,136,079 | <p>A cone $K$, where $K ⊆\Bbb R^n$ , is pointed; which means that it contains no line (or equivalently, $(x ∈ K~\land~ −x∈K) ~\to~ x=\vec 0$.</p>
| Fred | 380,717 | <p>Here is a picture (in 3D) of a cone which is not a pointed cone:</p>
<p><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/1024px-DoubleCone.png" rel="noreferrer">https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/1024px-DoubleCone.png</a></p>
<p>Here is a pictu... |
3,839,878 | <p>Am currently doing a question that asks about the relationship between a quadratic and its discriminant.</p>
<p>If we know that the quadratic <span class="math-container">$ax^2+bx+c$</span> is a perfect square, then can we say anything about the discriminant?</p>
<p>Specifically, can we be sure that the discriminant... | David Kipper | 764,256 | <p>Since the quadratic is a perfect square, this tells you that a root of that quadratic will have multiplicity 2 - you took a linear equation and squared it. If the quadratic vanishes at <span class="math-container">$x=t$</span>, then so does this linear equation, but we have that linear equation twice by squaring, he... |
3,839,878 | <p>Am currently doing a question that asks about the relationship between a quadratic and its discriminant.</p>
<p>If we know that the quadratic <span class="math-container">$ax^2+bx+c$</span> is a perfect square, then can we say anything about the discriminant?</p>
<p>Specifically, can we be sure that the discriminant... | sirous | 346,566 | <p>The quadratic has two equal roots like d if it is perfect square and we may write:</p>
<p><span class="math-container">$(x-d)^2=x^2-2dx +d^2$</span></p>
<p>Then:</p>
<p><span class="math-container">$\Delta=4d^2-4d^2=0$</span></p>
<p>That is the discriminant is zero when the quadratic is perfect square.</p>
|
977,232 | <p>We have in and out degree of a directed graph G. if G does not includes loop (edge from one vertex to itself) and does not include multiple edge (from each vertex to another vertex at most one directed edge), we want to check for how many of the following we have a corresponding graph. the vertex number start from 1... | Arthur | 15,500 | <p>You're on the right track, but you're not supposed to set the elements to be equal to one another. That would solve the problem "which $t$ makes this vector <em>equal</em> to that vector", which is, as you've discovered, impossible.</p>
<p>What you <em>should</em> do is set one vector equal to a scalar multiple of ... |
4,569,910 | <p><span class="math-container">$ABC$</span> is a right-angled triangle (<span class="math-container">$\measuredangle ACB=90^\circ$</span>). Point <span class="math-container">$O$</span> is inside the triangle such that <span class="math-container">$S_{ABO}=S_{BOC}=S_{AOC}$</span>. If <span class="math-container">$AO^2... | Alan | 175,602 | <p>First off, we know there has to be at least one white stone in each, or you would cap out at <span class="math-container">$\frac 2 3$</span> which is less than what you have.</p>
<p>We can limit what we look at by just considering how many boxes have black stones in them.</p>
<p>The first case is putting all the bad... |
4,314,162 | <p>Assume k is a finite field with n elements, how many elements are in the projective line <span class="math-container">$\mathbb{P}^{1}(k)$</span> and how do I work this out?</p>
<p>I know that an element of <span class="math-container">$\mathbb{P}^{1}(k)$</span> is represented by <span class="math-container">$[a, b]$... | Ethan Bolker | 72,858 | <p>If you want to think geometrically you can work it out this way.</p>
<p>The affine line over a field of cardinality <span class="math-container">$q$</span> is the one dimensional vector space over the field, so has <span class="math-container">$q$</span> elements. Add the point at infinity to construct the projectiv... |
4,314,162 | <p>Assume k is a finite field with n elements, how many elements are in the projective line <span class="math-container">$\mathbb{P}^{1}(k)$</span> and how do I work this out?</p>
<p>I know that an element of <span class="math-container">$\mathbb{P}^{1}(k)$</span> is represented by <span class="math-container">$[a, b]$... | Servaes | 30,382 | <p>Your description of a projective line can be paraphrased as follows:</p>
<blockquote>
<p>The projective line over a field <span class="math-container">$k$</span> consists of all nonzero points in <span class="math-container">$k^2$</span>, where two points are considered 'the same' if they are on the same line throug... |
579,907 | <p>Let $G$ be a group and let $H$ be a normal subgroup.</p>
<p>Prove that if $S\subseteq G$ generates $G$,
then the set $\{sH\mid s∈S\} ⊆ G/H$ generates $G/H$.</p>
<p>I have no idea how to deal with the question above.
Can somebody please give me some help?</p>
| Berci | 41,488 | <p>I think, the best to consider any quotient set $A/\sim$ as that it contains elements of the set $A$ itself but the <em>equality sign is replaced</em> by $\sim$ (we <em>consider</em> $a$ and $b$ equal in $A/\sim$ if $a\sim b$).</p>
<p>In your case the set is the group $G$ and $x\sim y$ iff $x^{-1}y\in H$, and replac... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | JTP - Apologise to Monica | 64 | <p>SOH-CAH-TOA </p>
<p>How to remember the ratios for the 3 main trig functions. </p>
|
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Valerij | 9,876 | <p>In the 11th grade our math teacher taught us following:</p>
<blockquote>
<p>was the maiden brave</p>
<p>the belly stays concave</p>
<p>but unprotected sex</p>
<p>makes the belly convex</p>
</blockquote>
|
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Rodrigo Zepeda | 5,922 | <p>This only makes sense in Spanish but it's pretty fun. For integration by parts,
$$
\int u dv = u v - \int v du
$$</p>
<blockquote>
<p><strong>S</strong>i <strong>u</strong>n <strong>d</strong>ía <strong>v</strong>i <strong>u</strong>na <strong>v</strong>aca <strong>menos</strong> <strong>s</strong>exy <strong>v</... |
1,657,694 | <p>The Algorithms course I am taking on Coursera does not require discrete math to find discrete sums. Dr. Sedgewick recommends replacing sums with integrals in order to get basic estimates.</p>
<p>For example: $$\sum _{ i=1 }^{ N }{ i } \sim \int _{ x=1 }^{ N }{ x } dx \sim \frac { 1 }{ 2 } N^2$$</p>
<p>How would I ... | Davey | 314,780 | <p>See <a href="http://www.redalyc.org/articulo.oa?id=46815211" rel="nofollow">http://www.redalyc.org/articulo.oa?id=46815211</a>. The answer is thus not necessarily. I haven't read the construction yet, but at least it's not in Spanish. Great question, by the way.</p>
|
1,451,745 | <p>Can someone check my logic here. </p>
<p><strong>Question:</strong> How many ways are there to choose a an $k$ person committee from a group of $n$ people? </p>
<p><strong>Answer 1:</strong> there are ${n \choose k}$ ways. </p>
<p><strong>Answer 2:</strong> condition on eligibility. Assume the creator of the comm... | Brian M. Scott | 12,042 | <p>It’s not clear whether your argument works or not, because you’ve not told us what you mean by <em>eligible</em>. Here is one way to make your argument valid.</p>
<p>Suppose that that the $n$ potential members all have different ages, and that the committee must be created by its oldest member. In other words, the ... |
546,276 | <p>Let $\{s_n\}$ be a sequence in $\mathbb{R}$, and assume that $s_n \rightarrow s$. Prove that $s^k_n\rightarrow s^k$ for every $k \in\mathbb{N}$</p>
<p>Ok, so we need $|s^k_n - s^k| < \varepsilon$. I rewrote this as</p>
<p>$$|s_ns^{k-1}_n - ss^{k-1}|=|(s_n-s)(s^{k-1}_n + s^{k-1}) -s_ns^{k-1}+ss_n^{k-1}|$$</p>
<... | Fly by Night | 38,495 | <p><strong>Hint</strong></p>
<p>Apply the chain rule and implicit differentiation to find the gradient. If you have the graph with equation $y=\operatorname{f}^{-1}(x)$ then it also has the equation $\operatorname{f}(y)=x$.</p>
|
2,262,167 | <p>The question in title has been considered for finite groups $G$ and $H$, but I do not know its situation, how far it is known whether $G$ and $H$ could be isomorphic. I have two simple questions regarding it.</p>
<p><strong>Q.0</strong> If $\mathbb{Z}[G]\cong \mathbb{Z}[H]$ then $|G|$ should be $|H|$; because, $G$ ... | Dietrich Burde | 83,966 | <p>This isomorphism problem was stated by Higman in this thesis $1940$:
$$
\mathbb{Z}[G]\cong\mathbb{Z}[H] \Longrightarrow G\cong H ?
$$
It is true for nilpotent groups, for metabelian groups (Whitcomp $1968$), and was disproved by Hertweck in $2001$, see <a href="https://math.stackexchange.com/questions/632372/minima... |
1,656,136 | <p>I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)</p>
| egreg | 62,967 | <p>The classical example is the ring of polynomials in a countable number of indeterminates: $k[x_1,x_2,\dots,x_n,\dotsc]$ and the chain is
$$
0\subsetneq (x_1)\subsetneq(x_1,x_2)\subsetneq\dotsb
$$</p>
<p>Note that $x_{n+1}\notin(x_1,\dots,x_n)$ by a standard argument: suppose
$$
x_{n+1}=f_1 x_1 + f_2x_2 + \dots + f_... |
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | amsmath | 487,169 | <p>Put $f(x) := 3\sqrt{x-1}+\sqrt{3x+1}$ on $[1,\infty)$. Then $f'(x) > 0$ and $f(1) = 2$.</p>
|
2,664,341 | <blockquote>
<p>Simplify $$\frac{1}{\sqrt[3]1+\sqrt[3]2+\sqrt[3]4}+\frac{1}{\sqrt[3]4+\sqrt[3]6+\sqrt[3]9}+\frac{1}{\sqrt[3]9+\sqrt[3]{12}+\sqrt[3]{16}}$$</p>
</blockquote>
<p>I have no idea how to do this. I tried using the idea of multiplying the conjugate to every term, but I seem to be getting no where. Is the... | drhab | 75,923 | <p>If $\nu(A):=\int_A f_+d\lambda$ and $\mu(A):=\int_A f_-d\lambda$ then $\nu$ and $\mu$ are both finite measures so that: $$\lim_{n\to\infty}\nu(E_n)=\nu(E)<\infty\text{ and }\lim_{n\to\infty}\mu(E_n)=\mu(E)<\infty$$
So: $$\lim_{n\to\infty}\int_{E_n} fd\lambda=\lim_{n\to\infty}[\nu(E_n)-\mu(E_n)]=\nu(E)-\mu(E)=... |
2,195,197 | <blockquote>
<p>A circle goes through $(5,1)$ and is tangent to $x-2y+6=0$ and $x-2y-4=0$. What is the circle's equation?</p>
</blockquote>
<p>All I know is that the tangents are parallel, which means I can calculate the radius as half the distance between them: $\sqrt5$. So my equation is
$$(x-p)^2+(y-q)^2=5$$
How ... | Parcly Taxel | 357,390 | <p>We can place an additional constraint on the circle centre $(p,q)$. It has to lie on the line parallel to the two tangents and equidistant from them:
$$p-2q+1=0\quad p=2q-1$$
Then since the circle passes through $(5,1)$:
$$(5-(2q-1))^2+(1-q)^2=5$$
$$25-10(2q-1)+(2q-1)^2+1-2q+q^2=5$$
$$25-20q+10+4q^2-4q+1+1-2q+q^2=5$... |
213,916 | <p>Let $ D\subset \mathbb{C}$ be open, bounded, connected and with smooth boundary. Let $f$ be a nonconstant holomorphic function in a neighborhood of the closure of $D$ , such that $|f(z)|=c \forall z\in \partial D$, show that $f$ takes on each value $a$, such that $|a| < |c| $ at least once in $D$.</p>
| froggie | 23,685 | <p>I should preface this by saying there <em>must</em> be a better solution than this.</p>
<p>Let $B$ be the open disk $B = \{z\in \mathbb{C} : |z|<c\}$. We want to show $B\subset f(D)$. Define $S = \{z\in B : z\notin f(D)\}$. First note that $S$ is closed in $B$, since by the open mapping theorem $f(D)$ is open, a... |
633,858 | <p>If G is cyclic group of 24 order, then how many element of 4 order in G?
I can't understand how to find it, step by step. </p>
| Xucheng Zhang | 119,182 | <p><strong>Lemma:</strong> If $o(g)=n$, then $o(g^k)=\dfrac{n}{gcd(n,k)}$.</p>
<p>Assume $G=\langle g \rangle$ and $o(g)=24$, notice that every element of $G$ has the form $g^k$ for some integer $k$, by lemma, $o(g^k)=\dfrac{24}{gcd(24,k)}$. Let $\dfrac{24}{gcd(24,k)}=4$, i.e., $gcd(24,k)=6$, we can easily conclude th... |
321,230 | <p>Suppose $Z$ is a topological space; and $X$ is dense in $Z$. Then do we have $W(X)= W(Z)$?
Where $W(X)$, $W(Z)$ denote the weight of the $X$ and $Z$ respectively. </p>
<p><strong>What I've tried:</strong> On one hand, $W(X)\le W(Z)$, clearly; On the other hand, for any open set $U$ of $Z$, we have $U\cap X$, an op... | Brian M. Scott | 12,042 | <p>Arthur’s example shows that $w(Z)$ can be as large as $2^{w(X)}$. If $Z$ is regular, this is the biggest possible value for $w(Z)$. In that case $Z$ has a base $\mathscr{R}$ of $w(Z)$ regular open sets, and if $U\subseteq Z$ is regular open, then $U=\operatorname{int}_Z\operatorname{cl}_Z(U\cap X)$, so each $U\in\ma... |
784,032 | <p>Find the remainder when $6!$ is divided by 7.</p>
<p>I know that you can answer this question by computing $6! = 720$ and then using short division, but is there a way to find the remainder without using short division?</p>
| lab bhattacharjee | 33,337 | <p>As $7$ is prime, use <a href="http://www.proofwiki.org/wiki/Wilson%27s_Theorem">Wilson's Theorem</a> $$(p-1)!\equiv-1\pmod p$$ for prime $p$</p>
<p>Now, $\displaystyle -1\equiv p-1\pmod p$</p>
|
2,136,937 | <p>Find $$\lim_{z \to \exp(i \pi/3)} \dfrac{z^3+8}{z^4+4z+16}$$</p>
<p>Note that
$$z=\exp(\pi i/3)=\cos(\pi/3)+i\sin(\pi/3)=\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}$$
$$z^2=\exp(2\pi i/3)=\cos(2\pi/3)+i\sin(2\pi/3)=-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}$$
$$z^3=\exp(3\pi i/3)=\cos(\pi)+i\sin(\pi)=1$$
$$z^4=\exp(4\pi i/3)=\cos(4\... | user414998 | 414,998 | <p>While both Spivak's and Apostol's books are rigorous in that they include complete proofs, Spivak's has a heavier emphasis on theoretical questions, and its exercises are much harder. Spivak's book also has a complete solution manual. Spivak's book can be considered one of the best introductions to rigorous mathemat... |
3,897,689 | <p>i have the equation:
<span class="math-container">$$y'+2y\:=1$$</span></p>
<p>and i solve it the regular way for first order differential equation:
<span class="math-container">$$y'\:=1-2y$$</span>
<span class="math-container">$$\frac{dy}{dx}=1-2y$$</span>
<span class="math-container">$$\int \:\frac{1}{1-2y}dy=\int ... | xpaul | 66,420 | <p>Since
<span class="math-container">$$\int\frac{1}{1-2y}dy=\int 1dx, \text{ or }-\int\frac{1}{2y-1}dy=\int1dx $$</span>
one has
<span class="math-container">$$ -\frac12\ln|2y-1|=x+C $$</span>
or
<span class="math-container">$$ \ln|2y-1|=-2x-2C$$</span>
So
<span class="math-container">$$ 2y-1=\pm e^{-2C}e^{-2x}$$</spa... |
4,264,558 | <p>I calculated homogenous already, I'm just struggling a bit with the right side. Would <span class="math-container">$y_p$</span> be <span class="math-container">$= ++e^x$</span> or <span class="math-container">$= ++e^{2x}$</span>?</p>
<p>Would the power in front of the root be the roots found from the homogenous part... | David P | 49,975 | <p><span class="math-container">$$\lim_{x\to\infty} x^2(1-\cos(1/x)) = \lim_{x\to\infty} \dfrac{1-\cos(1/x)}{1/x^2} \to {0\over 0}$$</span></p>
<p>Now you can apply L'Hopital's rule</p>
<p><span class="math-container">$$\lim_{x\to\infty} \dfrac{1-\cos(1/x)}{1/x^2} =\lim_{x\to\infty} = \dfrac{-\sin(1/x)/x^2}{-2/x^3} ... |
3,814,195 | <p>As an applied science student, I've been taught math as a tool. And although I've been studying <strong>a lot</strong> throughout the years, I always felt like I am missing depth. Then I read geodude's answer on this <a href="https://math.stackexchange.com/questions/721364/why-dont-taylor-series-represent-the-enti... | Community | -1 | <p>I found the book by Gamelin "Complex Analysis" together with "Visual Complex Analysis" by T. Needham to be a good combination.</p>
<p>Also if you read German, then I recommend chapters 1-4 from "Funktionentheorie" Busam/Freitag (not sure if it exists in English).</p>
<p>There is also &q... |
94,440 | <p>In Sean Carroll's <em>Spacetime and Geometry</em>, a formula is given as
$${\nabla _\mu }{\nabla _\sigma }{K^\rho } = {R^\rho }_{\sigma \mu \nu }{K^\nu },$$</p>
<p>where $K^\mu$ is a Killing vector satisfying Killing's equation ${\nabla _\mu }{K_\nu } +{\nabla _\nu }{K_\mu }=0$ and the convention of Riemann curvatu... | Zhen Lin | 5,191 | <p>Permit me the use of Latin indices instead of Greek indices and the convention <span class="math-container">$\nabla_a K_b=K_{b;a} $</span>. So we wish to prove
<span class="math-container">$\newcommand{\Tud}[3]{{#1}^{#2}_{\phantom{#2}{#3}}}$</span>
<span class="math-container">$$\Tud{K}{a}{;b c} = \Tud{R}{a}{b c d} ... |
2,468,326 | <p>I want to read <a href="https://www.amazon.co.uk/Introduction-Cyclotomic-Fields-Graduate-Mathematics/dp/0387947620" rel="noreferrer">Lawrence Washington's An <em>Introduction to Cyclotomic Fields</em></a>. However, my knowledge of algebraic number theory doesn't extend farther than what's found in <a href="https://w... | lhf | 589 | <p>The preface to the first edition of Washington's book says:</p>
<blockquote>
<p>The reader is assumed to have had at least one semester of algebraic
number theory (though one of my students took such a course
concurrently). In particular, the following terms should be familiar:
Dedekind domain, class number... |
4,228,512 | <p>The question is: does the sequence of characteristic functions <span class="math-container">$f_k(x) := \chi_{[-\frac{1}{k}, \frac{1}{k}]}(x)$</span> converge in distributional sense to the Dirac delta?</p>
<p>In order to answer I followed this approach, but I fear I'm neglecting something important in my lines:</p>
... | paul garrett | 12,291 | <p>Your argument is correct. Indeed, characteristic functions <span class="math-container">$\chi_\varepsilon$</span> of smaller and smaller intervals <span class="math-container">$[-\varepsilon,\varepsilon]$</span> do not converge to <span class="math-container">$\delta$</span> (in a distributional sense), but, rather,... |
330,991 | <p>Many things in math can be formulated quite differently; see the list of statements equivalent to RH <a href="https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis">here</a>, for example, with RH formulated as a bound on lcm of consecutive integers, as an integral equality, et... | none | 140,370 | <p><a href="https://www.cs.toronto.edu/~sacook/homepage/ptime.pdf" rel="nofollow noreferrer">https://www.cs.toronto.edu/~sacook/homepage/ptime.pdf</a></p>
<p>The above paper (1991) gives a syntactic method for enumerating all the PTIME functions. P != NP is the proposition that none of the functions in that enumerati... |
338,535 | <p>Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$.
Given that $f(1)= 2013$, find the value of $f(2013)$.</p>
| Eric Naslund | 6,075 | <p><strong>Hint:</strong> Try computing some small examples first. When $n=2$, we get that $$f(1)+2^{2}f(2)=2^{3}f(2) $$ $$\Rightarrow 2^2f(2)=f(1).$$ Using the previous case and the given equation, when $n=3,$ we have that </p>
<p>$$f(1)+2^2f(2)+3^{2}f(3)=3^{3}f(3)$$
$$\Rightarrow f(1)+f(1)+3^{2}f(3)=3^{3}f(3),$$ ... |
1,740,151 | <p>$$\lim_\limits {x \to \pi} \frac{(e^{\sin x} -1)}{(x-\pi)}$$</p>
<p>I found $-1$ as the answer and what I did was: </p>
<p>$\lim_\limits {x \to \pi} \frac{(e^{\sin x} -1)}{(x-\pi)}$ $\Rightarrow$ $\lim \frac{(f(x) - f(a))}{(x-a)}$ $\Rightarrow$ $f(x)=(e^{\sin x})$ </p>
<p>$f(a)=1$ </p>
<p>$x=x$ </p>
<p>and $a=... | Matthé van der Lee | 75,745 | <p>As to your question (1), the answer is: <em>all</em> of AC is needed.</p>
<p><strong>Lemma</strong>: if $f:A\times B\to C\cup D$ is injective, $C\cap D=\varnothing$, and at least one of $A,B,C,D$ is well ordered, we have $|A|\leq|C|\vee|B|\leq|D|$. (By symmetry, one has $|A|\leq|D|\vee|B|\leq|C|$ as well!).</p>
<p... |
3,453,408 | <p>I'm reading through some lecture notes and see this in the context of solving ODEs:
<span class="math-container">$$\int\frac{dy}{y}=\int\frac{dx}{x} \rightarrow \ln{|y|}=\ln{|x|}+\ln{|C|}$$</span> why is the constant of integration natural logged here?</p>
| Eric Towers | 123,905 | <p>In this form, it is evident that you can rewrite the result as <span class="math-container">$\ln \left| C x \right|$</span>. Perhaps this is less evident from the form <span class="math-container">$\ln |x| + C$</span>.</p>
|
238,547 | <p>I have a PDE like</p>
<pre><code>D[h[x1, x2], x1]*a[x1,x2]+D[h[x1,x2], x2]*b[x1,x2] + c[x1,x2] == h[x1,x2]
s.t. gradient(h(0,0))==0
</code></pre>
<p>where a,b,c are known functions of x1 and x2, and h are the function to be solved. x1 and x2 are both in [-2, 2].
For some selected a,b,c, DSolveValue can give me perfe... | bbgodfrey | 1,063 | <p>Linear PDEs typically are solved by the method of characteristics. For the PDE in the question, the ODEs of the characteristics are</p>
<pre><code>{x2'[s] == Sin[x1[s]] + x2[s], x1'[s] == x2[s], h'[s] == h[s]}
</code></pre>
<p>Attempting to solve these ODEs with <code>DSolve</code> is unsuccessful, returning the <c... |
403,631 | <p>$a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1 $
<br/>
Hint $u_{2n}$ = $u_{n}^2$</p>
<p>I have totally no idea how to prove this, this looks obvious but i found out proof is really hard...
I am doing a real analysis course and there's a lot of proving and I stuck there.
Any advices? Pra... | A.S | 24,829 | <p>Proof sketch: Maybe try showing that $|\frac{1}{a^n}| \to \infty$.</p>
<p>Fix $p = |\frac{1}{a}| > 1$, and let $p = ( 1 + b )$. Show by induction that $p^n \ge 1 + nb$, and conclude the statement above using the Archimedean property of the reals.</p>
|
403,631 | <p>$a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1 $
<br/>
Hint $u_{2n}$ = $u_{n}^2$</p>
<p>I have totally no idea how to prove this, this looks obvious but i found out proof is really hard...
I am doing a real analysis course and there's a lot of proving and I stuck there.
Any advices? Pra... | xavierm02 | 10,385 | <p>Let $u_n = a^n$ with $|a|< 1$</p>
<p>Let $v_n=|u_n|$</p>
<p>$v_{n+1} = a v_n < v_n$ so $(v_n)$ is decreasing</p>
<p>$0 \le v_n$ so $(v_n)$ is minored</p>
<p>Since $(v_n)$ is minored and decreasing, it converges.</p>
<hr>
<p>Now let $v_\infty=\lim\limits_{n\to \infty}v_n$</p>
<p>You have $v_{n+1} = a v_n... |
177,519 | <p>Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1}]$. In a sequence of papers, Kazhdan and Lusztig constructed a braided monoidal structu... | Makoto Yamashita | 9,942 | <p>This is not very sophisticated answer, but in a way there isn't much choose from as the representation categories of "deformations of $U(\mathfrak{g})$." For $\mathfrak{g}=\mathfrak{sl}_n$, this can be made precise by [1] as follows. Any semisimple $\mathbb{C}$-linear monoidal category with the fusion rule of $\math... |
599,126 | <p>Question is to check which of the following holds (only one option is correct) for a continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$.</p>
<ul>
<li>$f$ has to be uniformly continuous.</li>
<li>there exists a $x\in \mathbb{R}$ such that $f(x)=x$.</li>
<li>$f$ can not be increasing.</li>
<li>$\lim_{x... | Igor Rivin | 109,865 | <p>Well, for $x$ really, really large, what can you say about $f(x) - x?$ </p>
<p>For $x$ really, really small, what can you say about $f(x) - x?$</p>
|
1,512,171 | <p>I want to show that there exists a diffeomorphic $\phi$ such that the following diagram commutes:
$$
\require{AMScd}
\begin{CD}
TS^1 @>{\phi}>> S^1\times\mathbb{R}\\
@V{\pi}VV @V{\pi_1}VV \\
S^1 @>{id_{S^1}}>> S^1
\end{CD}$$
where $\pi$ is the associated projection of $TS^1$, and $\pi_1(x,y)=x$ is ... | dannum | 152,081 | <p>It depends if you're talking about a Riemann integral or a Lebesgue integral. </p>
<p>If we are talking about a Riemann integral, the answer is that we cannot define the integral because any sub-interval of $[0,1]$ - no matter how small - contains a rational and an irrational. For this reason the upper integral and... |
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