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 |
|---|---|---|---|---|
1,961,727 | <p>As far as I understood <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram–Schmidt orthogonalization</a> starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting vectors did.</p>
<p>I have no problem understand... | Community | -1 | <p>Orthonormal bases are nice because several formulas are much simpler when vectors are given wrt an ON basis. </p>
<p>Example: Let $\mathcal E = \{e_1, \dots, e_n\}$ be an ON basis. Then the Fourier expansion of any vector $v\in\operatorname{span}(\mathcal E)$ is just $$v = (v\cdot e_1)e_1 + (v\cdot e_2)e_2 + \cdo... |
2,007,373 | <p>At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. </p>
<p>For me the first instance that comes to memory was in 7th grade in a inner city USA school district. </p>
<p>Getting to the point, my geometry teacher taught,</p>
<p>"a p... | AnoE | 360,316 | <blockquote>
<p>Meanwhile I'm experiencing cognitive dissonance, how can anything with zero length or width be stacked on top of itself and build itself into something with width of length?</p>
</blockquote>
<p>That's not what happens, at all, when going from zero to one and then on to n dimensions. You are thinking... |
2,109,832 | <p>This is for beginners in probability!</p>
<p>Could someone give me a step by step on how to find the MGF of the binomial distribution?</p>
| David Holden | 79,543 | <p>for any $\theta$ we have
$$
\sin 2\theta = 2 \sin \theta\cos \theta
$$
if<br>
$$
\theta = \arcsin \frac35
$$
then, trivially,
$$
\sin \theta = \frac35
$$
using
$$
\cos^2 \theta +\sin^2 \theta = 1
$$
can you compute $\cos \theta$ to finish off? (how do you interpret the fact that there are two possible values of $\co... |
567,391 | <p>A bus follows its route through nine stations, and contains six passengers. What is the probability that no two passengers will get off at the same station? </p>
<p>no detailed solution is required here but an idea of the general line of thought could be nice...</p>
| Max Sherman | 36,053 | <p>Usually what you want to do is pick any arbitrary element of $A \cup B$, call it $x$. So $x \in A \cup B$. Then show that $x \in C$, using what you know about elements of $A$ and $B$.</p>
|
567,391 | <p>A bus follows its route through nine stations, and contains six passengers. What is the probability that no two passengers will get off at the same station? </p>
<p>no detailed solution is required here but an idea of the general line of thought could be nice...</p>
| amWhy | 9,003 | <p>To show $A\cup B \subseteq C$, we show that for any $x \in A\cup B$, it follows that $x \in C$.</p>
<hr>
<p>Assumptions (Givens): $A\subseteq C$, $B \subseteq C$.</p>
<p>Suppose $x \in A\cup B.\;\;$ Then $x \in A$, or $x\in B\;$ (from the definition of set union). </p>
<p>Now, use what you know from our givens ... |
1,976,382 | <p>Hölder's inequality for finite sums is given by
$$\sum_{k=0}^n|a_kb_k|\leq\left(\sum_{k=0}^n|a_k|^p\right)^{1/p}\left(\sum_{k=0}^n|b_k|^q\right)^{1/q},$$
where $1/p+1/q=1$, $p,q\in(1,\infty)$.</p>
<p>Is there a "similar" inequality which gives a lower bound for the left hand sum? I have searched, but found nothing ... | Mark Fischler | 150,362 | <p>the corresponding lower bound is
$$
\max(|a_k b_k|)
$$
which is not very interesting because it squelches $p$ and $q$, but it saturates so we can't produce a more interesting lower bound.</p>
|
9,934 | <p>I have been given some code with the following line</p>
<pre><code>PeriodicExtension[g_, x_] := If[Abs[x] < Pi, g[x], PeriodicExtension[g, x - 2 Sign[x] Pi]]
</code></pre>
<p>I do not understand the syntax. I would appreciate if someone can explain what this
code does for different values of <code>x</code>.</p>... | Dr. belisarius | 193 | <pre><code>PeriodicExtension[g_, x_] := If[Abs[x] < Pi, g[x], PeriodicExtension[g, x - 2 Sign[x] Pi]]
g[x_] := x
Plot[PeriodicExtension[g, x], {x, 0, 4 Pi}]
</code></pre>
<p><img src="https://i.stack.imgur.com/9Tgp0.png" alt="Mathematica graphics"></p>
|
3,534,364 | <blockquote>
<p><span class="math-container">$x^2y'^2 + 3xyy' +2y^2 = 0 $</span></p>
</blockquote>
<p>Usually, to solve an ODE with respect to <span class="math-container">$y'=p$</span>, we first isolate the <span class="math-container">$y$</span>, to get <span class="math-container">$y = f(x,p)$</span> and then dif... | Robert Israel | 8,508 | <p>You can factor your differential equation to get
<span class="math-container">$$ (x y' + y)(x y' + 2 y) = 0 $$</span></p>
<p>so either <span class="math-container">$x y' + y = 0$</span> or <span class="math-container">$x y' + 2 y = 0$</span>. One gives you <span class="math-container">$y = c/x$</span>, the other <... |
1,198,722 | <p>I am working with a standard linear program:</p>
<p>$$\text{min}\:\:f'x$$
$$s.t.\:\:Ax = b$$
$$x ≥ 0$$</p>
<p><strong>Goal:</strong> I want to enforce all nonzero solutions $x_i\in$ x to be greater than or equal to a certain threshold "k" if it's nonzero. In other words, I want to add a conditional bound to the LP... | bassen | 194,657 | <p>Like Rahul mentioned in a comment to your question, this is not possible (incidentally, I do not agree with TravisJ's comment that an ILP is a special case of an LP. Rather, an LP is a special case of a mixed-integer linear program, of which integer-linear programming is also a special case. But I do not have enoug... |
2,393,525 | <p>I have two questions which I think both concern the same problem I am having. Is $...121212.0$ a rational number and is $....12121212....$ a rational number? The reason I was thinking it could be a number is when you take the number $x=0.9999...$, then $10x=9.999...$ . Therefore, we conclude $9x=9$ which means $x=1... | Noah Schweber | 28,111 | <p>You write:</p>
<blockquote>
<p>I thought the real numbers were defined as the numbers on the number line.</p>
</blockquote>
<p>This isn't really a definition of the real numbers, since "number line" is a bit vague, but: a key fact about the number line as generally understood is that the distance between any two... |
3,558,784 | <p>A question is as follows:
Consider a open top cylinder with radius <span class="math-container">$R$</span> and height <span class="math-container">$H$</span> full of water and tilt the cylinder to pour the water until the water surface at the base of the cylinder intersects the diameter of the base.</p>
<p>Find the... | John Omielan | 602,049 | <p>Notice when you multiply out the second line you get <span class="math-container">$8 \times 65548 = 524384$</span> (you have a typo in your second line where it says <span class="math-container">$65568x^2$</span>) for the <span class="math-container">$x^3$</span> term and <span class="math-container">$-7 \times 4210... |
3,558,784 | <p>A question is as follows:
Consider a open top cylinder with radius <span class="math-container">$R$</span> and height <span class="math-container">$H$</span> full of water and tilt the cylinder to pour the water until the water surface at the base of the cylinder intersects the diameter of the base.</p>
<p>Find the... | zwim | 399,263 | <p>Another method is to set <span class="math-container">$y=tx$</span> to get the polynomial <span class="math-container">$f(t)=\sum\limits_{i=0}^3 a_it^i$</span></p>
<p><span class="math-container">$$f(t)=524384-27204t-40806t^2-294763t^3$$</span></p>
<p>Possible rational roots of <span class="math-container">$f$</sp... |
2,281,932 | <p>If Peano axioms uniquely determine the natural numbers, doesn't this mean that Peano axioms are categorical and hence complete?</p>
<p>If above is true, how is it explained by Goedel's incompleteness theorem?</p>
| M. Winter | 415,941 | <p>The Peano axioms do <em>not</em> pin down the natural numbers uniquely (see <a href="https://math.stackexchange.com/a/2251401/415941">this</a> amazing answer to some silimar question of mine).</p>
<p>As you said, there is Gödel's incompleteness theorem which prevents this from happening. Each approach to pin down t... |
1,699,833 | <p>I found a problem, I don't really know how to solve, although it should be something very easy, since it is stuff of Algebra I.</p>
<blockquote>
<p>Let $f= 29X^5−13X^4−44X^3+ 18X^2+ 35X+ 10\in\mathbb{Z}[X]$.</p>
<p>1) Decompose $f$ into irreducible factors in $\mathbb{F}_2[X]$ and $\mathbb{F}_3[X]$.</p>
... | Nikolas Wojtalewicz | 312,038 | <blockquote>
<p>$X^4+X^3+1$ is irreducible, because $0+0+1=1=1+1+1$ in $\mathbb{F}_2$</p>
</blockquote>
<p>Not true: or, at least, the part about being irreducible because it doesn't contain any linear factors. A polynomial can have no linear factors and still be reducible. Consider
\begin{align}
p = \left( X^2... |
1,699,833 | <p>I found a problem, I don't really know how to solve, although it should be something very easy, since it is stuff of Algebra I.</p>
<blockquote>
<p>Let $f= 29X^5−13X^4−44X^3+ 18X^2+ 35X+ 10\in\mathbb{Z}[X]$.</p>
<p>1) Decompose $f$ into irreducible factors in $\mathbb{F}_2[X]$ and $\mathbb{F}_3[X]$.</p>
... | user26857 | 121,097 | <p>We have $\bar{f}=X(X^4+X^3+1)\in\mathbb{F}_2[X]$ and $\bar f=2(X^2+1)(X^3+X^2+X+2)\in\mathbb F_3[X]$ (credit to Macavity) factorizations into irreducibles.<br>
If $f$ is reducible over $\mathbb Z$ then $f=gh$ with $g,h\in\mathbb Z[X]$. We have $\deg g+\deg h=5$, $1\le\deg g<5$ and $1\le\deg h<5$. Moreover, the... |
273,798 | <p>I am writing a large numerical code where I care a lot about performance, so I am trying to write compiled functions that are as fast as possible.</p>
<p>I need to write a function that does the following. Consider a list of positive integers, for example {5,3}, take its flattened binary form (with a given number of... | Syed | 81,355 | <p>Please investigate how this technique fares in your application. Let's say:</p>
<pre><code>SeedRandom[1];
binSequence = RandomInteger[{0, 1}, 20]
</code></pre>
<blockquote>
<p>{1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1}</p>
</blockquote>
<pre><code>indices = {5, 9, 15};
acc = Accumulate[binSequence... |
3,069,987 | <p>I know that whatever numbers you choose for x and y and their sum equals to 1 will satisfy the equation <span class="math-container">$x^2 + y = y^2 + x$</span></p>
<p>Algebraic proof: </p>
<p>Given: <span class="math-container">$x + y = 1$</span></p>
<p><span class="math-container">$$LS = x^2+ y
= (1-y)^2 + y ... | Jaap Scherphuis | 362,967 | <p>Here is a picture. The left shows <span class="math-container">$y^2+x$</span>, the right <span class="math-container">$x^2+y$</span>.</p>
<p><a href="https://i.stack.imgur.com/D7MKa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/D7MKa.png" alt="enter image description here"></a></p>
|
132,862 | <p>Is it true that given a matrix $A_{m\times n}$, $A$ is regular / invertible if and only if $m=n$ and $A$ is a basis in $\mathbb{R}^n$?</p>
<p>Seems so to me, but I haven't seen anything in my book yet that says it directly.</p>
| Trismegistos | 23,730 | <p>Way of proving some theories by indcution is very importand technique without it mathematics would not be the same so it was added to formalize our intuition. From what I know suprise was that this axiom was not consequence of rest of the axioms but is needed to be stated explicite.</p>
|
179,581 | <p><strong>Problem:</strong></p>
<p>(a). If $f$ is continuous on $[a,b]$ and $\int_a^x f(t) dt = 0$ for all $x \in [a,b]$, show that $f(x) = 0$ for all $x \in [a,b]$.</p>
<p>(b). If $f$ is continuous on $[a,b]$ and $\int_a^x f(t)dt = \int_x^b f(t)dt$ for all $x \in [a,b]$, show that $f(x)=0$ for all $x\in [a,b]$.</p>... | Kevin Arlin | 31,228 | <p>Yes, you've solved these accurately. As a small point, I wouldn't call these Leibniz' rule problems, as that relates to taking partial derivatives of multivariable functions in a direction orthogonal to the path of integration. This is an advanced calculus technique you may not have encountered yet. </p>
<p>Rather,... |
3,366,064 | <p>I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere. </p>
<p>My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with van... | Quanto | 686,284 | <p>It may be surprising that the problem actually admits an analytic solution.</p>
<p>A spherical cap is the difference between two overlapping cones, one with a spherical bottom and the other with a flat bottom, i.e.</p>
<p><span class="math-container">$$ V = \frac{2\pi}{3}r^2h - \frac{\pi}{3}(2rh-h^2)(r-h)
=\frac{\p... |
1,783,323 | <p>Given the transition matrix for a 2 state Markov Chain, how do I find the n-step transition matrix P^n? I also need to take n--> inf and find the invariant probability pi?</p>
| nicomezi | 316,579 | <p>A common way to find $P^n$ is to diagonalize your matrix.
Then you will have $P=MDM^{-1}$ with D a diagonal matrix, so $P^n=MD^nM^{-1}$.
So taking $n \rightarrow \infty$ will be easy.</p>
<p>Also, if $\mu$ is a measure of probability on your two states MC, $\underset{n\to \infty}\lim\mu P^n$, if it converges, is an... |
875,729 | <p>Prove that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $A$ as its diagonal elements.</p>
<p>I can see that all eigenvalues of $A$ are real, and the corresponding eigenvectors ... | Algebraic Pavel | 90,996 | <p>Provided that the Schur decomposition is an allowed tool:</p>
<p>Using the Schur decomposition, we have that there exists an orthogonal $Q$ and an upper triangular $R$ such that $A=QRQ^T$. Since $A$ is symmetric, $Q^TAQ=R$ is symmetric as well. Therefore $R$ is symmetric. A symmetric triangular matrix is necessaril... |
627,258 | <p>Helly everybody,<br>
I'm trying to find another approach to topology in order to justify the axiomatization of topology. My idea was as follows:</p>
<p>Given an <strong>arbitrary</strong> collection of subsets of some space: $\mathcal{C}\in\mathcal{P}^2(\Omega)$<br>
Define a closure operator by: $\overline{A}:=\big... | Dominik | 50,527 | <p>In fact, the consequences are of leaving out this axiom are rather dull.</p>
<p>Let $X$ be such an <em>almost topological space</em>. Now let $$Y\subset X$$ be the supset of all elements of $x$ such that there is an open set $U\ni x$. Taking the union of all such open sets $U$ you see that $Y$ is open and every ope... |
496,255 | <p>Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact? </p>
| Marquis Randell | 104,178 | <p>For any integer n, n =
0, 1, 2 or 3 (mod 4). So $n^{2}$ = 0 or 1 (mod 4). Then for
any integers a and b, $a^{2} + b^{2}$ = 0, 1 or 2 (mod 4). This means the sum of two squares can
only be in the form 4k, 4k+1 or 4k+2, but never 4k+3. Thus no integer of the form 4k+3
is the sum of two squares.</p>
|
1,665,833 | <p>Given that A $\in$ M $_{mxn}$ (<strong>R</strong>). Assume that {$v_1$...$v_n$} is a basis for $R^n$ such that {$v_1$...$v_k$} is a basis for Null(A). </p>
<p>How would I prove that {A$v_{k+1}$...A$v_n$} spans Col(A)?</p>
| Sam | 630,614 | <p>@henry and @user104111 I will share the same answer as the <a href="https://stats.stackexchange.com/questions/1624/based-on-z-score-is-it-possible-to-compute-confidence-without-looking-at-a-z-ta/389082#389082">thread here</a> because I understand what you're saying. You don't want a software or tool to build a table... |
3,565,015 | <p>I generated this polynomial after playing around with the golden ratio. I first observed that (using various properties of <span class="math-container">$\phi$</span>), <span class="math-container">$\phi^3+\phi^{-3}=4\phi-2$</span>. This equation has no significance at all, I just mention it because the whole problem... | Toby Mak | 285,313 | <p>Your original method is tedious but it can be done.</p>
<p>You can show that <span class="math-container">$(x^3+Ax^2+Bx+C)(x^2+Dx+E)$</span> is equal to:</p>
<p><span class="math-container">$$x^5+(D+A)x^4+(1+AD+B)x^3 + (AE+BD+C)x^2 + (BE+CD) + CE$$</span></p>
<p>so <span class="math-container">$A+D = 1, B+AD+1 = ... |
1,617,698 | <p>While I was trying to find the formula of something by my own means I came across this sum which I need to solve, however I don't know if there is a solution for it, maybe it doesn't mean anything and I made a mistake. However if there's an equation which can replace this sum I will appreciate it a lot if you show m... | N. S. | 9,176 | <p>Multiply your sum by $\sin\left(\frac{\pi}{4n}\right)$ and use the formula
$$\sin\left(\frac{i \pi}{2n}\right)\sin\left(\frac{\pi}{4n}\right)=\frac{1}{2}\left[\cos\left(\frac{i \pi}{2n}-\frac{\pi}{4n}\right)-\cos\left(\frac{i \pi}{2n}+\frac{\pi}{4n}\right)\right]$$</p>
|
1,562,010 | <p>Let $f(x) \in \mathbb Z[x]$ be an irreducible monic polynomial such that $|f(0)|$ is not a perfect square . Then is $f(x^2)$ also irreducible in $\mathbb Z[x]$ ?</p>
<p>( It is supposed to have an elementary solution , without using any field-extension etc. )</p>
| An Hoa | 41,874 | <p>Let $\alpha$ be a root of $f$. Let $K = \mathbb{Q}(\alpha)$ and $L = \mathbb{Q}(\sqrt{\alpha})$. We have tower of extension
$$L \supseteq K \supseteq \mathbb{Q}$$
By irreducibility of $f$, we know that $[K : \mathbb{Q}] = \deg f$ and obviously $[L : K] \leq 2$. If $[L : K] = 2$ then $[L : \mathbb{Q}] = 2 \deg f$ so ... |
313,030 | <p>I often find myself writing a definition which requires a proof. You are defining a term and, contextually, need to prove that the definition makes sense. </p>
<p>How can you express that? What about a definition with a proof?</p>
<p>Sometime one can write the definition and then the theorem. But often happens tha... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}$</span>
I think the notion of "well-defined" may not always be well defined and should perhaps be avoided. In your example, it may be unclear what exactly is being proved. </p>
<p>I also think it is all right to introduce notions ... |
4,608,805 | <p>Suppose that I have a class of 35 students whose average grade is 90. I randomly picked 5 students whose average came out to be 85. Assume their grades are i.i.d and of normal <span class="math-container">$N(\mu, \sigma^2)$</span>. From the example I have seen, <span class="math-container">$\mu$</span> is usually ca... | José Gabriel Astaíza-Gómez | 361,031 | <p>We do hypothesis testing when the sample size excedes the number of our observations. That is the meaning of <span class="math-container">$n\rightarrow \infty$</span> in, e.g. <span class="math-container">$\sqrt{n}(\bar{X}_n - \mu)\rightarrow_d N(0,\sigma^2)$</span> where <span class="math-container">$\bar{X}_n=\dfr... |
1,002,777 | <p>I want to convert this polynomoial to partial fraction.</p>
<p>$$
\frac{x^2-2x+2}{x(x-1)}
$$</p>
<p>I proceed like this:
$$
\frac{x^2-2x+2}{x(x-1)} = \frac{A}{x} + \frac{B}{x-1}
$$
Solving,
$$
A=-2,B=1
$$
But this does not make sense. What is going wrong?</p>
| Module | 114,669 | <p>$$\frac{x^2-2x+2}{x(x-1)}=1+\frac Ax+\frac B{x-1}$$</p>
<p>Now what you have to do to solve for A and B is to multiply both sides of your equation by $x(x-1)$, and that should give you something like.</p>
<p>$$\frac{x^2-2x+2}{1}=x^2-x +A(x-1)+Bx=x(A+B)-A=-x+2$$ From here on it's pretty easy to solve for both A and... |
2,136,024 | <p>I am having problems with this linear algebra proof:</p>
<blockquote>
<p>Let $ A $ be a square matrix of order $ n $ that has exactly one nonzero entry
in each row and each column. Let $ D $ be the diagonal matrix whose $ i^{th} $
diagonal entry is the nonzero entry in the $i^{th}$ row of $A$</p>
<p>For ... | Fimpellizzeri | 173,410 | <p><strong>Hint:</strong> For $(1)$, find a matrix $P(i,j)$ that swaps columns $i$ and $j$. Your permutation matrix will be a product of $P(i,j)$'s.</p>
<p>For $(2)$, try to convince yourself that when $D$ is diagonal, $D^{T}=D$. It's not too hard!</p>
|
2,165,213 | <p><strong>The Problem</strong></p>
<p>Let $V=k^3$ for some field $k$. Let $W$ be the subspace spanned by $(1,0,0)$ and let $U$ be the subspace spanned by $(1,1,0)$ and $(0,1,1)$. Show that $V= W \oplus U$. Explain your argument in detail.</p>
<hr>
<p><strong>What I Know</strong></p>
<ol>
<li><p>I know that a fi... | Kenny Wong | 301,805 | <p>I presume you're trying to identify $\mathbb R[x]/(x-5)$?</p>
<p>$\phi$ is a homomorphism of rings. As you mentioned, ${\rm ker} \phi = (x-5)$. It shouldn't be hard to see that ${\rm im} \phi = \mathbb R$. Now use the fact that if $\phi : R \to S$ is a ring homomorphism, then $R/{\rm ker } \phi \cong {\rm im} \phi$... |
3,060,742 | <p><span class="math-container">$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = 1.644934$</span> or <span class="math-container">$\frac{\pi^2}{6}$</span></p>
<p>What if we take every 3rd term and add them up? </p>
<p>A = <span class="math-container">$ \frac{1}{3^2} + \fra... | Jack D'Aurizio | 44,121 | <p>As a complement to Mark's answer, </p>
<p><span class="math-container">$$\sum_{n\geq 0}\frac{1}{(3n+1)^2}=-\int_{0}^{1}\sum_{n\geq 0} x^{3n}\log(x)\,dx=\int_{0}^{1}\frac{-\log x}{1-x^3}\,dx $$</span>
(and similarly <span class="math-container">$\sum_{n\geq 0}\frac{1}{(3n+2)^2}$</span>) can be expressed in terms of ... |
1,433,980 | <p>so the problem I m having deals with conditional probability. I am given so much information and don't know what to do with what. Here is the problem:</p>
<p>"A study investigated whether men and women place more importance on a mate's ability to express his/her feelings or on a mate's ability to make a good livin... | MegaboofMD | 269,606 | <p>$P(M' and F)=P(F)*P(M' given F)$ so $P(F)=0.71$ and $P(M'givenF)= 0.36/0.71$</p>
<p>Thus,
$P(M' and F)=0.71*0.36/0.71=0.36$</p>
<p>I edited, initially had the wrong value for M'givenF.</p>
|
1,433,980 | <p>so the problem I m having deals with conditional probability. I am given so much information and don't know what to do with what. Here is the problem:</p>
<p>"A study investigated whether men and women place more importance on a mate's ability to express his/her feelings or on a mate's ability to make a good livin... | M.K. | 124,485 | <p>It's sometimes easier to break this down into a table as follows:</p>
<p><a href="https://i.stack.imgur.com/lGPLF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lGPLF.png" alt="enter image description here"></a></p>
<p>As you can see the percentage that the are not male and consider feelings im... |
30,718 | <p>As we all know, questions lacking context are strongly discouraged on this site. This includes mainly "homework questions" that look a bit like:</p>
<blockquote>
<p>Prove that <span class="math-container">$\lim_{x\to0}x^2=0$</span> using <span class="math-container">$\epsilon$</span>-<span class="math-container">... | hardmath | 3,111 | <p>I often claim to have a <em>de minimus</em> requirement for context, in which even a slight indication of effort or interest in a problem suffices.</p>
<p>It is with the intent of offering an example that I feel <em>narrowly misses</em> this qualification that I post <a href="https://math.stackexchange.com/question... |
603,986 | <p>Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$</p>
<p>Any ideas how to prove it?</p>
| DBFdalwayse | 88,790 | <p>Consider counting the total number of monic polynomials with roots, for fixed degree : say for degree $3$ , you can have $(x-r_1)(x-r_2)(x-r_3)$ (notice that you need to
consider that different permutations of $r_1,r_2,r_3$ will generate the same polynomial). And the overall number of monic polynomials will be $x^3+... |
8,997 | <p>I have a set of data points in two columns in a spreadsheet (OpenOffice Calc):</p>
<p><img src="https://i.stack.imgur.com/IPNz9.png" alt="enter image description here"></p>
<p>I would like to get these into <em>Mathematica</em> in this format:</p>
<pre><code>data = {{1, 3.3}, {2, 5.6}, {3, 7.1}, {4, 11.4}, {5, 14... | LIU Qi | 1,902 | <p>Notice that you can use <code>Import["file.xlsx", {"Data",k,m,n}]</code> to import data at cell {m,n} on k-th sheet in the file.
To import a range of data, simply replace number <code>m</code> and <code>n</code> by a range list, e.g.</p>
<p><code>Import["file.xlsx", {"Data",1,Table[i,{i,3,8}],{1,2}}]</code> </p>
... |
481,167 | <p>Let $V$ be a $\mathbb{R}$-vector space.
Let $\Phi:V^n\to\mathbb{R}$ a multilinear symmetric operator.</p>
<p>Is it true and how do we show that for any $v_1,\ldots,v_n\in V$, we have:</p>
<p>$$\Phi[v_1,\ldots,v_n]=\frac{1}{n!} \sum_{k=1}^n \sum_{1\leq j_1<\cdots<j_k\leq n} (-1)^{n-k}\phi (v_{j_1}+\cdots+v_{j... | Ewan Delanoy | 15,381 | <p>SKETCH OF THE PROOF : Your big sums are always sums of (sums of sums of) terms of the form <span class="math-container">$\Phi(v_{k_1},v_{k_2},\ldots ,v_{k_n})$</span> for some tuples of indices <span class="math-container">$(k_1,k_2, \ldots ,k_n)$</span>. Thanks to the symmetry of <span class="math-container">$\Phi$... |
737,835 | <p>Why is $[0,1]$ not homeomorphic to $[0,1]^2$? It seems that the easiest way to show this is to find some inconsistency between the open set structures of the two. It is clear that the two share the same cardinality. Both are compact. Both are normal since they are metric spaces. However, where to find the open set s... | Siminore | 29,672 | <p>If you remove an <em>inner</em> point from $[0,1]$, the resulting space is disconnected. This is clearly false if you remove <em>any</em> point from $[0,1]\times [0,1]$.</p>
<p>This is a rather rough approach. A much more general one follows from Dimension Theory, or from the <a href="http://en.wikipedia.org/wiki/I... |
737,835 | <p>Why is $[0,1]$ not homeomorphic to $[0,1]^2$? It seems that the easiest way to show this is to find some inconsistency between the open set structures of the two. It is clear that the two share the same cardinality. Both are compact. Both are normal since they are metric spaces. However, where to find the open set s... | Stefano | 108,586 | <p>If $A$ is omeomorphic to $B$ through $f$, then $A \setminus \lbrace a \rbrace$ is omeomorphic to $B \setminus \lbrace f \left( a \right) \rbrace$ through $f$. Then pick $a= \frac{1}{2}$. $\left[ 0,1\right] \setminus \lbrace \frac{1}{2} \rbrace$ is disconnected, while $\left[ 0,1\right] \times \left[ 0,1\right] \setm... |
229,606 | <p>I need little help in proving the following result :</p>
<p>Consider the ring $R:=\mathbb{F}_q[X]/(X^n-1)$, where $\mathbb{F}_q$ is a finite field of cardinality $q$ and $n\in\mathbb{N}$. Then any ideal $I$ of $R$ is principle and can be written as $I=(g(X))$, such that $g(X)|(X^n-1)$.</p>
| Berci | 41,488 | <p><strong>Hint:</strong> Think about $R=\Bbb F_q[X]/(X^n-1)$ as the ring of polynomials of degree $<n$, and multiplication is '<em>modulo $(X^n-1)$</em>', meaning that $X^n=1$ is <em>the rule</em> to use in $R$.</p>
|
3,472,151 | <p>I find two main sources on how to compute the half-derivative of <span class="math-container">$e^x$</span>. Both make sense to me, but they give different answers.</p>
<p>Firstly, people argue, that
<span class="math-container">$$\begin{align}
\frac{\mathrm{d}}{\mathrm{d} x} e^{k x} &= k e^{k x} \\[4pt]
\frac{\... | Ninad Munshi | 698,724 | <p>Those two formulae are not as different as you think. Finish the computation:</p>
<p><span class="math-container">$$\frac{1}{\Gamma\left(\frac{1}{2}\right)}\frac{d}{dx}\left(e^{kx}\int_0^x u^{-\frac{1}{2}}e^{-ku}du\right) = \frac{1}{\Gamma\left(\frac{1}{2}\right)}\left(ke^{kx}\int_0^x u^{-\frac{1}{2}}e^{-ku}du + \f... |
2,277,115 | <p>I'm asking for examples of interesting categories in which there exist non-isomorphic objects $X$ and $Y$, a split monomorphism $f : X \to Y$, and a split epimorphism $g : X \to Y$. Spelled out, there should exist maps $f : X \leftrightarrow Y : f'$ such that $f'f = \mathrm{id}_{X}$ and maps $g : X \leftrightarrow Y... | Niels J. Diepeveen | 3,457 | <p>In topological terms, you are asking for two non-homeomorphic spaces, each
of which is homeomorphic to a retract of the other.
You can easily find examples of such spaces by stringing together an infinite chain of copies of two non-homeomorphic spaces. Taking a circle and a segment as an example, we get subspaces o... |
1,285,014 | <p>Let $R,S$ be commutative rings with identity.</p>
<p>Proving that $X \sqcup Y$ is an affine scheme is the same as proving that $Spec(R) \sqcup Spec(S) = Spec(R \times S)$.</p>
<p>I proved that if $R,S$ are rings, then the ideals of $R \times S$ are exactly of the form $P \times Q$, where $P$ is an ideal of $R$ and... | Mauro ALLEGRANZA | 108,274 | <p>Of course, we can use also the tableau method, obtaining the same result produced by the use of truth-table.</p>
<p>We have to apply the tableau to the original formula, checking its <em>satisfiability</em>.</p>
<p>Each open path defines a (set of) assignments to the <em>sentential variables</em> satisfying the fo... |
4,462,081 | <p>I actually already have the solution to the following expression, yet it takes a long time for me to decipher the first operation provided in the answer. I understand all of the following except how to convert <span class="math-container">$\left(1+e^{i\theta \ }\right)^n=\left(e^{\frac{i\theta }{2}}\left(e^{\frac{-i... | Mark Bennet | 2,906 | <p>If you had <span class="math-container">$1+x^2$</span> you could write it as <span class="math-container">$x\left(\dfrac 1x+x\right)$</span> if you wanted to.</p>
<p>This occurs occasionally with the trigonometric/complex exponential functions because, of course <span class="math-container">$e^{-ia}=\dfrac 1{e^{ia}}... |
612,827 | <p>I'm self studying with Munkres's topology and he uses the uniform metric several times throughout the text. When I looked in Wikipedia I found that there's this concept of a <a href="http://en.wikipedia.org/wiki/Uniform_space" rel="nofollow">uniform space</a>.</p>
<p>I'd like to know what are it's uses (outside poi... | arsmath | 4,880 | <p>Uniform spaces are right on the boundary of formalisms that are worth knowing. Topological groups are definitely worth knowing, and are central to analysis and its applications. The topology of a topological group is determined by the system of neighborhoods of the identity. They also have a natural notion of uni... |
2,541,044 | <p>I read this argument on the internet about how the solution to the sleeping beauty problem is $\frac{1}{3}$:</p>
<p>All these events are equally likely in the experiment : </p>
<ol>
<li><p>Coin landed Heads, it's Monday and Beauty is awake</p></li>
<li><p>Coin landed Heads, it's Tuesday and Beauty is asleep</p></l... | spaceisdarkgreen | 397,125 | <p>To add to Qiaochu's answer, the Monty Hall problem's solution depends on our understanding of the exact process the game follows. The assumption is that if you are standing in front of a door with the goat, Monty will always open the other door with the goat. What if instead the procedure that Monty will open either... |
1,666,615 | <blockquote>
<p>For the series $\sum_{k=1}^{\infty}a_k$, suppose that there is a number $r$ with $0\leq r<1$ and a natural number $N$ such that $$|a_k|^{1/k}<r\qquad\text{for all indices $k\geq N$}$$ Prove that $\sum_{k=1}^{\infty}a_k$ converges absolutely.</p>
</blockquote>
<p>Proof:</p>
<p>For a given $r\in... | ldiaz | 709,059 | <p>Since you are trying to prove that </p>
<p><span class="math-container">$$\sum_{k=0}^\infty a_k$$</span> converges absolutely, consider the sum <span class="math-container">$$|\sum_{k=0}^\infty a_k|$$</span> = <span class="math-container">$$|\sum_{k=0}^N a_k + \sum_{k=N}^\infty a_k|$$</span> <span class="math-conta... |
947,730 | <p>I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it.</p>
<p>Thanks :)</p>
<p>Find a polynomial that passes through the points (-2,-1), (-1,7), (2,-5), (3,-1). Present the answer in standard form.</p>
<p>What I've tried:</p>
<p><img src="https://i.sta... | André Nicolas | 6,312 | <p><strong>Hints:</strong> </p>
<p>Way 1: Consider the polynomial
$$\small A(x+1)(x-2)(x-3)+B(x+2)(x-2)(x-3)+C(x+2)(x+1)(x-3)+D(x+2)(x+1)(x-2).$$
You can find constants $A,B,C,D$ such that the above polynomial will do the job. For example, to make the polynomial be equal to $-1$ at $-2$, all we need to do is to make $... |
339,880 | <p>I'm interested in examples where the sum of a set with itself is a substantially bigger set with nice structure. Here are two examples:</p>
<ul>
<li><strong>Cantor set</strong>: Let <span class="math-container">$C$</span> denote the ternary Cantor set on the interval <span class="math-container">$[0,1]$</span>. The... | José Hdz. Stgo. | 1,593 | <p>I know you asked for examples of the "thin + thin = nice and thick" phenomenon but, since <a href="https://www.usatoday.com/story/news/nation/2019/09/10/palindrome-week-last-one-century/2273558001/" rel="noreferrer"><strong>Palindrome Week</strong></a> is all the rage these days, I can't avoid mentioning the followi... |
3,552,219 | <p>I come across an explanation of recursion complexity. This screenshot is in question:</p>
<p><a href="https://i.stack.imgur.com/ySKdo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ySKdo.png" alt="a"></a></p>
<p>How do you get this?</p>
<pre><code>T(n) = 3T(n/4) + n
</code></pre>
<p>The <spa... | marty cohen | 13,079 | <p><span class="math-container">$\begin{array}\\
T(n)
&=3T(n/4)+1\\
&=3(3T(n/16)+1)+1\\
&=9T(n/16)+4\\
&=9(3T(n/64)+1)+4\\
&=27T(n/64)+13\\
&.....\\
&=3^kT(n/4^k)+\sum_{j=0}^{k-1}3^j
\qquad\text{conjecture}\\
&=3^k(3T(n/4^{k+1})+1)+\sum_{j=0}^{k-1}3^j
\qquad\text{induction step}\\
&=... |
3,375,375 | <p>I noticed this issue was throwing off a more sophisticated problem I'm working on. When computing the indefinite integral </p>
<p><span class="math-container">$$ I(x) = \int \frac{dx}{1-x} = \log | 1-x | + C,$$</span></p>
<p>I realized I could equivalently write</p>
<p><span class="math-container">$$ I(x) = - \... | José Carlos Santos | 446,262 | <p>The first answer is wrong. Note that<span class="math-container">$$\int\frac{\mathrm dx}{a+bx}=\frac1b\log\lvert a+bx\rvert+C.$$</span>In particular<span class="math-container">$$\int\frac{\mathrm dx}{1-x}=-\log\lvert1-x\rvert+C.$$</span></p>
|
3,375,375 | <p>I noticed this issue was throwing off a more sophisticated problem I'm working on. When computing the indefinite integral </p>
<p><span class="math-container">$$ I(x) = \int \frac{dx}{1-x} = \log | 1-x | + C,$$</span></p>
<p>I realized I could equivalently write</p>
<p><span class="math-container">$$ I(x) = - \... | YiFan | 496,634 | <p>You forgot to use the chain rule when doing the first integral.
<span class="math-container">$$\int\frac{dx}{1-x}=-\log|1-x|+C,$$</span>
which is the same as the second one you gave.</p>
|
45,441 | <p>There is a method of constructing representations of classical Lie algebras via Gelfand-Tsetlin bases. It has also been applied to Symmetric groups by Vershik and Okounkov. Does anybody know of any application of the method to complex representations of $GL_n(\mathbb F_q)$? Or, at least, any results in this directio... | Matt Davis | 10,738 | <p>Not an answer either, but in response to Jim - Schur-Weyl duality doesn't always apply over finite fields. See <a href="http://www.ams.org/mathscinet-getitem?mr=2563588" rel="nofollow">http://www.ams.org/mathscinet-getitem?mr=2563588</a> for one result and some discussion of the related issues. </p>
|
1,281,507 | <blockquote>
<p>$$x*y = 3xy - 3x - 3y + 4$$</p>
<p>We know that $*$ is associative and has neutral element, $e$.</p>
<p>Find $$\frac{1}{1017}*\frac{2}{1017}*\cdots *\frac{2014}{1017}.$$</p>
</blockquote>
<p>I did find that $e=\frac{4}{3}$, and, indeed, $x*y = 3(x-1)(y-1)+1$. Also,it is easy to check that t... | Erick Wong | 30,402 | <p>One easy way to manufacture non-obvious associative/commutative operations is to take a known associative operation and <em>conjugate</em> it with an invertible function $f : \mathbb R \to \mathbb R$. In this case, taking $f(x) = 3(x-1)$ (with inverse function $f^{-1}(x) = 1 + \tfrac x3$), we can see that</p>
<p>$... |
3,954,410 | <p>I am solving exercises from Loring Tu.</p>
<p>Show that if <span class="math-container">$L : V \rightarrow V$</span> is a linear operator on a vector space V of dimension n, then the pullback <span class="math-container">$L^{\wedge} : A_n(V) \rightarrow A_n(V)$</span> is multiplication by determinant of L.</p>
<p>At... | ArsenBerk | 505,611 | <p><strong>HINT:</strong> Instead of induction here, we can use Fermat's Little Theorem as it says <span class="math-container">$n^7 \equiv n \mod 7$</span> for every positive integer <span class="math-container">$n$</span>.</p>
|
132,238 | <p>I'm trying to solve a maximization problem that apparently is too complicated (it's a convex function) and NMaximize just runs endlessly.</p>
<p>I'd like to have an approximate result, though. How can I tell <code>NMaximize</code> to just give up after $n$ seconds and give me the best it has found so far?</p>
| Michael E2 | 4,999 | <p>Each of the four methods in <code>NMinimize</code> has built-in hooks you can use to get the current values through <code>StepMonitor</code>. There are some great advantages to this approach over ones that hijack the user's objective function and only minor drawbacks:</p>
<ul>
<li>The regular <code>NMinimize</code... |
2,226,337 | <p>How many multiples of $5$ are greater than $60,000,$ and can be made from the digits:
$$0, 1, 2, 3, 4, 5, 6$$ </p>
<p>if <strong>all</strong> digits have to be used and each can only be used once with no repeats.</p>
<p>Am I looking at this in the wrong way too simplistically or is it simply $2 \times 6!$</... | Bernard | 202,857 | <p><strong>Hints:</strong></p>
<p>How do you know a number is divisible by $5$?</p>
<p>The notation of an integer can't begin with $0$.</p>
<p><em>Some details:</em></p>
<p>A multiple of $5$ ends in a $0$ or a $5$. So two cases:</p>
<ul>
<li>If it ends in a $0$, you have to consider the $6!$ permutations of the d... |
850,390 | <p>Let $f(x)$ be differentiable function from $\mathbb R$ to $\mathbb R$, If $f(x)$ is even, then $f'(0)=0$. Is it always true?</p>
| David K | 139,123 | <p>Trick question. Remember that "continuous" does not imply "differentiable".</p>
<p>If the function is <em>differentiable</em> at $0$, refer to the answer by @Brandon.</p>
<p><strong>Update:</strong> The question has been edited so that it says "differentiable" rather than "continuous". The answer above applied to ... |
104,626 | <p>I encountered the following differential equation when I tried to derive the equation of motion of a simple pendulum:</p>
<p>$\frac{\mathrm d^2 \theta}{\mathrm dt^2}+g\sin\theta=0$</p>
<p>How can I solve the above equation?</p>
| Peđa | 15,660 | <p>Use substitution : $\theta' =v$ ,therefore we have that :</p>
<p>$$\theta''=\frac{dv}{dt}\cdot \frac{dt}{d\theta}\cdot \frac{d\theta}{dt} \Rightarrow \theta''=\frac{dv}{d\theta}\cdot v \Rightarrow \theta''=v'\cdot v$$</p>
<p>where $v$ is function in terms of variable $\theta$ .So di... |
40,241 | <p>Let $N$ be a prime number. Let $J(N)$ be the jacobian of $X_\mu(N)$, the moduli space of elliptic curves with $E[N]$ symplectically isomorphic to $Z/NZ \times \mu_N$. Over complex numbers we get that J(N) is isogeneous to product of bunch of irreducible Abelian varieties. Is there a way of describing these Abelian v... | William Stein | 8,441 | <p>Ernst Kani was very interested in this and related questions around 2000. I remember implementing an algorithm for him in around 2000 when I visited Essen to compute a basis of $S_2(\Gamma(p))$ in terms of $\Gamma_1(p^2)$. I'm sure Kani knows the decomposition of $J(N)$ for small $N$, since I vaguely remember tal... |
2,403,404 | <p>I would be thankful if anyone can answer my question. This is a very basic question. Let's say we wish to minimise the quantity</p>
<p>$$\hat{h}= \|h-h_i\|+\lambda\|h-u\|,$$</p>
<p>where:</p>
<p>$$h=[13,17,20, 17, 20, 14, 17, 18, 16, 15, 15, 12, 19, 13, 17, 13]^\top,\\ h_i=[18, 17, 14, 13, 17, ... | pisco | 257,943 | <p>The locus of $C$ is two lines parallel to $AB$, one line lie above $AB$ by 4 cm, one line lie below $AB$ by 4 cm. We denote the upper line by $l$. We only need to consider the case when $C$ lies on $l$.</p>
<p>Let $B'$ be the reflection of $B$ along $l$, then $CA+CB = CA+CB'$, so $CA+CB$ is minimized when $C,A,B'$... |
2,766,879 | <p>Show that there are no primitive pythagorean triple $(x,y,z)$ with $z\equiv -1 \pmod 4$. </p>
<p>I once have proven that, for all integers $a,b$, we have that $a^2 + b^2$ is congruent to $0$, or $1$, or $2$ modulo $4$. I feel like it is enough to conclude it by considering $a=x$, $b=y$ and $\gcd(x,y)=1$. But I am n... | Robert Z | 299,698 | <p>Note that the given inequality is
$$f(x)-f(1/x)=\frac{(x+1)^3}{x(x-1)}\geq 8,$$
that is for $x>1$,
$$h(x)=(x+1)^3-8x(x-1)\geq 0$$
which holds because $h(1)=8$ and $h$ is strictly increasing since $$h'(x)=3(x+1)^2-16x+8=3x^2-10x+11>0 \quad (\Delta=10^2-12\cdot 11<0).$$</p>
|
4,090,970 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be independent exponential random random variables with common parameter <span class="math-container">$\lambda$</span> and let <span class="math-container">$Z = X + Y$</span>. Find <span class="math-container">$f_Z(z)$</span>.</p... | herb steinberg | 501,262 | <p>For <span class="math-container">$x\gt z$</span> the integrand <span class="math-container">$=0$</span> since <span class="math-container">$f_Y(y)=0$</span> for <span class="math-container">$y\lt 0$</span>.</p>
<p><span class="math-container">$f_Z(z)=\lambda^2 e^{-\lambda z}\int\limits_0^zdx=z\lambda^2 e^{-\lambda z... |
2,828,472 | <p>This question is regarding property of little o notation given in Apostol Calculus. The property is given on page 288 and stated as:</p>
<blockquote>
<p>Theorem 7.8 (c) As $x\to a$ we have $f(x)\cdot o (g(x)) = o(f(x)g(x))$.</p>
</blockquote>
<p>Here say $h(x) = o(g(x))$ then we have $f(x) \lim_{x\to a} \frac{h(... | hamam_Abdallah | 369,188 | <p>$$o(g(x))=g(x)\epsilon(x)$$ with $$\lim_{x\to a}\epsilon(x)=0$$</p>
<p>then</p>
<p>$$f(x)o(g(x))=\Bigl(f(x)g(x)\Bigr)\epsilon(x)=o(f(x)g(x))$$</p>
|
697,402 | <p>I have this limit:</p>
<p>$$ \lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2\sqrt{x^2+1}} $$ I tried to solve it by this:</p>
<p>$$ \lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2\sqrt{x^2+1}} = \lim_{x\to\infty}\frac{\frac{x^3}{x^3}+\frac{\cos x}{x^3}+\frac{e^{-2x}}{x^3}}{\frac{x^2\sqrt{x^2+1}}{x^3}} = \frac{0+0+0}... | Christian Blatter | 1,303 | <p>A hint: When $x\to\infty$ then numerator and denominator both have order of magnitude $x^3$. Therefore extract a factor $x^3\ne0$ on top and bottom, in the hope that now the numerator and denominator both have a finite limit when $x\to\infty$.</p>
|
66,000 | <p>In 2008 I wrote a group theory package. I've recently started using it again, and I found that one (at least) of my functions is broken in Mathematica 10. The problem is complicated to describe, but the essence of it occurs in this line:</p>
<pre><code>l = Split[l, Union[#1] == Union[#2] &]
</code></pre>
<p>He... | Simon Woods | 862 | <p>Another possible workaround is to wrap <code>Dispatch</code> with a memoized function, so that both expressions <code>a</code> and <code>b</code> contain references to the same internal dispatch table.</p>
<p>i.e. define</p>
<pre><code>mem : disp[x_] := mem = Dispatch[x]
</code></pre>
<p>then use <code>disp</code... |
2,952,392 | <p>Revisit the following discussion: </p>
<p><a href="https://math.stackexchange.com/questions/843909/prove-that-the-inverse-image-of-an-open-set-is-open">Prove that the inverse image of an open set is open</a></p>
<p>Obviously, the above discussion is based on Euclidean space (which is also a metric space, so the pr... | Shweta Aggrawal | 581,242 | <p>This statement is false in general. Consider the following polynomials in <span class="math-container">$\mathbb{C}[X,Y]$</span></p>
<p>The polynomials <span class="math-container">$ XY $</span> and <span class="math-container">$ X + Y $</span> have infinitely many zeros in <span class="math-container">$ \mathbb{C}^... |
2,468,329 | <p>Let F be a field and choose an element $u \in F$. Consider the function $\epsilon_u:F[x]\rightarrow F$ given by $$\epsilon_u(a_nx^n+...+a_0)=a_nu^n+...+a_0$$</p>
<p>I am asked to show that this is surjective but not injective, as well as finding its kernel.</p>
<p>My idea is that this function will just send every... | Bernard | 202,857 | <p>For a polynomial ring over a commutative ring $A$, you have the equivalence
$$\forall u\in A\;\forall f\in A [X],\quad(f(u)=0\iff f\;\text{is divisible by}\; X-u).$$</p>
<p>Indeed, let's divide $f(X)$ by $X-u$. We get
$$f(X)=q(X)(X-u)+r,\qquad r\in A. $$
Clearly, $\;f(u)=0\iff r=0$.</p>
|
697 | <p><a href="https://mathoverflow.net/questions/36307/why-cant-i-post-a-question-on-math-stackexchange-com">This question</a> was posted on MO about not being able to post on math.SE. While MO wasn't the right place for the question, I have to wonder what is. New users who are experiencing difficulty using math.SE can... | Community | -1 | <p>We need to allow people to post on meta with low reputation.</p>
|
1,113,415 | <p>Is there a website or a book with a calculus theorems list?
Or what are the ways remembering calculus theorems list?</p>
| A.D | 37,459 | <p>You know $\frac{1}{0}$ is undefined. So, $\frac{1}{0} - 0.5$ is also undefined. Since $\text{undefined} - 0.5$, is not possible to define. But main interesting thing is to know why $\frac{1}0$ is undefined. For that purpose see this link <a href="https://math.stackexchange.com/questions/26445/division-by-0">here</a>... |
3,005,208 | <p>I want to solve this polynomial analytically. I know the useful answer is between 0 and 1. Is there any way I can write the answer based on a, b, and c?
<span class="math-container">$$
6\cdot a \cdot x^4 + 2 \cdot b \cdot x^3-b \cdot c=0
$$</span>
Also, an approximate answer is acceptable, for example, an answer wit... | theREALyumdub | 175,429 | <p>Since you say an approximate answer is alright, up to 2% tolerance, it might be a good idea to use a numerical approximation like Newton's method.</p>
<p>The derivative works out to be
<span class="math-container">$$ f'(x) = 24ax^3 + 6bx^2 = 6x^2 (4ax + b) $$</span></p>
<p>So you can take an <span class="math-cont... |
3,005,208 | <p>I want to solve this polynomial analytically. I know the useful answer is between 0 and 1. Is there any way I can write the answer based on a, b, and c?
<span class="math-container">$$
6\cdot a \cdot x^4 + 2 \cdot b \cdot x^3-b \cdot c=0
$$</span>
Also, an approximate answer is acceptable, for example, an answer wit... | G Cab | 317,234 | <p>The exact solution would turn into a "complicated" expression in <span class="math-container">$a,b,c$</span>. </p>
<p>If you are looking for an approximated solution, and you know that a real root is near to <span class="math-container">$1$</span> (and in fact it is for "normal" positive value of the parameters) , ... |
86,067 | <p>So I am having an issue using <code>NDSolve</code> and plotting the function. So I have two different <code>NDSolve</code> calls in my plotting function. (They are technically the same, just have different names; but that can be changed back if at all possible because I want them to be the same.) But the second one ... | george2079 | 2,079 | <p>This is tedious.. manually drawing the axes.</p>
<pre><code> GraphicsRow[{Histogram[data],
Show[{Histogram[data , PlotRangePadding -> Scaled[.2],
Axes -> False,
PlotRange -> {{-3, 3}, {0, 100}}],
Graphics[{Line[{Scaled[{.2, .15}], Scaled[{.8, .15}]}],
... |
86,067 | <p>So I am having an issue using <code>NDSolve</code> and plotting the function. So I have two different <code>NDSolve</code> calls in my plotting function. (They are technically the same, just have different names; but that can be changed back if at all possible because I want them to be the same.) But the second one ... | Virgil | 27,697 | <p>This can be done more-or-less easily with a combination of options for <code>AxesOrigin</code>, <code>PlotRange</code>, and <code>PlotRangePadding</code> and the <a href="http://library.wolfram.com/infocenter/MathSource/5599/" rel="noreferrer"><code>CustomTicks</code> package</a> (for easy outward-facing ticks).</p>... |
1,722,226 | <p>How many solutions are there to the inequality
$x_1 + x_2 + x_3 ≤ 11$,
where $x_1, x_2$ and $x_3$ are non-negative integers? [Hint: Introduce
an auxiliary variable $x_4$ such that $x_1 + x_2 + x_3$ +
$x_4$ = 11.]</p>
<p>Would my reasoning be correct if I let $x_4 = x_1 + x_2 + x_3, x_4 = 11$</p>
<p>Then proceeded... | ashleydc | 323,353 | <p>If you use a "matchstick" approach - using 11 x's and 3 |'s, count the number of x's to the left of each pipe to determine $x_1$, $x_2$, and $x_3$:</p>
<p>|||xxxxxxxxxxx => 0 + 0 + 0 </p>
<p>xxx|||xxxxxxxx => 3 + 0 + 0 </p>
<p>x|x|x|xxxxxxxx => 1 + 1 + 1 </p>
<p>xxxxxxxxxxx||| => 11 + 0 + 0 </p>
<p>In the abo... |
1,860,267 | <blockquote>
<p>Prove the convergence of</p>
<p><span class="math-container">$$\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$$</span></p>
</blockquote>
<p>First I thought the integral does not converge because</p>
<p><span class="math-container">$$\int\limits_1^{\infty} -\frac{1}{x} \,\mathrm{d}x \le \int\lim... | Maman | 167,819 | <p><strong><em>Hint</em></strong>: Awful and tricky but as the problem is near <span class="math-container">$+\infty$</span>, write <span class="math-container">$$\int \limits_{\frac{\pi}{2}}^{N\pi+\frac{\pi}{2}}\frac{\cos(x)}{x}\mathrm{d}x= \sum \limits_{k=1}^{N}\left(\int\limits_{k\pi-\frac{\pi}{2}}^{k\pi+\frac{\pi}{... |
3,204,950 | <p>This question arose from Physics, where the force on an object attached on a spring is proportional to the displacement to the equilibrium (that is, the rest position). Also, if the displacement to the equilibrium is positive, the force will be negative, as it tries to pull the object back (i.e. if you pull a string... | Adam Latosiński | 653,715 | <p>Mathematically they are equivalent, but physycist may want to differentiate between a force in the same direction as the displacement, and a force opposite to the displacement, as they lead to physically different behavior of the system. Thus the minus sign matters. </p>
|
1,904,767 | <p>I'm trying to understand regularization in machine learning. one way of regularization is adding a l1 norm to the error function. This is said to produce sparsity. But I can't understand.</p>
<p>sparsity is defined as "only few out of all parameters are non-zero". But if you look at the l1 norm equation, it is the ... | Bill Yan | 362,324 | <p>It took me an hour yesterday to finally understand this. I wrote a very detailed blog to explain it.</p>
<p><a href="https://medium.com/@shiyan/l1-norm-regularization-and-sparsity-explained-for-dummies-5b0e4be3938a#.nhy58osj5" rel="noreferrer">https://medium.com/@shiyan/l1-norm-regularization-and-sparsity-explained... |
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | WalkingRandomly | 4,786 | <p>From a cold start, I would have written it like this:</p>
<pre><code> findNarc = Compile[{{stop, _Integer}, {pow, _Integer}},
Do[
If[Total[IntegerDigits[n]^pow] == n, Sow[n]]
, {n, 1, stop}
]
, RuntimeOptions -> "Speed", CompilationTarget -> "C"];
</code></pre>
<p>However, it is slower than... |
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | wolfies | 898 | <p>Not an answer <em>per se</em>, but two clarifications (which are too long for the comment box):</p>
<p>1) The Wiki definition you have linked to for a narcissistic number is not really apt. The Wiki page is actually describing the definition for an Armstrong Number, also known as pluperfect digital invariants, or <... |
3,680,864 | <p>I'm trying to understand the relation between the following conditions. I will assume that <span class="math-container">$X$</span> is a Hausdorff topological space and <span class="math-container">$A \subset X$</span>.</p>
<ol>
<li><span class="math-container">$\overline{A}$</span> is compact;</li>
<li>Every net <s... | Eric Wofsey | 86,856 | <p>I find these things easier to think about in the language of filters. Using the usual correspondence between nets and filters, (2) is equivalent to saying that every filter on <span class="math-container">$X$</span> containing <span class="math-container">$A$</span> has an accumulation point in <span class="math-co... |
644,057 | <p>I am having trouble with this problem:</p>
<p>Let $a_n$ be sequence of positive terms with $$\frac{a_{n+1}}{a_n}\lt \frac{n^2}{(n+1)^2}.$$
Then is the series $\sum a_n$ convergent?</p>
<p>Thanks for any help.</p>
| Community | -1 | <p>According to the Daniel Fischer's comment let
$$v_n=\frac{1}{n^2}$$
so we have
$$\frac{a_{n+1}}{a_n}<\frac{v_{n+1}}{v_n}\iff \frac{a_{n+1}}{v_{n+1}}<\frac{a_{n}}{v_{n}}$$
hence the sequence $\left(\frac{a_{n}}{v_{n}}\right)$ is decreasing and then
$$ \frac{a_{n}}{v_{n}}<\frac{a_{1}}{v_{1}}=C\iff a_n<C v... |
2,342,537 | <p>Suppose $f:\mathbb R\rightarrow \mathbb R$ s.t. $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb R$ and $f$ is not continuous on $\mathbb R$. Prove that</p>
<p>(a).$f$ is not bounded below (or above) on any subinterval $(a,b)$ of $\mathbb R$.</p>
<p>(b). $f$ is not monotone.</p>
<p>On plugging $x=y=0$; $f(0)=f(0)+f(0)$... | Elle Najt | 54,092 | <p>We can reduce all statements to studying behavior near zero, by the observation that $f(x + a) = f(x) + f(a)$, so if $x$ had P near zero , then it has $P$ near $a$, where $P$ is continuous, or monotone, or not bounded below on any interval.</p>
<p>Here is a technique to prove continuity from monotonicity.</p>
<p>S... |
376,600 | <p>$$\lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^n}\,dx $$</p>
<p>Mathematica tells me the answer is 0, but how can I go about actually proving it mathematically?</p>
| Community | -1 | <p>Simply use this inequality</p>
<p>$$0\leq\int_0^\infty\frac{1}{(1+x^2)^n}\,dx\leq \int_0^\infty\frac{1}{1+nx^2}\,dx=\frac{1}{\sqrt{n}}\int_0^\infty\frac{dt}{1+t^2}=\frac{\pi}{2\sqrt{n}}$$</p>
|
149,872 | <p>How would I show that $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$? </p>
<p>Im not sure how to begin, does it involve using $\sinh z=\frac{e^{z}-e^{-z}}{2}$ and $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$?</p>
| DonAntonio | 31,254 | <p>$$z=x+iy\Longrightarrow \sin z=\frac{e^{iz}-e^{-iz}}{2i}=\frac{e^{-y+ix}-e^{y-ix}}{2i}=$$$$=\frac{e^{-y}(\cos x+i\sin x)-e^y(\cos x-i\sin x)}{2i}=\frac{1}{2i}\left[i\sin x\left(e^y+e^{-y}\right)-\cos x\left(e^y-e^{-y}\right)\right]=$$$$=\sin x\cosh y+i\cos x\sinh y\Longrightarrow ...$$</p>
|
1,236,600 | <p>A dose of $D$ milligrams of a drug is taken every 12 hours. Assume that the drug's half-life is such that every $12$ hours a fraction $r$, with $0<r<1$ of the drug remains in the blood. Let $d_1= D$ be the amount of the drug in the blood after first dose.
It follows that the amount of the drug in the blood aft... | Vectornaut | 16,063 | <p>You defined $d_n$ to be the amount of drug in the blood just after the $n$th dose is taken. Where does the expression $$d_n = D\sum_{k = 0}^{n - 1} r^k$$ come from? The $m$th dose contributed $D$ milligrams of drug at the moment it was taken. The $n$th dose is taken $n - m$ time units later; by then, the contributio... |
2,483,611 | <p>I believe the answer is 13 * $13\choose4$ * $48\choose9$.</p>
<p>There are $13\choose4$ to draw 4 of the same cards, and multiply by 13 for each possible rank (A, 2, 3, ..., K). Then there are $48\choose9$ to choose the remaining cards.</p>
<p>One thing I am not certain of, is whether this accounts for the possibi... | Nick Pavlov | 477,185 | <p><strong>EDIT</strong>: I realized that calling Macavity's answer "working by accident" is not fair. What it does is use the general prescription for solving an inequality involving one irrational expression, just doesn't explicitly state that. In general, $f(x) \geq \sqrt{g(x)}$ is equivalent to $$f(x) \geq 0 \;\; \... |
1,903,717 | <p>This is actually from an Analysis text but i feel its a set theory question.</p>
<p>Proposition for ever rational number $\epsilon > 0$ there exists a non-negative number x s.t $x^2 < 2 < (x+ \epsilon )^2 $</p>
<p>It provides a proof that im having trouble understanding.</p>
<p>Proof: let $ \epsilon >... | Francesco Alem. | 175,276 | <p>The statement is true.
here's a constructive proof for ya.</p>
<p>let $\epsilon\in \mathbf{Q}\,|\, \epsilon >0$</p>
<p>choose $k \in \mathbf{N}$ such that $k>2\epsilon$ so that $\frac{\epsilon}{k}<\frac{1}{2}$</p>
<p>choose $x \in [\sqrt 2 -\frac{\epsilon}{k};\,\,\sqrt2 - \frac{\epsilon}{2k}] \cap\mathbf... |
1,171,150 | <p>I am struggling to figure out $$\lim\limits_{n \to \infty} \sqrt[n]{n^2+1} .$$ I've tried manipulating the inside of the square root but I cannot seem to figure out a simplification that helps me find the limit.</p>
| abel | 9,252 | <p>let $$y = \sqrt[n]{n^2 + 1}$$ then $$\ln y = \dfrac{\ln(n^2 + 1)}{n} = \dfrac{\ln n^2}{n} + \cdots = \dfrac{2 \ln n} n + \cdots \to 0 \text{ as } n \to \infty.$$ therefore
$$\lim_{n \to \infty} y = \lim_{n \to \infty}\sqrt[n]{n^2 + 1} = 1$$</p>
|
2,581,135 | <blockquote>
<p>Find: $\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$</p>
</blockquote>
<p>Question from a book on preparation for math contests. All the tricks I know to solve this limit are not working. Wolfram Alpha struggled to find $1$ as the solution, but the solution process pre... | Peter Szilas | 408,605 | <p>Let $y=√x$. $\lim x \rightarrow \infty =\lim y \rightarrow \infty. $</p>
<p>Numerator: $y$</p>
<p>Denominator:</p>
<p>$\sqrt {y^2 +\sqrt{y^2+y}}= \sqrt{y^2+y\sqrt{1+1/y}}=$</p>
<p>$y\sqrt{1+(1/y)\sqrt{1+1/y}}.$</p>
<p>$\lim_{y \rightarrow \infty} \dfrac{y}{y \sqrt{1+(1/y) \sqrt{1+1/y}}}= $</p>
<p>$\lim_{y \rig... |
1,346,073 | <p>$$100\frac{dy^2}{dx^2} + y = 0$$</p>
<p>Is this worked out by using the auxillary equation such that:</p>
<p>$$100m^2 + 1 = 0$$</p>
<p>so $m = \pm i\sqrt{1/100}$ ?</p>
<p>So the general solution would be
$y(x) = A cos (1/10) + B sin(1/10)$?</p>
<p>I am not sure if I've gone about this the right way.</p>
| ccorn | 75,794 | <p>If <em>lowest</em> means the same as <em>minimum</em> to you, then yes.
Here is a sloppy outline, I will leave it to you to fill in the gaps.</p>
<p>For a real symmetric matrix $n\times n$ matrix $A$, consider the Rayleigh quotient
$$R_A(u) = \frac{u^\top A\,u}{u^\top u}\quad\text{for}\quad u\in\mathbb{R}^n\setminu... |
1,496,651 | <p>I'm trying to solve a control problem involving a pendulum, in which the equation of motion is:</p>
<p>$ml^2\frac{d^2 \theta}{d\theta^2} = \tau -mgl cos(\theta)$</p>
<p>I need to linearize $\tau -mgl cos(\theta)$ for $\tau = \tau_0 + \delta \tau, \theta = \theta_0 + \delta \theta$.</p>
<p>The answer to this quest... | obareey | 111,671 | <p>I think you forgot to mention that $(\tau_0, \theta_0)$ is a fixed point, i.e. $\tau_0 - mgl \cos(\theta_0) = 0$. Now, we can use Taylor series expansion around $(\tau_0, \theta_0)$ to obtain</p>
<p>$$ml^2 \frac{d^2 (\theta_0 + \delta \theta)}{dt^2} = \tau_0 + \delta \tau - mgl [\cos(\theta_0) - \sin(\theta_0) \del... |
1,496,651 | <p>I'm trying to solve a control problem involving a pendulum, in which the equation of motion is:</p>
<p>$ml^2\frac{d^2 \theta}{d\theta^2} = \tau -mgl cos(\theta)$</p>
<p>I need to linearize $\tau -mgl cos(\theta)$ for $\tau = \tau_0 + \delta \tau, \theta = \theta_0 + \delta \theta$.</p>
<p>The answer to this quest... | JMJ | 295,405 | <p>First: $\tau$ is not a state variable but a forcing term (torque in the equation), so the perturbation $\tau = \tau_0 + \delta\tau$ is unnecessary. </p>
<p>Second: In principle, you need not necessarily choose a fixed point equilibrium to linearize about (though it certainly helps for any practical implementation o... |
636,730 | <p>Let $G$ be a group of infinite order . Does there exist an element $x$ belonging to $G$ such that $x$ is not equal to $e$ and the order of $x$ is finite?</p>
| user1729 | 10,513 | <p>Easy example: Take the cross product of $\mathbb{Z}$ with your favourite non-trivial finite group $H$, $G=\mathbb{Z}\times H$. It in infinite as it contains $\mathbb{Z}$ as a subgroup, but it contains elements of finite order as it contains your favourite non-trivial finite group $H$ as a subgroup.</p>
<p>For examp... |
3,224,745 | <p>Naive evaluation of <span class="math-container">$\sqrt{a + x} - \sqrt{a}$</span> when <span class="math-container">$|a| >> |x|$</span> suffers from catastrophic cancellation and loss of significance.</p>
<p>WolframAlpha gives the Taylor series for <span class="math-container">$\sqrt{a+x}-\sqrt{a}$</span> as:... | Robert Israel | 8,508 | <p>The Taylor series is</p>
<p><span class="math-container">$$ \sqrt{a+x} - \sqrt{a} = \sum_{k=1}^\infty (-1)^{k+1} \frac{(2k)!}{(k!)^2(2k-1)} 4^{-k} a^{1/2-k} x^k$$</span>
If <span class="math-container">$|x/a| < 1$</span>, the absolute values of the terms decrease, since if <span class="math-container">$c_k = (2k... |
1,618,411 | <p>I'm learning the fundamentals of <em>discrete mathematics</em>, and I have been requested to solve this problem:</p>
<p>According to the set of natural numbers</p>
<p>$$
\mathbb{N} = {0, 1, 2, 3, ...}
$$</p>
<p>write a definition for the less than relation.</p>
<p>I wrote this:</p>
<p>$a < b$ if $a + 1 <... | miracle173 | 11,206 | <p>How can you decide if $3<5$ using your definition?
You can say $3<5$ if $4<6$ if $5<7$ and so on, but this sequence will never end.</p>
<p>It works the other way round: </p>
<ul>
<li>if $b \ne 0$: $0 \lt b$ </li>
<li>if $a \lt b$: $a+1 \lt b+1 $ </li>
</ul>
<p>$2 \ne 0$ , so $0 \lt 2$, therefore $1... |
2,613,410 | <blockquote>
<p>What is the value of <span class="math-container">$2x+3y$</span> if</p>
<p><span class="math-container">$x+y=6$</span> & <span class="math-container">$x^2+3xy+2y=60$</span> ?</p>
</blockquote>
<p>My trial:
from given conditions: substitute <span class="math-container">$y=6-x$</span> in <span class="... | Dr. Sonnhard Graubner | 175,066 | <p>we get
$$x^2+3x(6-x)+2(6-x)=60$$
simplifying we obtain
$$-2x^2+16x-48=0$$
or
$$x^2-8x+24=0$$
can you solve this?
you will get
$$x_1=4+2\sqrt{2}i$$
or
$$x_2=4-2\sqrt{2}i$$ and
and
$$y_1=2-2\sqrt{2}i$$
$$y_2=2+2\sqrt{2}i$$ so $x+y=6$</p>
|
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