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 |
|---|---|---|---|---|
1,419,756 | <p>Let $A$ be a nonempty set in the metric space $(X,d)$ and, for $\epsilon>0$, define</p>
<p>$$A_\epsilon = \{x\in X: d(x,A) < \epsilon\}$$.</p>
<p>Then I want to prove that $A_\epsilon$ is open in $X$.</p>
<p>So what I have tried so far is that, well I want to prove that a set is open so I take $x \in A_\eps... | Plutoro | 108,709 | <p>The issue is that you do not know that $d(x,A)<\varepsilon/2$. Try this: pick $x\in A$ with $d(x,A)=\varepsilon_2<\varepsilon$. Let $0<\varepsilon_1<\varepsilon-\varepsilon_2$. Now try the ball centered at $x$ of radius $\varepsilon_1$. </p>
|
1,419,756 | <p>Let $A$ be a nonempty set in the metric space $(X,d)$ and, for $\epsilon>0$, define</p>
<p>$$A_\epsilon = \{x\in X: d(x,A) < \epsilon\}$$.</p>
<p>Then I want to prove that $A_\epsilon$ is open in $X$.</p>
<p>So what I have tried so far is that, well I want to prove that a set is open so I take $x \in A_\eps... | David Hill | 145,687 | <p><strong>Yet Another Approach:</strong> We show that $\displaystyle A_\epsilon=\bigcup_{a\in A}B_\epsilon(a)$.</p>
<p>Let $a\in A$. If $x\in B_{\epsilon}(a)$, then
$$d(x,A)\leq d(x,a)<\epsilon.$$
Therefore, $B_\epsilon(a)\subset A_\epsilon$, so $\displaystyle \bigcup_{a\in A}B_\epsilon(a)\subset A_\epsilon$.</p>
... |
3,642,479 | <p>What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?</p>
| dcromley | 157,712 | <p>From reals to complex you lose order. From complex to quaternions you lose commutativity. From quaternions to octonions you lose associativity. From octonions to ..? </p>
<p>I had written this as a comment, then I followed Noah Schweber's link, which essentially says this plus more. And pregunton's answer is... |
3,996,790 | <p>I just realized that there may be a case where L'Hopital's rule fails, specifically</p>
<p><span class="math-container">$$ \lim_{x \to \infty} \frac{e^x}{e^x}$$</span></p>
<p>which evaluates to an indeterminate form, specifically <span class="math-container">$\frac{\infty}{\infty}$</span>. Sure, we can cancel the <s... | José Carlos Santos | 446,262 | <p>In the first place, L'Hopital's Rule does not fail, in the usual sense of “fail”, only in the sense that you've added at the end of your question: you can apply it as many times as you wish without ever getting an answer.</p>
<p>And there are other cases:<span class="math-container">$$\lim_{x\to\infty}\frac{x^{-1}}{... |
2,271,204 | <p>How do I solve these two equations:
$\frac{x}{x+y} = \frac{1}{2}$ and $\frac{y}{x+y} = \frac{1}{6}$ ?</p>
<p>I tried reducing the first one and end up getting $x = y$, clearly my maths is pretty dusty </p>
| kingW3 | 130,953 | <p>Notice that $$\frac{x}{x+y}+\frac{y}{x+y}=\frac{x+y}{x+y}=1$$ but $$\frac{1}{2}+\frac{1}{6}\neq 1$$</p>
|
2,781,265 | <p>I am learning category theory as a hobby. In the book I am studying from, I was looking at an example of a functor. I want to understand what this functor looks like.</p>
<p>If $G$ is a group, a functor $F: G \to \mathbf{Set}$ picks out a set $A = F(\star)$, together with a homomorphism from $G$ to the group of per... | Kyle Miller | 172,988 | <p>Think about transformation groups. You have a space $X$ with symmetries $\operatorname{Aut} X$, and a transformation group is a subgroup $G\subset\operatorname{Aut} X$. Elements of $G$ are automorphisms $X\to X$.</p>
<p>You can think of an abstract group as being a transformation group for an abstract object $\st... |
2,339,479 | <p>I am an avid board game player, especially ones with miniatures that duke it out (Risk, Axis and Allies, etc.) and I'm trying to analyze efficiency of different units in one of my more recent pickups. </p>
<p>In the game (Battlelore Second Edition for those who are curious), you get a hit on either a 5 or a 6. This... | David K | 139,123 | <p>This is a binomial distribution with $p = \frac13,$ since that is the probability of rolling a $5$ or $6$ on one die.
The probability of exactly $k$ hits on $n$ dice is</p>
<p>$$
P_n(k) = \binom nk \left(\frac13\right)^k \left(\frac23\right)^{n-k}
$$</p>
<p>where $\binom nk$ is a binomial coefficient;
$$
\binom nk... |
2,153,128 | <p>I have been given an exercise to convert switch coordinated from cylindrical to rectangular ones. This task is easy but one of them is a strange looking. The point in cylindrical coordinates is $(0,45,10)$. This corresponds to $r=0$. What is this point? Is it not the origin? But why then the angle and z-coordinate. ... | Mitchell | 413,342 | <p>You can use the formula to do this.
But here is the concept behind it. </p>
<p>Suppose you want to choose 3 people from 11 people.</p>
<p>In order to do that, imagine that there are 3 chairs. There are 11 ways to fill chair 1. After filling chair one there are 10 ways to fill chair 2 and similarly, there are 9 way... |
1,013,776 | <p>I am looking at the following <strong>Theorem</strong>:</p>
<p>Let $\phi$ a type. We suppose that there is a set $Y$, such that $\forall x(\phi(x) \to x \in Y)$. Then, there is the set $\{ x: \phi(x) \}$.</p>
<p>and I try to understand its proof.</p>
<p>From the axiom shema of specification, there is the set $V=\... | Rene Schipperus | 149,912 | <p>You want that </p>
<p>$$\{x|\phi(x)\}=\{x\in Y|\phi(x)\}$$
And this follows directly from your assumptions.</p>
|
1,407,169 | <p>Let $A$ and $B$ be two matrices in $M_n$.
Is the following ture:</p>
<p>$A$ and $B$ are similar $\iff$ $A$ and $B$ have the same jordan canonical form.</p>
<p>Could someone explain?</p>
| Surb | 154,545 | <p>If $A=PJP^{-1}$ is the <a href="https://en.wikipedia.org/wiki/Jordan_normal_form" rel="noreferrer">Jordan decomposition</a> of $A$ and $M$ is an invertible matrix such that $A=M^{-1}BM$ then we have
$$ B = (MM^{-1})B(MM^{-1})=M(M^{-1}BM)M^{-1} =MAM^{-1}=M(PJP^{-1})M^{-1}=(MP)J(P^{-1}M^{-1})$$</p>
<p>and thus the Jo... |
2,338,199 | <p>Let $G$ and $H$ be two groups. Suppose that $M$ and $N$ be two normal subgroups of $G$ and $H$ respectively such that we have the following,</p>
<ul>
<li><p>$G$ and $H$ are isomorphic.</p></li>
<li><p>$M$ and $N$ are isomorphic.</p></li>
</ul>
<p>Then we know that $G/M$ and $H/N$ are isomorphic if for an isomorphi... | Hagen von Eitzen | 39,174 | <p>Not at all.</p>
<p>Just let $G$ and $H$ be arbitary group and $M=G$, $N=H$.</p>
<hr>
<p>Apart from this, $G\cong H$ and $M\cong N$ does <strong>not</strong> imply $G/M\cong H/N$. Just pick $G=H=\Bbb Z$, $M=2\Bbb Z$, $N=H$.</p>
|
2,338,199 | <p>Let $G$ and $H$ be two groups. Suppose that $M$ and $N$ be two normal subgroups of $G$ and $H$ respectively such that we have the following,</p>
<ul>
<li><p>$G$ and $H$ are isomorphic.</p></li>
<li><p>$M$ and $N$ are isomorphic.</p></li>
</ul>
<p>Then we know that $G/M$ and $H/N$ are isomorphic if for an isomorphi... | Dietrich Burde | 83,966 | <p>No, this is not true. Consider $G=GL_n(K)$ and $M=SL_n(K)$, and $H=K^{\times}$, $N=1$.
Then $G/M\cong H/N\cong K^{\times}$, but neither $G\cong H$ nor $M\cong N$.</p>
|
3,853,896 | <p>Do I consider the probability before drawing both cards or after?</p>
<p><strong>Question more clearly</strong>:
A single card is removed at random from a deck of <span class="math-container">$52$</span> cards. From the remainder we draw <span class="math-container">$2$</span> cards at random and find that they are ... | Brice Cawley | 833,121 | <p>You are calculating the probability of the first card being a spade, therefor the next two additional draws mean nothing. The chance of drawing a spade from a deck of 52 would be 25%. It's as simple as that.</p>
<p>I hope this helps.</p>
|
1,915,455 | <p>First of all they are all even... So it is at least $2$. But is it bigger than that? How do I find out?</p>
| GoodDeeds | 307,825 | <p>This is solved using combinatorics. Any divisor <span class="math-container">$x$</span> of <span class="math-container">$n$</span> will be of the form
<span class="math-container">$$x=p_1^{n_1}p_2^{n_2}\cdots p_k^{n_k}$$</span>
where <span class="math-container">$0\le n_1\le a$</span>, <span class="math-container">$... |
3,696,265 | <p>This is from Vakil's FOAG: exercise 2.5 C, part b. I understand how objects in the extension sheaf from a sheaf on a base <span class="math-container">$\mathcal B$</span> of a topology are created, but I am having trouble understanding how to produce a morphism of sheaves given a morphism of sheaves on a base. </p>
... | Brian M. Scott | 12,042 | <p>Suppose that you want to count the <span class="math-container">$i$</span>-element subsets of <span class="math-container">$[i]=\{1,2,\ldots,i\}$</span>. Of course there’s only one of them, but we can also count them by the following roundabout procedure. We first expand the set from which we’re drawing the <span cl... |
77,642 | <p>$$\frac{1}{\sin(z)} = \cot (z) + \tan (\tfrac{z}{2})$$</p>
<p>I did this: </p>
<p><strong>First attempt</strong>: $$\displaystyle{\frac{1}{\sin (z)} = \frac{\cos (z)}{\sin (z)} + \frac{\sin (\frac{z}{2})}{ \cos (\frac{z}{2})} = \frac{\cos (z) }{\sin (z)} + \frac{2\sin(\frac{z}{4})\cos(\frac{z}{4})}{\cos^{2}(\frac{... | J. M. ain't a mathematician | 498 | <p>I'll go backwards; I hope you don't mind.</p>
<p>$$\begin{align*}\cot\,z+\tan\frac{z}{2}&=\frac{\cos\,z}{\sin\,z}+\frac{\sin\,z}{1+\cos\,z}\\&=\frac{\sin^2 z+(1+\cos\,z)\cos\,z}{(1+\cos\,z)\sin\,z}\\&=\frac{\cos^2 z+\sin^2 z+\cos\,z}{(1+\cos\,z)\sin\,z}\\&=\frac{1+\cos\,z}{(1+\cos\,z)\sin\,z}\\&... |
17,772 | <p>I have an algorithm that segments depth images using surface fitting. At the moment the algothim uses least squares polymonial fitting, but polynomials are not powerful enough to fit the shapes that are in these images. I replaced the explicit polynomial $z = f(x,y)$, with the implicit fitting problem $f(x,y,z) = 0$... | fedja | 1,131 | <p>Asuming that your surfaces are not too wild, it seems logical to try to combine the implicit approximation with the explicit one, i.e., to solve 2 least square problems simultaneously: one for the implicit function and one for the explicit one. The explicit solution will give you a crude approximation for the root a... |
17,772 | <p>I have an algorithm that segments depth images using surface fitting. At the moment the algothim uses least squares polymonial fitting, but polynomials are not powerful enough to fit the shapes that are in these images. I replaced the explicit polynomial $z = f(x,y)$, with the implicit fitting problem $f(x,y,z) = 0$... | David Bar Moshe | 1,059 | <p>I think that it is better to use rational approximation for the fitting z = P(x,y)/Q(x,y), where P and Q are polynomials. This type of fitting is more flexible in general than polynomial fitting and doesn't have the multiple solution problem. There are many algorithms to establish this type of fitting, one possibili... |
255,230 | <p>A <em>complete linear hypergraph</em> is a <a href="https://en.wikipedia.org/wiki/Hypergraph" rel="nofollow noreferrer">hypergraph</a> $H=(V,E)$ such that </p>
<ol>
<li>$|e|\geq 2$ for all $e\in E$,</li>
<li>$|e_1\cap e_2|=1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$, and</li>
<li>for all $v\in V$ we have $|\{e\in... | Aaron Meyerowitz | 8,008 | <p>I'll conjecture that for $n \gt 20$, $\ell(n) \le \sqrt{2n}+3.$ </p>
<p>It seems optimal to have most points on only two lines. So consider first the case that there is at most one line with points on it having more than two lines. </p>
<p>Write $[a_1,a_2,\cdots,a_t]$ for the configuration of one line with $t$ poi... |
1,782,423 | <p>Gödel's completeness theorem says that for any first order theory $F$, the statements derivable from $F$ are precisely those that hold in all models of $F$. Thus, it is not possible to have a theorem that is "true" (in the sense that it holds in the intersection of all models of $F$) but unprovable in $F$.</p>
<p>H... | Carl Mummert | 630 | <p>The problem is not building the model, it's making a completion of a theory. This is related closely to the fact that the set of first-order valid formulas is semi-computable but not computable. </p>
<p>Reverse Mathematics <em>reveals</em> this particular quirk of the completeness theorem. The following are in Si... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| littleO | 40,119 | <p>Changing basis can make it easier to understand a given linear transformation.</p>
<p>Suppose $T:V \to V$ is a linear transformation. It may seem difficult to understand or to visualize what effect $T$ has when it is applied to a vector $x$.
However, suppose we are lucky enough to find a vector $v$ with the specia... |
1,910,030 | <p>Let $V$ be a vector space and $U,W$ be its subspaces. Prove that if the union $U\cup W$ is a subspace of $V$, then $W \subseteq U$ or $U \subseteq W$.</p>
<p>I'm not sure where to begin at all really.</p>
| Michael Curtis | 365,275 | <p>I believe the answer you are looking for lies in the rule for calculating the number of posibilites of two disjoint sets A and B being N(A, B) = N(A) * N(B) where N(A) is the number of possibilities in set A. In this case there are two independent attributes: suit and rank. This means once you calculate the number o... |
424,853 | <p>As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been a narrowing of focus, via a choice to ignore certain "pathological"—or at least intractably complicated—ph... | Mirco A. Mannucci | 15,293 | <p>Thanks to both Pavlov and White there is now an almost complete list of "critical points" in the history of homotopy theory.</p>
<p>There are a few items that perhaps should make it into the list, for instance the Archaic Period: Betti introduced Betti's numbers <a href="https://link.springer.com/article... |
3,611,845 | <p>For the Stable Matching algorithm by Gale-Shapley, how do I prove that at most one man will get his worst choice? </p>
<p>My intuition is that I have to use contradiction. Assume that there are two men who will get their worst preferences: <span class="math-container">$M_1$</span> with <span class="math-container">... | stewbasic | 197,161 | <p>I assume the basic setup as described on <a href="https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm" rel="nofollow noreferrer">wikipedia</a> (ie equal numbers of men and women, strict preferences). Assume <span class="math-container">$M_1$</span> and <span class="math-container">$M_2$</span> are paired wi... |
613,836 | <blockquote>
<p>Let $x,y,z$ be integers and $11$ divides $7x+2y-5z$. Show that $11$
divides $3x-7y+12z$.</p>
</blockquote>
<p>I know a method to solve this problem which is to write into $A(7x+2y-5z)+11(B)=C(3x-7y+12z)$, where A is any integer, B is any integer expression, and C is any integer coprime with $11$.</... | lab bhattacharjee | 33,337 | <p>The easiest way can be : eliminate one of the three variables like below</p>
<p>$$3(7x+2y-5z)-7(3x-7y+12z)=55y-99z=11(5y-9z)$$</p>
<p>$$\iff 7(3x-7y+12z)=3(7x+2y-5z)-11(5y-9z)$$</p>
<p>If $\displaystyle 11$ divides $\displaystyle 7x+2y-5z,11$ will divide $\displaystyle7(3x-7y+12z)$</p>
<p>But $(7,11)=1$</p>
|
613,836 | <blockquote>
<p>Let $x,y,z$ be integers and $11$ divides $7x+2y-5z$. Show that $11$
divides $3x-7y+12z$.</p>
</blockquote>
<p>I know a method to solve this problem which is to write into $A(7x+2y-5z)+11(B)=C(3x-7y+12z)$, where A is any integer, B is any integer expression, and C is any integer coprime with $11$.</... | AHH | 31,500 | <p>$ 7x+2y-5z \equiv 14x +4y-10z \equiv(4-1)x+(4-11)y-(0-1)z\equiv 3x-7y+z$
$ \equiv3x-7y+12z $ (mod 11)</p>
|
2,100,726 | <p><strong>Problem:</strong> In a binomial experiment with 45 trials, the probability of more than 25 successes can be approximated by $P(z > \frac{(25-27)}{3.29}$)</p>
<p>What is the probability of success of a single trial of this experiment? </p>
<p><strong>Choices:</strong> </p>
<ul>
<li>0.07</li>
<li>0.56</l... | Feynmaniac | 407,050 | <p>As mentioned, $\frac{1}{1-e^{\frac{1}{x}}} \xrightarrow[x \to \infty]{} -\infty$. As for the limit of $f(x) + x$, </p>
<p>$\frac{1}{1-e^{\frac{1}{x}}} + x = \frac{1}{1-(1+\frac{1}{x} +\frac{1}{2x^2} +\mathcal{O}(\frac{1}{x^3}))} + x$</p>
<p>$= \frac{-x}{1+\frac{1}{2x} + \mathcal{O}(\frac{1}{x^2})} + x = -x(1-\fra... |
4,045,943 | <p>I was requested to find the values of the parameter p for which the following series converges:</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \sqrt{n} \ln^{p} \left(1+ \frac{1}{\sqrt{n}}\right)$$</span></p>
<p>I tried using <a href="https://en.wikipedia.org/wiki/Root_test" rel="nofollow noreferrer">Cauchy'... | user | 293,846 | <p>Hint: use the fact that
<span class="math-container">$$
x-\frac12x^2\le\log(1+x)\le x.
$$</span></p>
<p>If required it can be proved integrating the inequality
<span class="math-container">$$
1-x\le\frac1{1+x}\le1,
$$</span>
obviously valid for <span class="math-container">$x\ge0$</span>.</p>
|
4,045,943 | <p>I was requested to find the values of the parameter p for which the following series converges:</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \sqrt{n} \ln^{p} \left(1+ \frac{1}{\sqrt{n}}\right)$$</span></p>
<p>I tried using <a href="https://en.wikipedia.org/wiki/Root_test" rel="nofollow noreferrer">Cauchy'... | Raffaele | 83,382 | <p><span class="math-container">$$\ln(1+x)\sim x;\;\text{ as }x\to 0$$</span>
<span class="math-container">$$\ln \left(1+\frac{1}{\sqrt{n}}\right)\sim \frac{1}{\sqrt{n}};\;\text{ as }n\to\infty$$</span>
<span class="math-container">$$\left(\ln \left(1+\frac{1}{\sqrt{n}}\right)\right)^p\sim \left(\frac{1}{\sqrt{n}}\righ... |
2,091,251 | <p>I want to be able to check if a function f is even, odd, or neither using Maple's symbolic math. Unfortunately I don't get a boolean return of 'true' on a function I know is even.</p>
<pre><code>g:=abs(x)/x^2
evalb(g(x)=g(-x))
false
</code></pre>
<p>Since my function is ev... | Galen | 230,586 | <p>I think I figured out what I was looking for. I needed a function that would evaluate g at a value, which I found in the <a href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=eval" rel="nofollow noreferrer">documentation</a>.</p>
<pre><code>evalb(eval(g, x = x) = eval(g, x = -x))
</code></pre>
|
2,091,251 | <p>I want to be able to check if a function f is even, odd, or neither using Maple's symbolic math. Unfortunately I don't get a boolean return of 'true' on a function I know is even.</p>
<pre><code>g:=abs(x)/x^2
evalb(g(x)=g(-x))
false
</code></pre>
<p>Since my function is ev... | T.V. | 580,653 | <p>More human readable method (especially in the part of declaring a function):</p>
<pre><code>h(x) := abs(x)/x^2;
evalb( h(x) = h(-x) );
</code></pre>
|
1,410,889 | <p>Consider the function with domain $A = \{ (x,y) \in \, \mathbb{R}^2: (x,y) \neq (0,0)\}$ given by</p>
<p>$$\frac{2x^2y}{x^4+y^2}$$</p>
<blockquote>
<p>Letting $(x,y)$ approach $(0,0)$ along the straight line $y=ax$ , where $a$ is a real constant, we find that the limit is zero. This is not enough to conclude tha... | Cameron Williams | 22,551 | <p>Here's a hint: what if you approach $(0,0)$ along the path $y = x^2$? Note that $f$ is only continuous at $(0,0)$ if <strong>every</strong> path to $(0,0)$ yields the same limit.</p>
|
1,787,744 | <p>There are two measurable spaces with Borel sigma-algebras on them $(X, \mathcal B (X)) $ and $(Y, \mathcal B (Y)) $. There is also a continuous function $f:(X, \mathcal B (X)) \to (Y, \mathcal B (Y)) $. Prove that this function is measurable with respect to Borel sigma-algebra. </p>
| Community | -1 | <p>If $(Z, \tau ) $ is topological vector space then the Borel algebra of sets is by definition the smallest $\sigma$- algebra that contains $\tau.$
So if </p>
<ol>
<li>A set $A\in B(Y) $ is open then by the contnuity of $f$ the set $f^{-1} (A) $ is open hence $f^{-1} (A)\in B(X).$</li>
<li>Let $\mathcal{M} =\{ A\in B... |
2,344,931 | <p>It is given that $a,b$ are roots of $3x^2+2x+1$ then find the value of:
$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3$$</p>
<p>I thought to proceed in this manner:</p>
<p>We know $a+b=\frac{-2}{3}$ and $ab=\frac{1}{3}$. Using this I tried to convert everything to <em>sum and product of roots</e... | farruhota | 425,072 | <p>The answer is $-10$.
Find the common denominator:
$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3=\frac{(1-ab-(a-b))^3+(1-ab+(a-b))^3}{(1+ab+(a+b))^3}=$$
$$\frac{2\cdot\left(\frac23\right)^3+2\cdot 3 \cdot\left(\frac23\right)\cdot (a-b)^2}{(\frac{2}{3})^3}=\frac{2\cdot\left(\frac23\right)^3+4\cdot ... |
2,344,931 | <p>It is given that $a,b$ are roots of $3x^2+2x+1$ then find the value of:
$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3$$</p>
<p>I thought to proceed in this manner:</p>
<p>We know $a+b=\frac{-2}{3}$ and $ab=\frac{1}{3}$. Using this I tried to convert everything to <em>sum and product of roots</e... | serg_1 | 450,580 | <p>From sum and product, we have:$$a+b+ab+ab=0$$ $$a(b+1)=-b(a+1)$$$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3=\left(1-\dfrac{2a}{1+a}\right)^3+\left(1-\dfrac{2b}{1+b}\right)^3=\left(1-\dfrac{2ab}{b(1+a)}\right)^3+\left(1-\dfrac{2b}{1+b}\right)^3=\left(1+\dfrac{2b}{1+b}\right)^3+\left(1-\dfrac{2b}... |
303,283 | <p>A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as:
\begin{equation*}
u - \epsilon \Delta u + H(Du) = f(x)
\end{equation*}
or
\begin{equation*}
u_{t} - \epsilon \Delta u + H(Du) = f(x)
\end{equation*}
with or without boundary conditions in the stationary case, or the... | Luc Guyot | 84,349 | <p>Yes, this is <strong>true</strong> and it follows immediately from</p>
<blockquote>
<blockquote>
<p><strong>[1, Corollary 4]</strong> Two numerical semigroups with multiplicity three are equal if and only if they have the same Froebenius number and the same gender.</p>
</blockquote>
</blockquote>
<p>togeth... |
807,025 | <p>What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$.
Of course I know what it is but I don't know what's part of its definition and what's part of its properties anymore.</p>
<p>My aim is to prove that this type of reflection is linear and that $R(u).R(u)=u.u$... | Moishe Kohan | 84,907 | <p>It is the unique isometry of $R^3$ whose fixed point set is exactly the plane $a\cdot r=0$.</p>
|
807,025 | <p>What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$.
Of course I know what it is but I don't know what's part of its definition and what's part of its properties anymore.</p>
<p>My aim is to prove that this type of reflection is linear and that $R(u).R(u)=u.u$... | Will Jagy | 10,400 | <p>Given a dot product and a nonzero vector $a,$ the reflection $\tau_a$ is given by
$$ \tau_a (x) = x - \left( \frac{2 a \cdot x}{a \cdot a}\right) a $$
Straightforward to calculate $\tau_a (x) \cdot \tau_a(x).$ </p>
<p>Note that a reflection, along with being an isometry, is also self adjoint, meaning its matrix w... |
1,673,277 | <p>In what scenarios is Goldbach's conjecture, that all even numbers greater than 4 are the sum of two prime numbers; a natural conjecture to research?</p>
| hmakholm left over Monica | 14,366 | <p>In Mallory's immortal words: <strong>Because it's there.</strong></p>
<p>After the proof of Fermat's last theorem, Goldbach's conjecture is probably the leading example of a conjecture which is <em>extremely simple to explain</em> -- even a grade schooler will be able to understand what the <em>question</em> is -- ... |
4,549,781 | <p>suppose we have a convex continuous function from<span class="math-container">$\mathbb{R}$</span> to <span class="math-container">$\mathbb{R^+}$</span>
<span class="math-container">$f:\mathbb{R}\to\mathbb{R^+}$</span> such that<br />
<span class="math-container">$f(-x)=f(x),lim_{x\to 0}\frac{f(x)}{x}=0 \text{ and } ... | G Frazao | 544,105 | <p>I think I found an answer:</p>
<p>First, use the <a href="https://en.wikipedia.org/wiki/Gershgorin_circle_theorem" rel="nofollow noreferrer">Gershgorin circle theorem</a> as in <a href="https://math.stackexchange.com/a/4549760/544105">Gregory's answer</a> to show that the eigenvalues of <span class="math-container">... |
1,209,233 | <p>How can I find all of the transitive groups of degree $4$ (i.e. the subgroups $H$ of $S_4$, such that for every $1 \leq i, j \leq 4$ there is $\sigma \in H$, such that $\sigma(i) = j$)? I know that one way of doing this is by brute force, but is there a more clever approach? Thanks in advance!</p>
| Nishant | 100,692 | <p>Let <span class="math-container">$G$</span> be a transitive subgroup of <span class="math-container">$S_4$</span>. Since the orbit of <span class="math-container">$1$</span> under the action of <span class="math-container">$G$</span> is <span class="math-container">$\{1, 2, 3, 4\}$</span>, the order of <span class="... |
1,080,746 | <p>I want to calculate the limit of this sum :</p>
<p>$$\lim\limits_{x \to 1} {\left(x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb\right)}$$</p>
<p>My efforts to solve the problem are described in the <a href="https://math.stackexchange.com/a/1308281/202081">self-answer below</a>.</p>
| RE60K | 67,609 | <p>From <a href="http://www.math.harvard.edu/~elkies/Misc/sol8.html" rel="noreferrer">here</a> is the amazing solution:</p>
<blockquote>
<p>Since $S$ satisfies the functional equation
$$S(x) = x − S(x^2)$$
it is clear that if $S(x)$ has a limit as $x$ approaches 1 then that limit must be $1/2$. One might guess t... |
4,586,314 | <p><em>A three-sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region, as shown in the diagram below. The enclosed area is to equal <span class="math-container">$1800 m^2$</span> and the fence running parallel to the river must be set back at least <span ... | Carl Christian | 307,944 | <p>There is a two-sided linear transformation that accomplishes your goal. Specifically, let <span class="math-container">$$E = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$</span> be the matrix such that <span class="math-container">$EA$</span> isolates the first row of <s... |
1,649,997 | <p>I'm having a hard time coming up with two unbounded sequences where their difference yields $0$ when $n\rightarrow\infty$. Any ideas?</p>
| Michael Hardy | 11,667 | <p>$n$ and $n+\dfrac 1 n$. $\qquad\qquad\qquad$</p>
|
1,649,997 | <p>I'm having a hard time coming up with two unbounded sequences where their difference yields $0$ when $n\rightarrow\infty$. Any ideas?</p>
| Jakub Konieczny | 10,674 | <p>Perhaps the simplest example:
$$
a_n = n,
\qquad
b_n = \begin{cases}
n, & n > 1\\
0, & n = 1
\end{cases}
$$
for $n = 1,2, \dots$ These are (obviously) different, and their difference is (obviously) convergent to $0$:
$$
a_n - b_n = \begin{cases}
0, & n > 1\\
1, & n = 1.
\end{cases}
$$
Arguably... |
1,191,176 | <p>I know how to prove the result for $n=2$ by contradiction, but does anyone know a proof for general integers $n$ ?</p>
<p>Thank you for your answers.</p>
<p>Marcus</p>
| Nicky Hekster | 9,605 | <p>FLT is quite a cannon to settle the proof. One can also follow the "classical" approach: assuming WLOG $\gcd(p,q)=1$, if $p^n=2q^n$, then $2 \mid p^n$, implying $2 \mid p$, say $p=2r$. But then $2^{n-1}r^n=q^n$, and since $n \gt 2$, we see that $2 \mid q^n$. It follows that $2 \mid q$, contracting the fact that $p$ ... |
126,897 | <p>It is an exercise in a book on discrete mathematics.How to prove that in the decimal expansion of the quotient of two integers, eventually some block of digits repeats.
For example:
$\frac { 1 }{ 6 } =0.166\dot { 6 } \ldots$ and $\frac { 217 }{ 660 } =0.328787\dot { 8 } \dot { 7 } \ldots$</p>
<p>How to think of thi... | JeremyFR | 23,961 | <p>Assume the decimal is infinite (otherwise the 0 repeats). Imagine evaluating the quotient by long division. After each step after the numerator has become all 0's, you have to carry over something less than the denominator. Eventually you'll have to carry something you carried before, because you carry something on ... |
4,383,098 | <p>This was a question I had ever since I started studying Formal mathematics. Take ZFC for example, in it the axioms tell us 'tests' to check if something is a set or not and how the object, if they are set, behave with some other operations defined on the set.</p>
<p>My question is how exactly do we find objects do f... | boaz | 83,796 | <p>Hint: try using the identities
<span class="math-container">$$
X\cup(Y\cap Z)=(X\cup Y)\cap(X\cup Z)
$$</span>
and
<span class="math-container">$$
X\cap(Y\cup Z)=(X\cap Y)\cup(X\cap Z)
$$</span></p>
|
2,391,505 | <p>Most ODE textbooks provide the following steps to the solution of a separable differential equation (here the exponential equation is used as an example):</p>
<p>$$\frac{dN}{dt}=-\lambda N(t) \Rightarrow \frac{dN}{N(t)}=-\lambda dt\Rightarrow \int\frac{1}{N}dN=-\lambda\int dt \Rightarrow ln\mid N\mid = -\lambda t+C... | Eric | 309,041 | <p>To be more clear, write $N(t)$ instead of $N$, write $\frac{dN(t)}{dt}$ instead of $\frac{dN}{dt}$. Also, it is more ideal to write $N'(t)$ than $\frac{dN(t)}{dt}$ in the answer, since we are talking the annoying Leibniz's mis-leding notation, right?</p>
<blockquote>
<p>$\begin{alignedat}{3}\frac{dN(t)}{dt}=-\lam... |
3,059,617 | <p>Note: I apologize in advance for not using proper notation on some of these values, but this is literally my first post on this site and I do not know how to display these values correctly.</p>
<p>I recently was looking up facts about different cardinalities of infinity for a book idea, when I found a post made sev... | J.G. | 56,861 | <p>Further to existing answers, let's clarify how transfinite recursion defines <span class="math-container">$\aleph_\alpha$</span> for arbitrary ordinals <span class="math-container">$\alpha$</span>. Fr simplicity I'll identify each aleph with the least ordinal of the intended cardinality. We have <span class="math-co... |
4,219,689 | <p>I was trying to derive the perimeter of a circle using vector algebra and came across this.....</p>
<p>Taking a circle of radius <span class="math-container">$r$</span> with center at the origin, the position vector of a particular point on the circle making an angle <span class="math-container">$t$</span> with the ... | peta arantes | 419,513 | <p><span class="math-container">$\frac{AA_1+DD_2}{2}=d =2d(I)\\
\frac{E_1E_2+B_1B}{2}=d=2d(II)\\
\frac{FF_1+CC_1}{2}=d=2d(III)\\
Sum (I+II+III) =18\rightarrow 6d = 18 \therefore \boxed{\color{red}d = 3} $</span></p>
|
1,670,816 | <p>Given <span class="math-container">$a, b \in \Bbb R$</span>, consider the following large tridiagonal matrix</p>
<p><span class="math-container">$$M := \begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
0 & b & (a+2)^2 &am... | Moo | 295,756 | <p>We are given the function:</p>
<p>$$f(x) = (x-1) (x-2) (x-3) (x-4)-\dfrac{x^6}{1000000}$$</p>
<p>A local plot of the function shows:</p>
<p><a href="https://i.stack.imgur.com/xqqhn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xqqhn.png" alt="enter image description here"></a></p>
<p>We clea... |
176,054 | <p>I have a friend who wants to study something applied to neurosciences. He is going to begin his grad studies in mathematics.
He asked me which areas of mathematics could be applied to neurosciences.
Since I don't know the answer, I thought mathoverflow would be the right place to ask.
I mean, there are many areas o... | Piyush Grover | 30,684 | <p>The site scholarpedia.org is a good start. <a href="http://www.scholarpedia.org/article/Encyclopedia:Computational_neuroscience" rel="nofollow">http://www.scholarpedia.org/article/Encyclopedia:Computational_neuroscience</a></p>
<p>It was started by a computational/mathematical neuroscientist, and has good articles ... |
6,661 | <p>Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical tran... | Ronny | 101,741 | <p>Think that the laplace transformation is a kind of a machine, the machine eats function of t f(t) out comes F(s). you do a transformation from time to frequency. Inside the machine you have this integral expression that you already know. it is similar when you transform from one vector space to another. for instance... |
384,991 | <p>If $f$ is defined as a function of real variables to real values, and $c \in cl(Domain)$ as its limit value (i.e. $\lim_{x \to c} {f(x)} = 0 $) how to prove that this implies: $\lim_{x \to c} {1 \over f(x)} = \infty$.</p>
<p>It seems logical that the values will be always bigger, but when tried to construct a cont... | Thomas | 26,188 | <p>Let me just assume that $c = 0$. And as pointed out in the comment above, you have to be careful. Take the following just as an outline of a general approach. As pointed out in the excellent answer by Cameron, this doesn't always work. You have to make some assumptions. But maybe it can be helpful?</p>
<p>But, I as... |
2,292,711 | <p>Show that the vector field $F(x,y)=(2x-x^5-xy^4,y-y^3-x^2y)$ defined in $R^2$ does not have periodic orbits; the Bendixson criterion is not useful.</p>
| dantopa | 206,581 | <p>@Robert Israel provides the best answer. This post is an elaboration of his remarks.</p>
<h1>Dynamical System</h1>
<p>Find the periodic solution for the dynamical system:
<span class="math-container">$$
%
\begin{align}
%
\dot{x} &= -x \left(x^4+y-2\right) \\
%
\dot{y} &= -y \left(x^2+y^2-1\right) \\
%
\e... |
1,621,275 | <blockquote>
<p>Show that $a_n=\frac{n+1}{2n}a_{n-1}+1$ given that:</p>
<p>$a_n=1/{{n}\choose{0}}+1/{{n}\choose{1}}+...+1/{{n}\choose{n}}$</p>
</blockquote>
<p>The hint says to consider when $n$ is even and odd. When $n=2k$ I get:</p>
<p>$$a_{n}=1/{{2k}\choose{0}}+1/{{2k}\choose{1}}+...+1/{{2k}\choose{2k}}$$
$... | heropup | 118,193 | <p>First, observe that the given recursion can be written as $$2n(a_n - 1) - (n+1)a_{n-1} = 0.$$ Second, observe that $$a_n - 1 = \sum_{k=0}^{n-1} \binom{n}{k}^{\!-1},$$ since the final term of $a_n$ is always $1$. Therefore, the relationship to be proven is equivalent to showing $$0 = S_{n-1} = \sum_{k=0}^{n-1} \fra... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Michaël Le Barbier | 38,203 | <p>I really like <em>Concrete Mathematics</em> by Knuth, Graham and Patashnik, and the introductions to number theory by Rose and by Hardy&Wright: you will find there many interesting school-like problems (but the whole books may not be suitable).</p>
<p>In geometry, I can suggest Hartshorne's <em>Geometry: Euclid... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Viktor | 40,294 | <p>I always enjoyed "How to Solve It: A New Aspect of Mathematical Method" by G. Pólya.</p>
<p>It doesn't really cover all that much mathematics, it just helps you structure your thoughts in a mathematical sense. </p>
<p>But it depends a lot on your actual needs. </p>
|
269,769 | <p>Is the following true always for a matrix norm </p>
<p>$$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$</p>
<p>Related to this
given $r$ is positive constant, $H$ is symmetric positive definite
is the following true :</p>
<p>$$\lVert (rI - H)(rI + H)^{-1}\rVert < 1 $$</p>
<p>or</p... | dineshdileep | 41,541 | <p>Matrix norms which satisfy the condition (in addition to others) you mentioned are referred to as sub-multiplicative norms. A well-known example of a matrix norm which doesn't satisfy this condition is the max-norm defined as
\begin{align}
||A||_\max=\max_{i,j}|a_{ij}|
\end{align}
Let $H=UDU^T$ where $U$ is the mat... |
3,247,982 | <p>If <span class="math-container">$f:X \to Y$</span> is a function and <span class="math-container">$U$</span> and <span class="math-container">$V$</span> are subsets of <span class="math-container">$X$</span>, then <span class="math-container">$f(U \cap V) = f(U) \cap f(V)$</span>.</p>
<p>I am a little lost on this ... | Daniel Sehn Colao | 649,328 | <p>You can't prove the equality unless <span class="math-container">$f$</span> is injective.</p>
<p>If you prove something by Double-Inclusion, then you get the equality. However, at this case you will just get the equality right if <span class="math-container">$f$</span> is injective.</p>
<p>I will do one inclusion
<s... |
1,273,086 | <p>Sum of three dice (six faced) throws is $15$. What is the probability that first throw was $4$?</p>
<p>The way I thought of solving this was...
- Given - sum of second and third throw is $11$
- Probability of getting first throw = $4$ is $1$ out of $6$, that is $\frac{1}{6}$</p>
<p>Is this correct?</p>
| Masclins | 238,119 | <p>Joffan pretty much pointed how to find the answer.</p>
<p>$P(D_{1}=4\mid Sum=15) = \frac {ways\ to\ get\ 11\ with\ 2\ dices}{ways\ to\ get\ 15\ with\ 3\ dices}$ (Note that getting 11 with 2 dices is like getting 4 with the first one).</p>
<p>$ways\ to\ get\ 11\ with\ 2\ dices = 2$ (both $(5,6)$ and $(6,5)$)</p>
... |
11,368 | <p>I am interested in improving this plot:</p>
<p><img src="https://i.stack.imgur.com/qvPAJ.png" alt="contour plot"></p>
<p>which I produced with the following command:</p>
<pre><code>dat // ListContourPlot[#, ContourShading -> False,
ContourStyle -> ColorData[10] /@ Range[10], Contours -> Range[21]/8,
... | Guillochon | 805 | <p>How about this:</p>
<pre><code>{d1, d2} = Dimensions[dat];
xvals = Range[-15, 15, (15 + 15)/(d1 - 1)];
yvals = Range[-7, 2, (2 + 7)/(d2 - 1)];
dat2 = Flatten[Join[Table[{yvals[[j]], xvals[[i]]}, {i, d1}, {j, d2}],
Table[Partition[dat[[i]], 1], {i, d1}], 3], 1];
dat2 = Extract[dat2, Position[dat2[[All, 3]], x_ ... |
2,958,566 | <p>Expressing <span class="math-container">$\frac{t+2}{t^3+3}$</span> in the form <span class="math-container">$a_o+a_1t+...+a_4t^4$</span>, where <span class="math-container">$t$</span> is a root of <span class="math-container">$x^5+2x+2$</span>.</p>
<p>So i can deal with the numerator, but how do I get rid of the de... | user | 505,767 | <p>We need to proceed by</p>
<ul>
<li>base case: <span class="math-container">$n=0 \implies ab^0+c0+d=a+d$</span></li>
<li>induction step: assume <span class="math-container">$m|ab^n+cn+d$</span> then</li>
</ul>
<p><span class="math-container">$$ab^{n+1}+c(n+1)+d=ab^{n+1}+bcn+bd-bcn-bd+c(n+1)+d=$$</span></p>
<p><spa... |
1,787,072 | <p>Boolean algebras aren't algebras (to the best of my understanding).</p>
<p>So why are they called algebras?</p>
<p>Wouldn't it make more sense to call them a "Boolean system" or a "Boology" or something else like that?</p>
| zyx | 14,120 | <p>Because Boole himself introduced the word "algebra" into the subject.</p>
<p>The term "algebra of logic" appears in Boole's 1854 book on Laws of Thought:</p>
<blockquote>
<p>Let us conceive, then, of an Algebra in which the symbols x, y, z, etc. admit indifferently of the values 0 and 1, and of these values alon... |
167,049 | <p>$p(z),t(z)$ are two mutually defined functional equations, while $\widehat{G}(z)$ is the exponential generation function of <a href="https://oeis.org/A182173" rel="nofollow noreferrer">A182173</a> (maybe, I am not sure...lol):</p>
<p>$$\begin{cases}
p(z)=e^{t(z)}-t(z)+2 z-1\\
t(z)=2\left(e^{p(z)}-e^{p(z)/2}+z\right... | J. M.'s persistent exhaustion | 50 | <p>As Akku14 notes in his answer, one can determine from the conditions that both functions $p,t$ have a zero constant term. With that additional piece of information, we can use <code>SolveAlways[]</code> to determine the rest of the required coefficients:</p>
<pre><code>n = 10; (* desired expansion order *)
p = Sum[... |
3,996,665 | <p>I am really confused about below questions. I am new both in this platform and below topics. I hope, I was clearly explain the questions.</p>
<p>For ∀n∈<span class="math-container">$\mathbb{Z}$</span><sup>+</sup></p>
<p>Based on <span class="math-container">$B_n$</span> = {n, n+1, n+2...}, <strong>B</strong> = {<spa... | Henno Brandsma | 4,280 | <p><span class="math-container">$(B_n), n \in \Bbb Z^+$</span> is a base for a topology, because their union is <span class="math-container">$\Bbb Z^+$</span> and the collection is closed under finite intersections.</p>
<p>The space is not Hausdorff, because there is no basic element that contains <span class="math-con... |
3,387,049 | <p>Say a deck of cards is dealt out equally to four players (each player receives 13 cards).</p>
<p>A friend of mine said he believed that if one player is dealt four-of-a-kind (for instance), then the likelihood of another player having four-of-a-kind is increased - compared to if no other players had received a four... | Ben | 526,995 | <p>Without even doing the calculations you can see intuitively that this must be true. If one player has four-of-a-kind then they have taken all four cards with the same number/face out of the deck, which means that the remaining deck now has heavier concentrations of each of the remaining numbers/faces. Conditioning... |
4,048,158 | <p>Given data points <span class="math-container">$\{(x_i,f(x_i)\}_{i=0}^{m}$</span>, if we define divided difference recursively as:</p>
<p><span class="math-container">$$f[x_0,\cdots,x_{k+1}] = \frac{f[x_1,\cdots,x_{k+1}]-f[x_0,\cdots,x_k]}{x_{k+1}-x_0} \text{ with the definition that} f[x_0] = f(x_0)$$</span></p>
<p... | Yuval Peres | 360,408 | <p>This proof was more delicate than I expected.
First observe that by an easy induction, for all <span class="math-container">$x_0,\ldots, x_m$</span> we have
<span class="math-container">$$f[x_0,\ldots, x_m]=f[x_m,\ldots,x_0]\,. \quad \quad (1)$$</span>
Recall <span class="math-container">$L_m$</span> is defined to... |
4,468,883 | <p>In the book "<em>Primes of the form <span class="math-container">$x^2+ny^2$</span></em>", David Cox had shown that:
<span class="math-container">$$p=x^2+14y^2 \Longleftrightarrow (-14/p)=1 \;\text{and}\; (x^2+1)^2=8 \mod p \; \text{has an integer solution.} $$</span>
Is this imply that there are infinitely... | Piquito | 219,998 | <p>COMMENT.- I bet there is an infinity of primes and I rely for this on the following:</p>
<p>for instance,the four identities <span class="math-container">$$(a+3)^2+14(3b\pm2)^2=6(6a^2+6a+21b^2\pm28+11)-1\\(6a+1)^2+14(3b)^2=6(6a^2+2a+21b^2)+1\\(6a+5)^2+14(3b)^2=6(6a^2+10a+21b^2+4)+1$$</span> There are an infinity of ... |
2,727,598 | <p>Given p is a prime number greater than 2, and</p>
<p>$ 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{p-1} = \frac{N}{p-1}$ </p>
<p>how do I show, $ p | N $ ???</p>
<p>The previous part of this question had me factor $ x^{p-1} -1$ mod $p$. Which I think is just plainly $(x-1) ... (x-(p-1))$ </p>
| Malcolm | 12,559 | <p>Working $\bmod{p}$:
$$
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p-1}\\
\equiv 1 + 2^{-1} + 3^{-1} + \cdots + (p-1)^{-1}\\
\equiv 1 + 2 + \cdots + (p-1) \ \text{ reordered}
$$
which you have shown is $0\pmod{p}$ when you factored $x^{p-1} -1\pmod{p}$.</p>
<p>So if $\displaystyle 1 + \frac{1}{2} + \frac{1}{3... |
1,356,524 | <p>EDIT: the OP has since edited the question fixing all the issues mentioned here. Yay!</p>
<p>There was a <a href="https://puzzling.stackexchange.com/questions/17627/knights-knaves-and-normals-smullyans-error/17628">question asked on Puzzling recently</a>, titled <code>Knights, Knaves and Normals "(Smullyan's Er... | Community | -1 | <p>Since <a href="https://puzzling.stackexchange.com/posts/17627/revisions">OP fixed the question along the lines from my question here</a> , I think I can assume the reasoning proposed in the question was correct. Prof. Smullyan is now officially "no longer in error" <em>chuckle</em>.</p>
|
3,855,521 | <p>Suppose <span class="math-container">$f:A\bigoplus M \to A\bigoplus N$</span> and <span class="math-container">$g: A \to A$</span>, where <span class="math-container">$g = \pi_A\circ f\circ l_A$</span>, are isomorphisms of <span class="math-container">$R$</span>-module. Prove that <span class="math-container">$M\con... | Intelligenti pauca | 255,730 | <p>This is a rectangle, because for <span class="math-container">$x=0$</span> we get <span class="math-container">$|y|=1$</span>, but for <span class="math-container">$y=0$</span> we obtain
<span class="math-container">$$
x=\root{10^{12}}\of{\pi^2\over4}\approx
1.0000000000009031654.
$$</span>
For a square, you'd bette... |
3,855,521 | <p>Suppose <span class="math-container">$f:A\bigoplus M \to A\bigoplus N$</span> and <span class="math-container">$g: A \to A$</span>, where <span class="math-container">$g = \pi_A\circ f\circ l_A$</span>, are isomorphisms of <span class="math-container">$R$</span>-module. Prove that <span class="math-container">$M\con... | David K | 139,123 | <p>This is related to what happens with the graphs of very high powers of <span class="math-container">$x,$</span>
which in turn is related to exponential growth and decay.</p>
<p>Graph <span class="math-container">$y = x^2.$</span> Notice that the curve goes through <span class="math-container">$(0,0)$</span> at its l... |
373,397 | <p>the question just like in the title: </p>
<p>How to solve $x^3-bx^2+c=0$ for x analitically, when b and c are constants, and all (coefficients and variables) are reals.</p>
| John Gowers | 26,267 | <p>The first step when solving an arbitrary cubic is to get rid of the quadratic term (the term in $x^2$). We do this by 'completing the cube' - a process analogous to completing the square for quadratics. </p>
<p>Specifically, we make the substitution $y=x-\frac{1}{3}b$. Then: </p>
<p>$$
y^3 = \left(x-\frac{1}{3}... |
2,177,619 | <p>Find a solution to the boundary value problem \begin{align}y''+ 4y &= 0 \\ y\left(\frac{\pi}{8}\right) &=0\\
y\left(\frac{\pi}{6}\right) &= 1\end{align}
if the general solution to the differential equation is $y(x) = C_1 \sin(2x) + C_2 \cos (2x)$.</p>
<p>I was able to compute the following equations:
\b... | Hmath | 262,582 | <p>You have a system of 2 -equations to 2 unknowns!! $C1+ C2 = 0$ and thus $C1 = -C2$.</p>
<p>Substitute in the second equations and solve!!</p>
|
1,678,687 | <p>Find the point $(x_0, y_0)$ on the line $ax + by = c$ that is closest to the origin. </p>
<p>According to this <a href="http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-line.php" rel="nofollow noreferrer">source</a>, I thought that $\left( -\frac{ac}{a^{2}+b^{2}}, -\frac{bc}{a^{2}+b^{... | Ng Chung Tak | 299,599 | <p>\begin{array}{cccc}
ax+by &=& c & \cdots \cdots (1) \\
bx-ay &=& 0 & \cdots \cdots (2)
\end{array}</p>
<p>$(1)$ is equation of the given line,</p>
<p>$(2)$ is the equation of the normal passing through the origin</p>
<p>$(1)\cap(2)$ is the required point.</p>
<p>$(1)\times a+(2)\times... |
1,586,607 | <p>This is something I've been thinking about lately;</p>
<p>$$\int_0^1 \int_0^1 \frac{1}{1-(xy)^2} dydx$$
Solutions I've read involve making the substitutions: $x= \frac{sin(u)}{cos(v)}$ and $y= \frac{sin(v)}{cos(u)}$. This reduces the integral to the area of a right triangle with both legs of length $\frac{\pi}{2}$... | Vivek Kaushik | 169,367 | <p>You can also do this, and this one does not involve any fancy tricks: </p>
<p>Consider the double integral: $$I=\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{(1+x)(x+y^2)}dydx.$$ </p>
<p>Integrate with respect to $y$ first and recognize this is :</p>
<p>$$I=\int_{0}^{\infty} \frac{1}{1+x}\lim_{y \rightarrow \infty} ... |
3,772,193 | <p>If I put the two vectors 1,0,0 and 0,1,0 next to each other</p>
<p><span class="math-container">$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$$</span></p>
<p>I can see that they are independent since I cannot write <span class="math-container">$(1,0,0)^t$</span> as <span class="math-container">... | gt6989b | 16,192 | <p>I think you may be confusing two notions.</p>
<p>The vectors <span class="math-container">$(1,0,0)^T$</span> and <span class="math-container">$(0,1,0)$</span> are linearly independent, since the <em>columns</em> in the matrix are independent.</p>
<p>The <em>rows</em> in the matrix may be linearly dependent, since th... |
3,772,193 | <p>If I put the two vectors 1,0,0 and 0,1,0 next to each other</p>
<p><span class="math-container">$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$$</span></p>
<p>I can see that they are independent since I cannot write <span class="math-container">$(1,0,0)^t$</span> as <span class="math-container">... | mathreadler | 213,607 | <p>If you look at row vectors: [0,0] must be a linear combination of [1,0] and [0,1].</p>
<p>Because the number of vectors is larger than the number of variables in each vector.</p>
<p>But if you look at columns it's the opposite relationship.</p>
<p>Too few columns in relation to elements in each column.</p>
|
57,508 | <p>While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question: </p>
<p>Is it possible to prove that $\mathbb A^2-point$ is not an affine variety, if you don't know that the polynomia... | Hailong Dao | 2,083 | <p>Yes, you can do it over any field.</p>
<p>First, it is enough to show $\mathcal O(Y) = k[x,y]$ ($Y=A^2-0$). If that is true and $Y$ is affine, then the embedding $Y \to A^2$ must correspond to some $k$-algebra map $k[x,y] \to k[x,y]$, which is absurd.</p>
<p>The key point now, as in Guillermo's post, is to show th... |
3,823,048 | <p>My questions needs more context than what can fit into the title so let me elaborate. In pretty much all the art textbooks I am reading on linear perspective they state that the correct way of placing an ellipse is to imagine the minor axis of an ellipse as an axel of a wheel running to the opposite vanishing point.... | Théophile | 26,091 | <p>You are absolutely right; the authors of these textbooks are mistaken, and your diagrams show why.</p>
<p>I happen to have watched a fair number of instruction videos on perspective drawing, and to my frustration, I haven't yet found a single one that addresses this mistake, even from the most meticulous and formal ... |
1,968,267 | <p>When doing induction should you always try to put your final answer as the "<em>desired</em> " form? For example if: $$\sum^{n}_{k=1}(k+2)(k+4) = \frac{2n^{3} + 21n^{2} + 67n}{6}$$ we ought to give the final answer as $$\frac{2(k+1)^{3} + 21(k+1)^{2} + 67(k+1)}{6}?$$</p>
<p>I just expanded both the $\text{LHS}_{k+1... | Iuli | 33,954 | <p>$(k+2)(k+4)=k^2+6k+8$</p>
<p>$$\sum^{n}_{k=1}{(k+2)(k+4)}=\sum^{n}_{k=1}{k^2}+6\sum^{n}_{k=1}{k}+\sum^{n}_{k=1}{8}=\frac{n(n+1)(2n+1)}{6}+6 \cdot \frac{n(n+1)}{2}+8n$$</p>
|
3,184,887 | <p>I want to divide a prize, say 1000, among an unknown number of contestants, as determined by their rank in a marathon race. I want the smallest portion to be above zero (so it is worthwhile to compete). So the higher the rank: the higher the prize.</p>
<p>PS, feel free to explain in detail as I am a math noob. Than... | celnaFR | 625,259 | <p>I think it might be interesting to formalize your thought: in fact, you want to distribute a certain amount a money, let's say 1000 (€!). </p>
<p>As you noticed in your previous message, it is strictly equivalent to create numbers between 0 and 1 and then multiply it by the total amount of money you want to distrib... |
1,252,326 | <p>"A student sits 6 examination papers, each worth 100 marks. In how many possible ways can he score 40% of the total possible marks?" I could not think of a way to attack this from first principles and thought there must be some theoretical generality that would make it quite easy. Haven't done any serious mathematic... | Tryss | 216,059 | <p>If we $e_i$ iqs the score in exam i, we want to know the numbers of combinaisons $(e_1,\cdots,e_6)$ such that :</p>
<p>$$\forall i, 0\leq e_i \leq 100$$</p>
<p>$$\sum_{i=1}^6 e_i = 240$$</p>
<p>And Here I guess, the best way is to consider the general problem with $n$ exams and $p$ points.</p>
<p>You then have t... |
2,598,912 | <p>Let $f$ be some function in $L_{loc}^1(\mathbb{R})$ such that, for some $a \in \mathbb{R}$,</p>
<p>$$\int_{|x| \leq r} |f(x)|dx \leq (r+1)^a$$</p>
<p>for all $r \geq 0$. Show that $f(x)e^{-|tx|} \in L^1(\mathbb{R})$ for all $t \in \mathbb{R} \setminus \{0\}$. </p>
<p>I'm having a hard time finding use of the bou... | Julián Aguirre | 4,791 | <p>A different proof. Assume $t>0$.
\begin{align}
\int_{\Bbb R}|f(x)|\,e^{-t|x|}\,dx&=\sum_{n=0}^\infty\int_{n\le|x|<n+1}|f(x)|\,e^{-t|x|}\,dx\\
&\le\sum_{n=0}^\infty e^{-t|n|}\int_{|x|<n+1}|f(x)|\,dx\\
&\le\sum_{n=0}^\infty e^{-t|n|}(2+n)^a\\
&<\infty.
\end{align}</p>
|
1,809,500 | <p>Prove that $k!>(\frac{k}{e})^{k}$. </p>
<p>It is known that $e^{k}>(1+k)$. So if we multiply $k!$ on both sides, we get $k!e^{k}>(k+1)!$. Also $k^k>k!$. Now how to proceed ?</p>
| Dark | 208,508 | <p>Work by induction.</p>
<ul>
<li>For $k=1$ it is true as $e >1$</li>
<li>Assume it is true for $k$. Then $(k+1)! = (k+1) k! > k k!> k \frac{k^k}{e^k} = e \frac{k^{k+1}}{e^{k+1}} > \frac{k^{k+1}}{e^{k+1}}$ because $e>1$.</li>
</ul>
|
2,577,634 | <p>My goal is to compute $$I=\int_{0}^{+∞}\frac{\cos{ax}}{1+x^2}dx$$ where $a>0$. </p>
<p>$$I=\frac{1}{2}\int_{-∞}^{+∞}\frac{\cos{ax}}{1+x^2}dx=\frac{1}{2}Re\bigg(\int_{-∞}^{+∞}\frac{e^{iax}}{1+x^2}dx\bigg)$$. </p>
<p>Let $f(z)=\frac{e^{iaz}}{1+z^2}$.</p>
<p>By Residue Theorem, $\int_{-R}^{R}\frac{e^{iax}}{1+x^2}... | Robert Z | 299,698 | <p>Note that for $z=R(\cos(t)+i\sin(t))$ with $R>1$ and $t\in [0,\pi]$
$$\left|\frac{e^{iaz}}{1+z^2}\right|=\frac{e^{-aR\sin(t)}}{R^2-1} \leq \frac{1}{R^2-1}.$$
Hence, as $R\to +\infty$,
$$\left|\int_{\gamma_R}\frac{e^{iaz}}{1+z^2}dz\right|\leq \frac{|\gamma_R|}{R^2-1}=\frac{\pi R}{R^2-1}\to 0.$$
P.S. This is a part... |
3,779,736 | <p>Let <span class="math-container">$(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$</span>. Find <span class="math-container">$E\left[\sum \frac{X_i }{X_{(n)}}\right]$</span> where <span class="math-container">$X_{(n)} = \max_i X_i$</span>.</p>
<p>It was suggested to use Basu's Theorem which I am unfamiliar ... | StubbornAtom | 321,264 | <p>Assuming <span class="math-container">$X_1,\ldots,X_n$</span> are independent and identically distributed,</p>
<p><span class="math-container">$$E\left[\sum_{i=1}^n \frac{X_i}{X_{(n)}}\right]=\sum_{i=1}^n E\left[\frac{X_{i}}{X_{(n)}}\right]=nE\left[\frac{X_1}{X_{(n)}}\right]$$</span></p>
<p>This can be evaluated by ... |
1,550,736 | <p>Consider Z a Normal (Gaussian) random variable with mean 0 and variance 1.<br>
It has density
$$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} \text{for all x real numbers}$$
We consider $X=2Z+1$. Write the CDF of X in terms of the one of Z and take the derivative to get that the density of X is
$$f_X(x)=\frac{1}{2... | Milen Ivanov | 294,209 | <p>Let $$A = U \Sigma V^*$$ be the SVD of A. Then, because $ U$ is orthogonal, $$ ||Ax||= ||\Sigma V^* x|| $$ If $V^*x = y$ , $ ||y||=||x||=1$ we are left with
$$||Ax||^2=||\Sigma y||^2 = \sum _i |\sigma_iy_i|^2 \geq \sum _i |\sigma_qy_i|^2 = \sigma_q^2 $$ So, the inequality holds. If $x$ is a right singular vector ... |
159,495 | <p>I have a text file where each line is a data point in the form:</p>
<pre><code>[ -495.01172, -158.35966, 2705.0 ]
[ -489.15576, -127.229675, 2673.0 ]
[ -487.6918, -97.679855, 2665.0 ]
[ -487.32578, -68.4594, 2663.0 ]
[ -485.86182, -39.19415, 2655.0 ]
[ -485.3128, -10.12311, 2652.0 ]
[ -484.03183, 18.853745, 2645.0 ... | Nasser | 70 | <p>This works, but may be there is easier way</p>
<pre><code>SetDirectory[NotebookDirectory[]]
data = Flatten@Import["input.txt","CSV"];
data2 = If[StringQ[#],ToExpression[StringDelete[#,{"[","]"}]],#]&/@data;
data2 = ArrayReshape[data2,{Length@data2/3,3}]
</code></pre>
<p>now</p>
<pre><code>MatrixForm[data2]
<... |
572,429 | <p>If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?</p>
<p>I don't really know where to start with this one. I know that $p(0)=det(0*I_{n}-A)=det(-A)=(-1)^{n}*det(A)$, thus if both $p(0)$ and $det(0) = 0$ then $0$ is an eigenvalue... | copper.hat | 27,978 | <p>If you have $v\neq 0$ such that $A v = \lambda v$, and $A$ is invertible, then $v = \lambda A^{-1} v$.</p>
<p>What does that tell you about $\lambda$?</p>
<blockquote class="spoiler">
<p> $\lambda$ must be non-zero, otherwise this would mean $v=0$, a contradiction.</p>
</blockquote>
<p>How do you use the above ... |
4,218,944 | <p>A very simple question. Saw this formula many places earlier, but how do we prove it?
<span class="math-container">$$ax^2+bx+c=a(x-r_1)(x-r_2)$$</span> Where <span class="math-container">$r_1$</span> and <span class="math-container">$r_2$</span> are the roots of the quadratic.</p>
<p>P.S.- I have seen a "proof&... | Bernard Pan | 800,888 | <p>For quadratic equation <span class="math-container">$$0=ax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a^2},$$</span> its roots are <span class="math-container">$$r_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$</span> Depending on the sign of <span class="math-container">$b^2-4ac$</span>, the roots could be dist... |
85,651 | <p>Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.</p>
<p>How does the lower bound for the length of primitive geodesics on $\Gamma \backslash \mathbb{H}$ depending on $N \rightarrow \infty$?</p>
<p>Any suggestions?</p>
| kassabov | 13,992 | <p>Let $\Gamma$ be the congruenece group $\ker SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/N\mathbb{Z})$.<br>
The length of the shortest geodesic is of the order of $\log N$.
The upper bound follows from a counting argument, and the lower
bound comes from the observation that a product of $c\log N$
generators of $SL_2(\mathb... |
4,172,955 | <p>Find all <span class="math-container">$|z|=1$</span> such that <span class="math-container">$|z^4+4| = \sqrt{5}.$</span></p>
<hr />
<p>I've tried doing
<span class="math-container">$$|z^4+4|^2 = 5 \implies (z^4+4)(\overline{z^4}+4) = 5 \implies |z|^8 + 4(z^4+\overline{z^4})
+11=0,$$</span>
but i'm not sure how to ... | Harsh joshi | 940,974 | <p>This problem can be solved by simple geometry.</p>
<p>Let <span class="math-container">$z^4=w$</span></p>
<p>Using <span class="math-container">$|z^4|=|z|^4$</span>, as <span class="math-container">$|z|=1$</span> then <span class="math-container">$|w|=1$</span>.</p>
<p>Now <span class="math-container">$w$</span> sat... |
2,531,523 | <p>Knowing that $S_3=\{\text{id},\sigma,\sigma^2,\tau,\tau\circ\sigma,\tau\circ\sigma^2\}$ ($\tau=(1 2), \sigma=(123)$), why does a group homomorphism $f:S_3\to\mathbb{Z}/2\mathbb{Z}$ satisfy $f(a)=0$ for all $a\in S_3$ such that $a^2\neq e$?</p>
| Barry Cipra | 86,747 | <p>For $a\gt0$, we have $u=x^a\to0^+$ as $x\to0^+$, and therefore</p>
<p>$$\lim_{x\to0^+}x^a\ln x={1\over a}\lim_{x\to0^+}x^a\ln x^a={1\over a}\lim_{u\to0^+}u\ln u$$</p>
<p>so it remains to show that $\lim_{x\to0^+}x\ln x=0$ without resorting to L'Hopital. My favorite way of doing this is to use the integral definit... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | amWhy | 9,003 | <p>I wanted to revisit a very old question of mine, asked when I was relatively new to math.se., if nothing else, to remind myself why I participate here (and why I returned after a very long sabbatical from math.se). Two points:</p>
<p>$(1)$ I participate and returned because I learn a lot here (or rather, math.se): ... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | Ben Blum-Smith | 13,120 | <p>I would like to add:</p>
<p>I participate in math.SE because</p>
<p>(a) I have learned an incredible amount here by reading answers to others' questions and especially by asking questions. From <a href="https://math.stackexchange.com/questions/50362/homology-why-is-a-cycle-a-boundary">my very first question</a>, i... |
18,421 | <p>I am teaching 4th-grade kids. The topic is Fraction. Basic understanding of a fraction as a part of the whole and as part of the collection is clear to the kids. Several concrete ways exist to teach this basic concept. But when it comes to fraction addition/subtraction I could not find a way that teaches it concrete... | WeCanLearnAnything | 7,151 | <p>Suggestion: Do not bother having students do calculations with <em>two</em> fractions until they can do a comparable operation with a fraction and a whole number.</p>
<p>Example: Before teaching, say, <span class="math-container">$\frac{3}{2}-\frac{4}{3}=\_\_$</span>, ensure they can answer something like <span cla... |
1,703,140 | <p>Let $n$ be composite. If $a$ is coprime to $n$ such that $a^{n-1} \equiv 1 \pmod n$ then $a$ is called a Fermat liar. If $a^{n-1} \not\equiv 1 \pmod n$, then $a$ is a Fermat witness to the compositeness of $n$.</p>
<p>Assume the witnesses or liars are themselves prime. If two Fermat liars are multiplied together... | mvw | 86,776 | <p>I assume you mean a number with $N$ digits in decimal base and having the digit sum $S$.</p>
<p>It runs down to partition the number $S$ into $N$ parts (if possible at all!), where the parts are in the range $\{ 0, \dotsc, 9 \}$ and sorting it such that either the largest parts come first for a maximal number, or s... |
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