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 |
|---|---|---|---|---|
2,734,338 | <p>I would like to show that
$$\forall n\in\mathbb{N}^*, \quad \sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$$</p>
<p>I'm interested in more ways of proofing this.</p>
<p>My method :</p>
<p>suppose that $\sqrt{\frac{n}{n+1}}\in \mathbb{Q}$ then there exist $(p,q)\in\mathbb{Z}\times \mathbb{N}^*$ such that $\sqrt{\frac{n}... | Community | -1 | <p>Another approach: If $n/(n+1)$ is a square, then both $n$ and $n+1$ are squares, since they are relatively prime. Therefore, there would be two squares, which difference is $1$:</p>
<p>$$(p+q)(p-q)=p^2-q^2=1$$</p>
<p>Therefore, $p+q=1$ and $p-q=1$. It follows that $p=1$ and $q=0$. Therefore, $n=0$.</p>
|
4,038,904 | <p>Your statistics teacher challenges you to write a mathematics paper, which depending on your time spent, will earn you a cash reward. You're given the paper and a candle stands on the table in front of you. The life span <span class="math-container">$X$</span>, in minutes, of the candle is a continuous random variab... | Misha Lavrov | 383,078 | <p>About <span class="math-container">$8$</span> years ago, not too long after the linked question was asked, Code Golf StackExchange <a href="https://codegolf.stackexchange.com/questions/10739/find-largest-prime-which-is-still-a-prime-after-digit-deletion?noredirect=1&lq=1">took up the challenge</a>. The winning a... |
258,704 | <p>How can I solve a system of linear congruences as such?</p>
<p><span class="math-container">$$\begin{align*}
3x+2y+28z &= 9 \pmod {29} \\
5x+27y+z &= 9 \pmod {29} \\
2x+y+z &= 6 \pmod {29}
\end{align*}$$</span></p>
<p>I tried it this way as a system of equations, but no luck:</p>
<pre><code>eqn1 = Full... | yarchik | 9,469 | <pre><code>Solve[{3*x + 2*y + 28*z == 9 && 5*x + 27*y + z == 9 &&
2*x + y + z == 6}, Modulus -> 29]
(* {{x -> 24, y -> 23, z -> 22}} *)
Solve[3*x == 5, Modulus -> 6]
(* {} *)
Solve[x + x^2 == 2, Modulus -> 8]
(* {{x -> 1}, {x -> 6}} *)
Solve[5 x == 1, Modulus -> 4... |
3,017,928 | <p>Doing some self study from the text <em>Basic Mathematics</em> by Serge Lang
I ran into an exercise question which I can't seem to wrap my head around.
The question is:</p>
<p>Express the following expressions in the form <span class="math-container">$2^m3^na^rb^s$</span> ,where <span class="math-container">$m,n,r,... | JDMan4444 | 620,745 | <p>The key idea here is that multiplication is both commutative and associative, so we may multiply in any order and in any groupings we wish, as well as "break up" products into any groupings and any order we wish.</p>
<p>The expression <span class="math-container">$8a^2b^3(27a^4)(2^5ab)$</span>, by associativity and... |
3,017,928 | <p>Doing some self study from the text <em>Basic Mathematics</em> by Serge Lang
I ran into an exercise question which I can't seem to wrap my head around.
The question is:</p>
<p>Express the following expressions in the form <span class="math-container">$2^m3^na^rb^s$</span> ,where <span class="math-container">$m,n,r,... | Blg Khalil | 572,713 | <p>You just apply basic power rules:
<br>
<span class="math-container">$a^na^m = a^{n+m}$</span> etc ...
<br>
Let's apply them now to your example:
<span class="math-container">$$8a^2b^3(27)a^2b^3a^4ab$$</span>
<span class="math-container">$$8 (27) 2^5 a^2 b^3 a^4 ab$$</span>
<span class="math-container">$$2^3 2^5 (27)... |
2,508,381 | <p>Starting from the idea that $$\sum_{n=1}^\infty n = -\frac{1}{12}$$
It's fairly natural to ask about the series of odd numbers $$\sum_{n=1}^{\infty} (2n - 1)$$
I worked this out in two different ways, and get two different answers. By my first method
$$\sum_{n=1}^{\infty} (2n - 1) + 2\bigg( \sum_{n=1}^\infty n \bi... | Misha Lavrov | 383,078 | <p>If we replace the sum $\sum_{n=1}^\infty n$ by $\sum_{n=1}^\infty n^{-s}$ then (where it converges) we have $$\sum_{n=1}^\infty n^{-s} = \zeta(s)$$ and when $s=-1$ (outside the region where the sum converges) we have $\zeta(-1) = -\frac1{12}$. Since setting $s=-1$ turns the terms of the second sum back into the term... |
46,772 | <p>I have the following to set into mathematica. </p>
<p>Assume a stress(sigma) applied on a specimen which has a number of fibers within it. All fibers that have a strength less than the applied stress should fail. I set a cumulative distribution of the fiber strengths with a mean and standard deviation.</p>
<p>I wa... | phosgene | 11,679 | <p>Here's an implementation showing the load-carrying capacity of the bundle of fibers as the weakest remaining fiber breaks:</p>
<pre><code>nFibers = 1000;
mean = 100;
stdev = 20;
fibers = Sort@Abs@RandomVariate[NormalDistribution[mean, stdev], nFibers];
pull[{load_, fibers_}] := {
First@fibers*Length@fibers,(* loa... |
1,363,213 | <p>I am given a chessboard of size $8*8$. In this chessboard there are two holes at positions $(X1,Y1)$ and $(X2,Y2)$. Now I need to find the maximum number of rooks that can be placed on this chessboard such that no rook threatens another. </p>
<p>Also no two rooks can threaten each other if there is hole between the... | Asinomás | 33,907 | <p>Here is the drawing I meant to make for ten rooks.(holes are yellow and rooks are black)
<img src="https://i.stack.imgur.com/pc0G5.png" alt="enter image description here"></p>
|
983,566 | <p>Here is the problem in full:</p>
<blockquote>
<p>A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more than 1 marble and form two non-empty heaps from it. One will contain $n$ marbles and the other $m$ ... | Brian M. Scott | 12,042 | <p>Let’s dispose of the technical point first. By straight algebra,</p>
<p>$$k+\binom{k}2=k+\frac{k(k-1)}2=\frac{2k+k^2-k}2=\frac{k(k+1)}2=\binom{k+1}2\;.$$</p>
<p>Alternatively,</p>
<p>$$k+\binom{k}2=\binom{k}1+\binom{k}2=\binom{k+1}2$$</p>
<p>by the Pascal’s triangle identity.</p>
<p>You’re quite right that it’s... |
983,566 | <p>Here is the problem in full:</p>
<blockquote>
<p>A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more than 1 marble and form two non-empty heaps from it. One will contain $n$ marbles and the other $m$ ... | André Nicolas | 6,312 | <p><strong>Hint:</strong> We give a part-solution, to give you an opportunity to do some of the details.</p>
<p>Yes, it is by strong induction. </p>
<p>Suppose that $x=m+n$, and we break the pile of $x$ into parts of size $m$ and $n$. </p>
<p>Then we write down $mn$, and proceed to break up the $m$-pile and the $n$-... |
3,991,072 | <p>Consider, two planar vectors:</p>
<p><span class="math-container">$$V= a \hat{x} + b \hat{y}$$</span></p>
<p>And
<span class="math-container">$$ U = a' \hat{x} + b' \hat{y}$$</span></p>
<p>These are analogous to the complex numbers:</p>
<p><span class="math-container">$$ v = a + bi$$</span></p>
<p>and,</p>
<p><span... | Eric M. Schmidt | 48,235 | <p>Your proposed product is geometrical in a sense, but there is a crucial difference between it and the dot and cross products. Imagine you have a table, or some other flat horizontal surface, and two arrows drawn on it that represent vectors. Here is a question: What is the "complex number" product of these... |
2,003 | <p>I use some custom shortcut keys in <code>KeyEventTranslations.tr</code>. One is for the <code>Delete All Output</code> function: </p>
<pre><code>Item[KeyEvent["w", Modifiers -> {Control}],
FrontEnd`FrontEndExecute[FrontEnd`FrontEndToken["DeleteGeneratedCells"]]]
</code></pre>
<p>or simply:</p>
<pre><code>... | bobknight | 8,565 | <p>An alternative is to customize the file MenuSetup.tr that is in the same folder of KeyEventTranslations.tr. The difference is that there you can use KernelEvaluate to execute a NotebookDelete. Here is an example of a new menu item. Insert this code in the list of Menu into the MenuSetup.tr file and restart Mathemati... |
439,745 | <blockquote>
<p>Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$</p>
</blockquote>
<p>example1: $|x-1|+|x-2|\geq 1$</p>
<p>my solution：(substitution)</p>
<p>$x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$</p>
<p>square,</p>
<p>$t^2-2t+1\geq 1-2|t|+t^2,\text{Since} -t\leq -|t|,$</p>
<p>so proved.</p>
<p><em>ques... | marty cohen | 13,079 | <p>You want
$S(x)
=\sum_{i=1}^n |x-i|$.</p>
<p>I will show</p>
<p>$\begin{cases}
\text{if } x \le 1&S(x) \ge \dfrac{n(n-1)}{2}\\
\text{if } x \ge n &S(x) \ge \dfrac{n(n-1)}{2}\\
\text{otherwise} &S(x) \ge \dfrac{n^2-1}{4}
\end{cases}
$</p>
<p>Exact results are</p>
<p>$\begin{cases}
\text{if } x \le 1, ... |
3,484,136 | <blockquote>
<p>Show that <span class="math-container">$n^2+n$</span> is even for all <span class="math-container">$n\in\mathbb{N}$</span> by contradiction.</p>
</blockquote>
<p>My attempt: assume that <span class="math-container">$n^2+n$</span> is odd, then <span class="math-container">$n^2+n=2k+1$</span> for all <... | James Turbett | 721,003 | <p>Suppose <span class="math-container">$n^2 + n$</span> is odd for all natural numbers <span class="math-container">$n$</span>. Then <span class="math-container">$n(n+1)$</span>, the product of two consecutive integers, is also odd. However, an odd number can not have any even factors. Either <span class="math-contain... |
3,484,136 | <blockquote>
<p>Show that <span class="math-container">$n^2+n$</span> is even for all <span class="math-container">$n\in\mathbb{N}$</span> by contradiction.</p>
</blockquote>
<p>My attempt: assume that <span class="math-container">$n^2+n$</span> is odd, then <span class="math-container">$n^2+n=2k+1$</span> for all <... | fleablood | 280,126 | <p>To be different. Suppose there some natural number where this is true. But it isn't true for <span class="math-container">$1$</span> or <span class="math-container">$2$</span>. But if there is some value where it <em>is</em> true there must be some <em>first</em> value, <span class="math-container">$n$</span> whe... |
4,572,336 | <p>This is the
<a href="https://i.stack.imgur.com/BRjsf.jpg" rel="nofollow noreferrer">question</a></p>
<p>I have managed to find the formula for part a) which is n(n+1) for all natural numbers n, but I'm not sure how to prove it directly. Can I just say that the summation of 2i is the same as the sums of the summation... | Sri-Amirthan Theivendran | 302,692 | <p>It suffices to prove that
<span class="math-container">$$
1+2+\dotsb+n=\frac{n(n+1)}{2} = \binom{n+1}{2}
$$</span>
The RHS counts the <span class="math-container">$2$</span> element subsets of <span class="math-container">$A=\{0,1,\dotsc, n\}$</span>. We note that there are <span class="math-container">$i$</span> su... |
1,534,981 | <p>I am trying to verify my procedure to find if the extrema is correct for a function
$u \left( x,y \right) ={x}^{2}-{y}^{2}$ on the set $\left( \mathop{\rm D}~~=~\left\{\left(x ,y \right)
\in \mathbb{R}^{2}{\it} | x ^{2}+y ^{2}
\le 1\right\} \right)
$</p>
<p>By the closed interval method, to find the absolute maximu... | Nizar | 227,505 | <p>I would advise you, in a similar strategy, to use <a href="https://en.wikipedia.org/wiki/Lagrange_multiplier" rel="nofollow">Lagrange multiplier</a>. </p>
<p>So you first study the max and min of $f$ in the open set $x^2+y^2< 1$, as you see in your case there is no min no max of $f$ in this region, bec... |
4,050,336 | <p>I am familiar with the Negative Binomial distribution <span class="math-container">$NB(p, k)$</span>, which gives the number of failures before <span class="math-container">$k$</span> successes occur in a Bernoulli process with parameter <span class="math-container">$p$</span>. I am wondering, however, if there the ... | Richard Jensen | 658,583 | <p>You are on the right track assuming it's irreducible, but you have not reached a contradiction yet. This is because you need to use what elements in <span class="math-container">$\mathbb{Q}(i\sqrt{2})$</span> are. Notice that any element in <span class="math-container">$\mathbb{Q}(i\sqrt{2})$</span> can be written a... |
4,638,393 | <p><a href="https://i.stack.imgur.com/SgVlq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SgVlq.png" alt="enter image description here" /></a></p>
<p>How does the author obtain formula (4)? From formula (2), I only get that <span class="math-container">$u\left(\frac{k}{n}+\frac{1}{n},\cdot\right)=\... | Alex Ortiz | 305,215 | <p>Suppose that for some <span class="math-container">$k\ge 1$</span>, we have
<span class="math-container">$$
u(\frac kn, \cdot) \approx (1-\frac{c}{n}\partial_x)^kg(\cdot).\tag{$\ast$}
$$</span>
The case <span class="math-container">$k = 1$</span> holds by (2). Again by the approximation (2) and our assumption <span ... |
1,159,155 | <p>If we define <strong>open</strong> as: A set $O⊆R$ is open if for all points $a∈O$ there exists an
$\epsilon$-neighborhood $V_\epsilon(a)⊆O$.</p>
<p>Where $V_\epsilon(a) = \{x \in \mathbb{R}: | x - a | < \epsilon\}$
Now consider some open interval: </p>
<p>$(c,d) = \{x \in \mathbb{R} : c<x<d \}$</p>
<p>T... | layman | 131,740 | <p>You correctly focused on one of the end points. For every point that is not an end point in the interval $[1,10]$, your argument for finding a ball around that point contained in $[1,10]$ works.</p>
<p>Now, if $x = 10$, we need to show that <em>every</em> ball around $10$ is <em>not</em> contained in $[1, 10]$. O... |
4,544,787 | <p>These sums showed up in a probability problem I was working on. They're not quite the Stirling numbers of the first kind since it's possible to have e.g. <span class="math-container">$i_1 = i_2$</span>. Denoting the sum by <span class="math-container">$(k\mid n)$</span> we have the recurrence relation</p>
<p><span c... | Marko Riedel | 44,883 | <p>We get from first principles the generating function where we mark
multisets containing some number of instances of <span class="math-container">$q$</span> giving the factor
being contributed and the exponent of <span class="math-container">$z$</span> how often it occurs:</p>
<p><span class="math-container">$$[z^k]... |
3,511,696 | <p>First off all, i looked to n=1. So, we had <span class="math-container">$\epsilon<x_1<1$</span>. Then, i'm start to think that the <span class="math-container">$\sqrt[n]{x_n}$</span> is in <span class="math-container">$(\epsilon,1)$</span> interval and is monotonous crescent(so, converges). But it's not so cle... | Mercy King | 23,304 | <p>For every positive integer <span class="math-container">$n$</span> the function <span class="math-container">$f_n$</span> defined by
<span class="math-container">$$f_n(x)=\sqrt[n]{x}$$</span>
is increasing over <span class="math-container">$(0,\infty)$</span>, therefore from the inequalities
<span class="math-conta... |
3,511,696 | <p>First off all, i looked to n=1. So, we had <span class="math-container">$\epsilon<x_1<1$</span>. Then, i'm start to think that the <span class="math-container">$\sqrt[n]{x_n}$</span> is in <span class="math-container">$(\epsilon,1)$</span> interval and is monotonous crescent(so, converges). But it's not so cle... | Hagen von Eitzen | 39,174 | <p>Let <span class="math-container">$h>0$</span>. Suppose <span class="math-container">$\sqrt[n]{x_n}>1+h$</span> for infinitely many <span class="math-container">$n$</span>. Then for these <span class="math-container">$n$</span>, <span class="math-container">$$\begin{align}x_n-n^k&>(1+h)^n-n^k\\&\ge ... |
4,517,429 | <p>Let <span class="math-container">$ {\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$</span> be a sequence of independent random variables, each of those random variable follow a Gamma distribution.</p>
<p>For the summation of those random variable:</p>
<p><span class="math-container">$ {\displaystyle {\bar {X}}_{n}\equi... | Henry | 6,460 | <p>Assuming the rate (or scale) parameters are all the same, then you are correct:</p>
<ul>
<li>the sum of independent gamma distributed random variables has a gamma distribution, with shape parameter equal to the sum of the shape parameters of the individual distributions</li>
</ul>
<p>But gamma distributions with a l... |
1,527,891 | <p>Determine (up to a constant multiplier) the polynomial with a maximum at $(-1,1)$, a minimum $(1,-1)$ and no other critical points.</p>
<p>The only thing I can think of is coming up with an equation with roots $1$ and $-1$ and then integrating it but I don't think that will work.</p>
| Balloon | 280,308 | <p><strong>Hint :</strong> try with your idea : you know that the derivative of your polynomial is $k(x-1)(x+1)=k(x^2-1).$ Integrate this and get that your polynomial is $k(\frac{1}{3}x^3-x)+c.$ Then use your informations about the values of your polynomial at $-1$ and $1$ to find $k$ and $c$ and finally get $P=\frac{1... |
3,566,603 | <p>I need a simple way to show <span class="math-container">$\mathbb R^2$</span> is not isomorphic <span class="math-container">$\mathbb{R}[x]/(x^2)$</span>. Both are not integral domains, and both are not fields, so I’m not sure how to go about it.</p>
| Vincent | 101,420 | <p>Everything you want to know about <span class="math-container">$\mathbb{R}^2$</span> can be understood through understanding the two elements <span class="math-container">$e_1 = (1, 0)$</span> and <span class="math-container">$e_2 = (0, 1)$</span>. So let's look at their properties. Both satisfy <span class="math-co... |
3,566,603 | <p>I need a simple way to show <span class="math-container">$\mathbb R^2$</span> is not isomorphic <span class="math-container">$\mathbb{R}[x]/(x^2)$</span>. Both are not integral domains, and both are not fields, so I’m not sure how to go about it.</p>
| Andrea Mori | 688 | <p>I think that there are several ways to see this.</p>
<p>One way is to look at ideals and for you may recall that given a quotient <span class="math-container">$A/I$</span> there is a one to one correspondence between ideals of <span class="math-container">$A/I$</span> and ideals of <span class="math-container">$A$<... |
13,843 | <p>We have a natural number $n>1$. We want to determine whether there exist
natural numbers $a, k>1$ such that $n = a^k$. </p>
<p>Please suggest a polynomial-time algorithm.</p>
| Pace Nielsen | 3,199 | <p>This can be done in "essentially linear time." Check out Daniel Bernstein's website: <a href="http://cr.yp.to/arith.html">http://cr.yp.to/arith.html</a></p>
<p>Especially note his papers labeled [powers] and [powers2].</p>
|
1,212,336 | <p>Let R be the set of all real numbers. Is $\{\mathbb R^+,\mathbb R^−,\{0\}\}$ a partition of $\mathbb R$? Explain your answer.</p>
<p>My answer is no because of $\{0\}$. I am confused with $\{0\}$. please help.</p>
| N. S. | 9,176 | <p><strong>Hint 1:</strong> $\int_1^n \log x \, dx =\sum_{k=2}^n \int_{k-1}^k \log x \, dx $</p>
<p><strong>Hint 2:</strong> If $x \in [k-1,k]$ then $\log(x) \leq \log(k)$.</p>
|
3,959,178 | <p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$\mathbb k$</span> a field. Then, let <span class="math-container">$\mathfrak 1$</span> be the trivial <span class="math-container">$\mathbb k G$</span>-module.</p>
<p>According to Lorenz's A Tour of Representation Theory, he sta... | Jendrik Stelzner | 300,783 | <p>One can see in the following way that every <span class="math-container">$G$</span>-representation (over <span class="math-container">$\mathbb{k}$</span>) is projective.</p>
<p><strong>Step 1.</strong>
For every two representations <span class="math-container">$V$</span> and <span class="math-container">$W$</span> o... |
656,701 | <p>Suppose we have:</p>
<p>$A = \{(x,v,w):x+v=w\}$</p>
<p>$B = \{(x,v):x=v\}$</p>
<p>$C = \{(w,u):\exists x 2x=w\}$</p>
<p>Can we say that $C = A \cup B$?</p>
| Unwisdom | 124,220 | <p>I don't really understand what you are asking here. </p>
<p>The usual way to define a set with this kind of "set builder notation" is to specify a universe $U$, and then use the following form:
$$
\{x : \phi(x)\}
$$
where $\phi$ is a formula with one free variable that is either true or false at each $x\in U$. </p... |
907,893 | <p>I wanted to know about this convention :</p>
<p>By rate of growth of R, we normally mean : (change in R) / (change in Time)</p>
<p>But Rate of growth of a geometric sequence "a(1+r)^n" is r, which is strange i feel</p>
<p>I am kind of confused, can anyone clear it </p>
| dylan7 | 163,751 | <p>Since the geometric series, or their partner the continuous exponential have varying rates of change, it is nice to find something consistant within them we can call a constant rate. The actual rate of change is the derivative $ df/dx $ which is discrete in the sense of a series. These rates of change constantly cha... |
109,989 | <p>A problem in my book is:</p>
<blockquote>
<p>Let the edges of $K_7$ be colored with the colors red and blue. Show that there are at least four subgraphs $K_3$ with all three edges the same color (monochromatic triangles). Also show that equality can occur.</p>
</blockquote>
<p>By the <a href="http://en.wikipedia... | Gerry Myerson | 8,269 | <p>A "biangle" is a triple of vertices $(a,b,c)$ where the edge joining $a$ to $b$ is not the same color as the edge joining $b$ to $c$. We call $b$ the "apex" of the biangle. </p>
<p>A vertex with 3 red and 3 blue edges is the apex of 9 biangles; any other color distribution leads to fewer biangles. Thus, the whole g... |
109,989 | <p>A problem in my book is:</p>
<blockquote>
<p>Let the edges of $K_7$ be colored with the colors red and blue. Show that there are at least four subgraphs $K_3$ with all three edges the same color (monochromatic triangles). Also show that equality can occur.</p>
</blockquote>
<p>By the <a href="http://en.wikipedia... | Josias Oberlin | 829,443 | <p>Gerry Myerson's answer is very good. However, I wanted to offer a different, more simple-minded answer.</p>
<p>To begin with, my answer requires that I show every <span class="math-container">$K_6$</span> has two monochromatic triangles. The proof is as follows:</p>
<p>Suppose there is only one monochromatic triangl... |
1,290,516 | <p>Find the values of $m$ if the line $y=mx+2$ is a tangent to the curve $x^2-2y^2=1$.</p>
<p>My working:</p>
<p>First we differentiate $x^2-2y^2=1$ with respect to $y$ to get the gradient. We get $y^2=\frac{1}{2}x^2-\frac{1}{2}\implies y=\pm\sqrt{\frac{1}{2}x^2-\frac{1}{2}}$.</p>
<p>We take the positive one for dem... | Ross Millikan | 1,827 | <p>You have $$(1-2m^2)x^2=-2m^2\\(1-2m^2)x^2=9+8mx\\-2m^2=9+8mx\\x=\frac{-2m^2-9}{8m}\\x^2=\frac{-2m^2}{1-2m^2}\\\frac{4m^4+36m^2+81}{64m^2}=\frac{-2m^2}{1-2m^2}$$Now you can use the quadratic equation in $m^2$</p>
|
521,740 | <p>$x$,$y$ are real numbers satisfying $(x-1)^{2}+4y^{2}=4$<br>
find the maximum of $xy$ and justify it without calculus.<br>
Does there exist a tricky solution using elementary inequalities (AM-GM or Cauchy-Schwarz) ?</p>
<p>I tried and got it's when $x=\dfrac{3+\sqrt{33}}{4}$</p>
| Betty Mock | 89,003 | <p>This is a circle of radius 2 centered at (1,0). If you let x-1 = 2cosu and y = 2sinu then xy - (x-1)y + y =4(cosu)(sinu) +2sinu = 2sin2u + 2 sinu = 2(sin2u + sinu).</p>
<p>Since sin2u < 0 for $\pi/2 < u <\pi$ and sinu = < 0 for $\pi < u < 2\pi$, the maximum is going to occur in the first quadra... |
1,131,622 | <p>The question itself is a very easy one:<br/></p>
<blockquote>
<p>Somebody has got two kids, one of whom is a girl. Then what's the probability that he's got <strong>at least</strong> one boy?</p>
</blockquote>
<p>My answer is that, since he's already got a girl, then "he's got at least one boy" amounts to "the o... | turkeyhundt | 115,823 | <p>Imagine all the people in the world with two kids separated into 4 groups. Those with (B,B), (B,G), (G,B), (G,G). </p>
<p>How many of them could say one of their kids is a girl? (75%) And of those, what is the probability that the other child is a boy? You see that it is $\frac{2}{3}$</p>
<p>ETA: Note your li... |
73,238 | <p>How can I calculate the solid angle that a sphere of radius R subtends at a point P? I would expect the result to be a function of the radius and the distance (which I'll call d) between the center of the sphere and P. I would also expect this angle to be 4π when d < R, and 2π when d = R, and less than 2π when d ... | Narasimham | 95,860 | <p>Using standard spherical coordinates for a sphere surface $( a,\theta,\phi)$ we have parametrization for axis on z-axis:</p>
<p>$$ (x,y,z) = (a \cos \phi \cos \theta +h, a \cos \phi \sin \theta+k, a \sin \phi+l) $$</p>
<p>for a sphere whose center is displaced from origin to $(h,k,l).$</p>
<p>A great circle is f... |
1,601,970 | <p>I want to use proof by contradiction.</p>
<p>Suppose that real numbers are bounded, then according to the axiom of continuity, there exists a least upper bound $b$.</p>
<p>But if $x\in \Bbb R$, then $x+1\in \Bbb R$ because of the inclusion property of real numbers.</p>
<p>But $x+1\in \Bbb R\Longrightarrow x+1\leq... | CommonerG | 74,357 | <p>Suppose there is such upper bound $r\in\mathbb{R}$. </p>
<p>$r+1>r$ and $r+1\in \mathbb{R}$ since $\mathbb{R}$ is closed under addition. </p>
|
1,601,970 | <p>I want to use proof by contradiction.</p>
<p>Suppose that real numbers are bounded, then according to the axiom of continuity, there exists a least upper bound $b$.</p>
<p>But if $x\in \Bbb R$, then $x+1\in \Bbb R$ because of the inclusion property of real numbers.</p>
<p>But $x+1\in \Bbb R\Longrightarrow x+1\leq... | mvw | 86,776 | <p>The Archimedian axiom states:</p>
<blockquote>
<p>For every $x, y \in \mathbb{R}$ with $0 < x < y$ there exists a $n \in
\mathbb{N}$ such that $y < n x$.</p>
</blockquote>
<p>Of course $n x \in \mathbb{R}$, so there is no largest real number $y$.</p>
|
2,708,891 | <p>Let $R$ be a ring and consider $f = r_nx^n + 1.x^{n-1} + \cdots + rx + r_0\; \in R[X]$ such that $r^n = 0$ for all $r \in R$. Then can I call $f$ a monic polynomial in $R[X]$ (assume $r_n$ is non-invertible)?</p>
| David C. Ullrich | 248,223 | <p>No. A polynomial over $R$ is not the same thing as the function from $R$ to $R$ defined by a polynomial. In particular, if we assume that $r^3=0$ for every $r\in R$ then $$p(x)=2x^3+x^2$$and $$q(x)=x^2$$ define the same function, but (assuming that $2\ne0$ in $R$) they are different polynomials, and in fact $q$ is m... |
858,494 | <p>Where does the definition of the $L_\infty$ norm come from?</p>
<p>$$\|x\|_\infty=\max \{|x_1|,\dots,|x_k|\}$$</p>
| robjohn | 13,854 | <p><strong>General Approach</strong></p>
<p>As shown in <a href="https://math.stackexchange.com/a/833397">this answer</a>,
$$
\lim_{p\to\infty}\left(\int_A|f(x)|^p\,\mathrm{d}x\right)^{1/p}=\sup_{x\in A}|f(x)|\tag{1}
$$
Using a discete measure, $(1)$ shows that
$$
\lim_{p\to\infty}\left(\sum_{k=1}^n|x_k|^p\right)^{1/p... |
858,494 | <p>Where does the definition of the $L_\infty$ norm come from?</p>
<p>$$\|x\|_\infty=\max \{|x_1|,\dots,|x_k|\}$$</p>
| copper.hat | 27,978 | <p>The following adds nothing to the above proofs other than a slightly geometric flavour. It is too long for a comment, so I am including it as an answer.</p>
<p>I am taking the space to be $\mathbb{C}^n$.</p>
<p>It is not hard to show (see <a href="https://math.stackexchange.com/a/424335/27978">https://math.stackex... |
3,066,020 | <p>I’m reading Hans Kurzweil ‘s “The Theory of Finite Groups”, where it says</p>
<blockquote>
<p>1.6.4 Let <span class="math-container">$N_1, . . . , N_n$</span> be normal subgroups of <span class="math-container">$G$</span>. Then the mapping <span class="math-container">$$α: G→G/N_1\times ··· \times G/N_n$$</span> ... | Community | -1 | <p>Given two groups <span class="math-container">$G_1$</span> and <span class="math-container">$G_2$</span>, you can form their direct product: <span class="math-container">$G_1×G_2$</span>, to be <span class="math-container">$\{(g_1,g_2)\mid g_1\in G_1,g_2\in G_2\}$</span>, with the group operation defined as <span ... |
3,329,363 | <p>Sorry for the long text; this is a nebulous question that has always been in the back of my mind, and I've had trouble putting into a short form.</p>
<hr>
<p><strong>"Natural" Definition</strong></p>
<p>If someone on the street hears the word "permutation," I think they will naturally assume that a permutation:</... | Aloizio Macedo | 59,234 | <p>"Rearranging" objects is a procedure that is not inherently related with order. The relation with order is more due to how we usually list things than the essence of permutation itself.</p>
<p>For instance, I have three pockets in my pants, on which I put my cell phone, keys and wallet, one on each pocket. I someti... |
1,239,211 | <p>I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask about.</p>
<p>The first one is probably straightforward and I'm guessing is related to Whitehead-type theorems. Still, I... | Qiaochu Yuan | 232 | <p>I think the most basic things you can say here have nothing much to do with homotopy theory in particular. It's a general feature of many familiar categories that every object is a filtered colimit of "small" objects which are easier to understand: for example, every group is a filtered colimit of finitely generated... |
2,263,759 | <p>Let $E$ be a vector space and $\varphi: E \to E$ be a linear map. Let $x, y \in E \setminus \{0\}$ and $\lambda, \mu \in F$ such that
$\varphi(x) = \lambda x$ and $\varphi(y) = \mu y$. Prove that if $\lambda \neq \mu$ then $\{x, y\}$ is linearly independent.</p>
<p>This proof seems like it should be on the simpler ... | florence | 343,842 | <p>Suppose that $\{x,y\}$ is linearly dependent. Then either $x$ is a multiple of $y$ or vice versa. Without loss of generality, assume that $x = cy$. Then
$$\lambda x = \phi(x) = \phi(cy) = c\phi(y) = c\mu y = \mu (cy) = \mu x$$
Thus, we have that $\mu x = \lambda x$. Since $x\neq 0$, it follows that $\mu = \lambda$. ... |
920,429 | <p>Given that $x(t)=(c_1+c_2 t + c_3 t^2)e^t$ is the general solution to a differential equation, how do you work backwards to find the differential equation? </p>
| AgentS | 168,854 | <p>Say $f(t) = c_1 + c_2t + c_3t^2$<br>
the given general solution is $$x(t) = fe^t$$
Since you have $3$ arbitrary constants, the required DE must be of order $3$. So you need to differentiate exactly 3 times :
$$\begin{align} x' &= (f'+f)e^t \\ x''&=(f''+2f'+f)e^t \\x'''&=(f''' + 3f''+3f'+f)e^t\end{alig... |
844,355 | <p>Let $T\colon V \rightarrow W$ a linear transformation between the real vector spaces $V$ and $W$ both with finite dimension.</p>
<p>How can i prove that $\dim(V) = \dim T(V) + \dim T^{-1}(0)$.</p>
<p>I can't understand this problem and how to solve it , if you can help me please.</p>
| egreg | 62,967 | <p>Hint: </p>
<ol>
<li><p>consider $\mathcal{A}=\{v_1,v_2,\dots,v_k\}$ such that $\{T(v_1),T(v_2),\dots, T(v_k)\}$ is a basis of $T(V)$;</p></li>
<li><p>consider a basis $\mathcal{B}=\{u_1,u_2,\dots,u_h\}$ of $\ker T=T^{-1}(0)$;</p></li>
<li><p>prove that $\mathcal{A}$ and $\mathcal{B}$ are disjoint;</p></li>
<li><p>p... |
2,200,034 | <p>Suppose the points A and B are connected by two roads of the length $S_1$ and $S_2$. Cars can drive from A to B on either of the two roads. They start at point A and must decide, <strong>one after the other</strong> which road to take. They know how many cars already chose to drive on each road.</p>
<p>The speed of... | samerivertwice | 334,732 | <p>$$\lim_{N\to\infty}\frac{1}{\sqrt{N}}=0$$</p>
<p>Therefore the cars are stationary</p>
<p>But they are arriving infinitely many per second, so there is a pile up of infinitely many cars every instant.</p>
<p>100% logical drivers cannot make an instant decision in the face of an illogical universe containing an in... |
3,103,991 | <p>Suppose <span class="math-container">$X$</span> is a topological space, <span class="math-container">$C$</span> is a closed subset of <span class="math-container">$X$</span>, <span class="math-container">$U$</span> is an open subset of <span class="math-container">$X$</span> and <span class="math-container">$U$</spa... | Henno Brandsma | 4,280 | <p>No, take <span class="math-container">$C=\mathbb{Z}$</span> (closed in <span class="math-container">$X=\mathbb{R}$</span> (usual topology)) and <span class="math-container">$U = \mathbb{R}\setminus{Z}$</span> which is open and dense. <span class="math-container">$C \cap U=\emptyset$</span> is not dense in <span clas... |
1,919,159 | <p>I don't get this step in proof of <a href="https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)" rel="nofollow">Carathéodory's theorem (convex hull)</a>
Why:</p>
<blockquote>
<p>Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points $x_2 − x_1, ..., x_k − x_1$ are linearly... | Ben Grossmann | 81,360 | <p>Note that $k > d+1$, and that our points are vectors in $\Bbb R^d$. In $\Bbb R^d$ (or any $d$-dimensional vector space), any set consisting of more than $d$ vectors is linearly dependent.</p>
|
622,090 | <p>We are asked to solve the following linear system</p>
<p>$$x_1-3x_2+x_3=1$$
$$2x_1-x_2-2x_3=2$$
$$x_1+2x_2-3x_3=-1$$</p>
<p>by using gauss-jordan elimination method. The augmented matrix of the linear system is $$\left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\2 & -1 & -2 & 2 \\1 & 2 & -... | William Ballinger | 79,615 | <p>You can conclude that the system is inconsistent, because the last row of your matrix implies that $0x_1+0x_2+0x_3=-2$, which cannot be satisfied.</p>
|
1,265,531 | <p>I understand the question but I am not sure how to solve it. For example, if we flip HHHTTTTT then the next three must be heads because of the question. This however seems counterintuitive. I believe that there are $2^{10}$ possible strings, but I am unsure of how to count all possible strings that begin with HHH.</... | user3856370 | 237,743 | <p>There are $\frac{10!}{5!5!}$ ways of getting an equal number of heads and tails. The first three are fixed at heads so we need 2 more heads and 5 tails in the remaining seven tosses this can be done in $\frac{7!}{2!5!}$ ways. The probability is therefore
$$
{\frac{7!}{2!5!}\over\frac{10!}{5!5!}} = \frac{1}{12}
$$</p... |
1,874,914 | <p>in order to find $e^{AT}$ We can't just take the exponential of A as we would do in its diagonalized form. We need to diagonalize $A=S^{-1}e^{\delta(t)}S$ in order to find $e^{AT}$ why is this the case? I know we can't take the exponential of the matrix right away, do we need to take the exponential of the diagonal ... | John Hughes | 114,036 | <p>In general,
$$
e^M = I + M + \frac{M^2}{2!} + \frac{M^3}{3!} + \ldots
$$
but computing this sum involves multiplying $M$ by itself many many times and summing up the results and is generally computationally horrible, even if we only go out 10 terms or so. </p>
<p>But observe that
\begin{align}
Q^{-1}M^2Q
&= Q^... |
299,170 | <p>Given the following integer programming formulation, how can I specify that the variables are unique and none of them has the same value as the other one. basically <code>x1</code>, <code>x2</code>, <code>x3</code> , and <code>x4</code> need to get only one unique value from 1, 2, 3 or 4. and same applies to <code>y... | dexter04 | 48,371 | <p>For every pair of variables $x_i,x_j$, define a new variable $z_{ij} = |x_i-x_j|$.</p>
<p>Note that the conditions $$z_{ij} \ge x_i-x_j \\z_{ij} \ge x_j-x_i$$ are enough to define $z_{ij}$. </p>
<p>Now, impose the extra condition $z_{ij}>0$ to keep $x_i,x_j$ distinct. You have to do this for all possible pairs ... |
299,170 | <p>Given the following integer programming formulation, how can I specify that the variables are unique and none of them has the same value as the other one. basically <code>x1</code>, <code>x2</code>, <code>x3</code> , and <code>x4</code> need to get only one unique value from 1, 2, 3 or 4. and same applies to <code>y... | Erick Wong | 30,402 | <p>One way to make this work is to change the underlying representation. Instead of your basic variables being $x_i$ and $y_i$, use binary indicator variables to indicate which of $\{1,2,3,4\}$ is assigned to $x_i$.</p>
<p>For instance, with four variables $b_{i1}, b_{i2}, b_{i3}, b_{i4}$, if we constrain each $0 \le... |
995,775 | <p>So we have the regular $\delta$-$\epsilon$ definition of continuity as: </p>
<p>(1) For all $\epsilon>0$, there exists a $\delta>0$ such that, if $|x-a|<\delta$, then $|f(x)-f(a)|<\epsilon$.</p>
<p>My question is why is the following definition incorrect?</p>
<p>(2) There exists a $\delta>0$ such t... | rocinante | 117,337 | <p>The definition is incorrect because the order in which the values are picked matters. I'm glad you asked this because this is not nearly hammered as much as it should be to get people to understand and appreciate what is going on the first time around.</p>
<p>First you pick $\epsilon$. THEN you pick $\delta$. The $... |
2,706,776 | <p>In solving the wave equation
$$u_{tt} - c^2 u_{xx} = 0$$
it is commonly 'factored'</p>
<p>$$u_{tt} - c^2 u_{xx} =
\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg)
\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)
u = 0$$</p>
<p>to get
$$u(x,t) = f(x+ct) + g(x-c... | Community | -1 | <p>1) differentials are linear operators. if you are working on a space of differntiable functions they satisfy $\frac{\partial}{\partial x}\frac{\partial}{\partial y} = \frac{\partial}{\partial y} \frac{\partial}{\partial x}$. so yes, this is legit.</p>
<p>2) to prove they can be individually set to zero, put them se... |
2,416,446 | <p>I need to prove the following inequality: </p>
<blockquote>
<p>$$\sum_{n=m+1}^\infty \frac{1}{n^2}\leq \frac1m$$</p>
</blockquote>
<p>But I'm stuck with it. I found online geometric justifications for this but I'd really appreciate to see actual proof. Any hints? </p>
| Tsemo Aristide | 280,301 | <p>Hint: Compare with the integral $\int_{m+1}^{\infty} {1\over x^2}dx$.</p>
|
2,767,679 | <blockquote>
<p>Let $A$ be an $m \times n$ matrix and let $B, C$ be $n \times p$ matrices. Prove that $A(B + C) = AB + AC$</p>
</blockquote>
<p>I know it's obvious that it is and that every mathematician takes this for granted but I've been asked to prove it and I don't know how to do it without just multiplying out... | José Carlos Santos | 446,262 | <p>Well, what I would do would be tu compute $A(B+C)$ and $AB+AC$ and to see that the result is the same.</p>
<p>Otherwise, you can use the fact that $A$, $B$, and $C$ are matrices of linear maps $f_A$, $f_B$, and $f_C$, that$$f_C\circ(f_B+f_C)=f_A\circ f_C+f_A\circ f_C$$and that the matrix of the composition of two l... |
4,032,983 | <p>I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.</p>
<p>Let me give my websites and why I use them:</p... | kimchi lover | 457,779 | <p>I have found these extremely useful, from time to time:</p>
<p><a href="https://oeis.org/" rel="noreferrer">Online Encyclopedia of Integer Sequences</a></p>
<p><a href="https://dlmf.nist.gov/" rel="noreferrer">NIST Digital Library of Mathematical Functions</a></p>
<p>They both provide facts of a specialized nature, ... |
4,032,983 | <p>I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.</p>
<p>Let me give my websites and why I use them:</p... | Danilo Gregorin Afonso | 225,911 | <p>There is a website devoted to counterexamples:
<a href="http://www.mathcounterexamples.net/?fbclid=IwAR0dZKurrSUV1ft8AeW-Gh189x58BdjGZ6M2KzeeyiHJQ6grdCuCgLdGUgw" rel="noreferrer">http://www.mathcounterexamples.net/?fbclid=IwAR0dZKurrSUV1ft8AeW-Gh189x58BdjGZ6M2KzeeyiHJQ6grdCuCgLdGUgw</a></p>
<p>The Encyclopedia of Tr... |
4,032,983 | <p>I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.</p>
<p>Let me give my websites and why I use them:</p... | MathIsFun7225 | 424,950 | <p><a href="https://proofwiki.org/wiki/Main_Page" rel="noreferrer">ProofWiki</a> is an online collection of proofs that I have found to be useful at times. Each page contains links to definitions and lemmas that are used in the proof, as well as a list of external sources on the topic. More importantly, there is a list... |
1,996,290 | <p>I don't know how to solve the following integral:</p>
<p>$$ \frac{2\pi}R\int_{r_1}^{r_2}r(r+R-|r-R|)dr $$</p>
<p>I have solved it when $R \le r_1$ and $R \ge r_2$ but I need the answer for $ r_1\lt R \lt r_2$. <br/>
R is a constant as well as $r_1$ and $r_2$.</p>
<p>I appreciate your help.</p>
| Dean C Wills | 366,201 | <p>A good thing to notice is $$r+R-|r-R|=2\min(r,R)$$.</p>
|
2,193,550 | <p>Prove that $G$ acts faithfully on $X$ when there are no two elements of $G$ acting the same way on an element $X$.</p>
<p>So I don't have much of a proof, but here's what I'm thinking. I know that for $G$ to act faithfully on $X$, the identity is the only element that fixes every element in $X$, so $\forall x \in ... | Jack D'Aurizio | 44,121 | <p>$-2$ is not a quadratic residue $\!\!\pmod{5}$, hence $x^2+2$ is irreducible over $\mathbb{F}_5$ and $\mathbb{F}_5[x]/(x^2+2)$ is a finite field isomorphic to $\mathbb{F}_{5^2}$. The multiplicative part of a finite field is a cyclic group, hence every element of $\mathbb{F}_5[x]/(x^2+2)$ has an order that is a divis... |
3,234,217 | <p>Let <span class="math-container">$a,b,c \in \mathbb{R},$</span> <span class="math-container">$\vec{v_1}=\begin{pmatrix}1\\4\\1\\-2 \end{pmatrix},$</span> <span class="math-container">$\vec{v_2}=\begin{pmatrix}-1\\a\\b\\2 \end{pmatrix},$</span> and <span class="math-container">$\vec{v_1}=\begin{pmatrix}1\\1\\1\\c \en... | egreg | 62,967 | <p>The matrix
<span class="math-container">$$
\begin{pmatrix}
1 & -1 & 1 &0\\
4 & a & 1 &0\\
0& b+1 & 0 &0\\
0 & 0 & c+2 &0\\
\end{pmatrix}
$$</span>
is not in row echelon form. Sum to the second column the first multiplied by <span class="math-container">$-4$</sp... |
122,776 | <p>Let $\Gamma$ be a lattice in a (real or p-adic) Lie group.
Is it true that for a given natural number $n$ there exists a finite index subgroup $\Sigma\subset\Gamma$ such that each $\sigma\in\Sigma$ is an $n$-th power of some element of $\Gamma$?</p>
<p>In other words, is it true that for given $\sigma\in\Sigma$ the... | Venkataramana | 23,291 | <p>The answer is no, even in higher rank groups. For example, take $\Gamma = SL_3({\mathbb Z})$. If such a $\Sigma $ existed for any $n$, then its completion in the profinite (same as congruence) completion of $\Gamma$ would have this property that every element in $\Sigma$ would be an $n$-th power. But the completion ... |
3,531,693 | <p>Let <span class="math-container">$A \subset \mathbb{R}^n$</span> be a compact set with positive Lebesgue measure on <span class="math-container">$\mathbb{R}^n$</span>. Can we find an open set <span class="math-container">$B \subset \mathbb{R}^n$</span> such that <span class="math-container">$B \subset A$</span>?</p>... | Thorgott | 422,019 | <p>Check out the <a href="https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set" rel="noreferrer">Smith-Volterra-Cantor set</a> or variations thereof. Note that <span class="math-container">$B=\emptyset$</span> is, of course, always possible.</p>
|
147,095 | <p>I was wondering if there is any stationary distribution for bipartite graph? Can we apply random walks on bipartite graph? since we know the stationary distribution can be found from Markov chain, but we have two different islands in bipartite graph and connections occur between nodes from different groups. </p>
| Ben Barber | 25,485 | <p>There is a distribution such that, if you start a random walk in that distribution, then it will remain there for all time: the usual formula, $\pi(v) = d(v)/2e(G)$, still works. What is no longer true is that the random walk will converge to the stationary distribution as time tends to infinity independent of the ... |
3,072,720 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be independent random variables with uniform distribution between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>, that is, have joint density <span class="math-container">$f_{xy}(x, y) = 1$</s... | Rohit Pandey | 155,881 | <p><span class="math-container">$$X = 2W-Y$$</span></p>
<p>So, </p>
<p><span class="math-container">$$X_c = (X|W=w) = (2w-Y_c) $$</span></p>
<p>Hence, <span class="math-container">$X_c$</span> should also be uniform from <span class="math-container">$[\max(2w-1,0),\min(2w,1)]$</span>.</p>
<hr>
<p>EDIT: Here is an ... |
3,550,942 | <p>I am making a symbolic computation library which supports symbolic polynomials (both univariate and multivariate) and among other things I would like to support (possibly truncated) nth root (radical) computation of a (univariate or multivariate) polynomial since this computation is needed for some other things.</p>... | Community | -1 | <p><strong>Suggestion:</strong></p>
<p>Write <span class="math-container">$p(x)$</span> as <span class="math-container">$p_0(1+x\,q(x))$</span> and compute</p>
<p><span class="math-container">$$\sqrt[n]{p(x)}=\sqrt[n]{p_0}\sum_{k=0}^d\binom{1/n}kx^k(q(x))^k.$$</span></p>
<p>You don't need to fully compute the powers... |
3,550,942 | <p>I am making a symbolic computation library which supports symbolic polynomials (both univariate and multivariate) and among other things I would like to support (possibly truncated) nth root (radical) computation of a (univariate or multivariate) polynomial since this computation is needed for some other things.</p>... | Nikos M. | 139,391 | <p>I managed to decipher the <strong>square root algorithm</strong> referenced in the question and <strong>generalise it to n-th root algorithm</strong>.</p>
<p><strong>Algorithm: Compute nth root/radical of polynomial <span class="math-container">$p(X) = \sum_{i}a_iX^i$</span></strong></p>
<p><em>Preliminary:</em> A... |
4,586,527 | <p>Consider the following problem.</p>
<p><a href="https://i.stack.imgur.com/uzgHX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uzgHX.png" alt="enter image description here" /></a></p>
<p>I'm fairly new to probability theory, but have some experience with combinatorics. For that reason, after fail... | user2661923 | 464,411 | <p>Shortcut.</p>
<p>With a sample space of <span class="math-container">$~\displaystyle \binom{4}{2}~$</span> there is only one way (out of <span class="math-container">$6$</span>) that exactly <span class="math-container">$2$</span> tests are made.</p>
<p>There is a <span class="math-container">$~\displaystyle \frac{2... |
15,237 | <p><a href="https://matheducators.stackexchange.com/questions/176/knowing-mathematics-does-not-translate-to-knowing-to-teach-mathematics-why">A question</a> has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with... | kcrisman | 1,608 | <p>I will speak purely anecdotally.</p>
<blockquote>
<p>However, how does having a master's or doctorate degree help one's teaching in calculus, if any at all?</p>
</blockquote>
<p>It isn't the degree per se that helps, but rather the process of having to learn, relearn, and reformulate gargantuan amounts of mathem... |
285,227 | <p>I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$.</p>
<p>I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$
I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{n=0}^\infty c_n(x-a)^n$ and $g(x) = \sum_{k=0}^\infty d_n(x-a)^n$ then
$$
f(x)g(x) = \sum_{n=0}^\infty e_n(x-a... | pre-kidney | 34,662 | <p>\begin{align} \exp(x+y)&=\sum_n\frac{(x+y)^n}{n!} \\\\ &=\sum_{n}\frac{1}{n!}\sum_{a+b=n} {n \choose a} x^ay^b \\\\
&= \sum_{n}\frac{n!}{n!}\sum_{a+b=n}\frac{x^a}{a!}\frac{y^b}{b!} \\\\
&= \sum_{a,b} \frac{x^a}{a!}\frac{y^b}{b!} \\\\
&= \exp(x)\cdot\exp(y)
\end{align}</p>
|
285,227 | <p>I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$.</p>
<p>I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$
I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{n=0}^\infty c_n(x-a)^n$ and $g(x) = \sum_{k=0}^\infty d_n(x-a)^n$ then
$$
f(x)g(x) = \sum_{n=0}^\infty e_n(x-a... | kalalele | 460,776 | <p>Assume the function $\frac{exp(x+y)}{exp(x)}$, where $y$ is some constant. Differentiate with respect to $x$ and find that the derivative is zero. Therefore $\frac{exp(x+y)}{exp(x)}=c\in\mathbb{R}$. From the fact that $exp(0)=1$ we get $exp(y)=c$ which proves what was asked.</p>
|
3,395,910 | <p>I understand how to apply the trapezoidal rule to approximate the area under a curve.</p>
<p>But I'm not sure how to apply it when approximating areas between two functions. </p>
<ul>
<li>Do you use the formula like how you normally would, except apply it to the <strong>first function - the second function</strong... | Fred | 380,717 | <p>Let <span class="math-container">$(x_n)$</span> be a convergent sequence in <span class="math-container">$T$</span> with limit <span class="math-container">$x_0$</span>. You have to show that <span class="math-container">$x_0 \in T$</span>.</p>
<p>To this end use</p>
<ol>
<li><p><span class="math-container">$f(x_n... |
3,395,910 | <p>I understand how to apply the trapezoidal rule to approximate the area under a curve.</p>
<p>But I'm not sure how to apply it when approximating areas between two functions. </p>
<ul>
<li>Do you use the formula like how you normally would, except apply it to the <strong>first function - the second function</strong... | Peter Szilas | 408,605 | <p><span class="math-container">$h(x):=f(x)-g(x)$</span>, <span class="math-container">$h$</span> is continuous.</p>
<p><span class="math-container">$T=$</span>{<span class="math-container">$x| h(x)=0$</span>}.</p>
<p>Then <span class="math-container">$T= h^{-1}$</span>{<span class="math-container">$0$</span>} is cl... |
3,756,649 | <p>If <span class="math-container">$f:\mathbb{R}\to \mathbb{R}$</span> such that <span class="math-container">$\lim\limits_{x \to \infty}xf(x)=L$</span>. Then <span class="math-container">$\lim\limits_{x \to \infty}f(x)=0$</span>.</p>
<p>My proof is as follows:</p>
<p>Let <span class="math-container">$g(x)=xf(x)$</span... | Moe Sarah | 787,944 | <p><span class="math-container">$lim_{x\rightarrow \infty} f(x)=L\in \mathbb{R}$</span> if and only if</p>
<p>for all <span class="math-container">$\epsilon>0$</span> there exists <span class="math-container">$\delta>0$</span> such that for all <span class="math-container">$x>\delta$</span>, <span class="math-... |
7,268 | <p>I'm a private tutor working with a 7th grader who is struggling with solving equations. Given a simple equation, he is able to solve it using a formulaic procedure, but it is very obvious that he has no idea what the solution really means. Hence, if he gets a problem that's slightly different from ones he's solved b... | Aeryk | 401 | <p>I don't know how feasible this is or what kind of resources you have, but try finding a balance scale, some standard weights, and a collection of identical objects. You can set up the scale to represent a problem and ask the student to determine the unknown weight. The only rule is that the scale can never become un... |
7,268 | <p>I'm a private tutor working with a 7th grader who is struggling with solving equations. Given a simple equation, he is able to solve it using a formulaic procedure, but it is very obvious that he has no idea what the solution really means. Hence, if he gets a problem that's slightly different from ones he's solved b... | Joseph O'Rourke | 511 | <p>You may find this <a href="http://myplace.frontier.com/~paulgriffith2/mathmodels/math_models.shtml" rel="nofollow noreferrer">Mathematics Models I webpage</a> by Paul Griffith useful, although it targets a lower educational level:
<hr />
<img src="https://i.stack.imgur.com/z5eZi.gif" alt="Factor... |
3,080,005 | <p>So, I was trying to understand the "Group action" theory. I read the definition of <span class="math-container">$Stab_G$</span> and I was trying to solve some basic questions.</p>
<p>I came across with the following question: </p>
<blockquote>
<p>Let <span class="math-container">$S_7$</span> be a group that on i... | Rylee Lyman | 447,318 | <p>One way to proceed is to try to improve our understanding of the action. For instance, if we write <span class="math-container">$y = (1\ 2\ 3)(4\ 5\ 6)$</span>, what is <span class="math-container">$(x\cdot y)(x(1))$</span>, the image of <span class="math-container">$x(1)$</span> under <span class="math-container">$... |
342,064 | <p>We know that if an estimator, say $\widehat{\theta}$, is an unbiased estimator of $\theta$ and if its variance tends to 0 as n tends to infinity then it is a consistent estimator for $\theta$. But this is a sufficient and not a necessary condition. I am looking for an example of an estimator which is consistent but ... | KalEl | 1,310 | <p>If the parameter space has only one element, then it's kinda boring. To make it a little more interesting, let's say parameters space is $\Theta=\{0,1\}$. And the random variable $X$ is normally distributed with mean $\theta$, i.e. $$X \sim N(\theta,1)$$</p>
<p>The natural estimate to consider is $t_n=\frac{\sum x_... |
4,374,307 | <p>Problem:<br />
Suppose there are <span class="math-container">$7$</span> chairs in a row. There are <span class="math-container">$6$</span> people that are going to randomly
sit in the chairs. There are <span class="math-container">$3$</span> females and <span class="math-container">$3$</span> males. What is the pro... | Math Lover | 801,574 | <p>Here is how I would think about it. There are seven of them - three male, three female and an empty chair (say, a ghost). In other words, for any given chair, there are seven equally likely possibilities.</p>
<p>So, <span class="math-container">$ \displaystyle P = {3 \choose 2} / {7 \choose 2} = \frac 17$</span></p>... |
1,521,518 | <p>Determine if the given vectors span $\mathbb{R}^4$</p>
<p>${(1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), (0, 0, 0, 1)}$.</p>
<p>I'm completely confused on this question. My textbook gives a different problem but in $\mathbb{R}^3$. How would i go about this?</p>
| MrMazgari | 284,607 | <p>You want to find constants $a,b,c,d \in \mathbb{R}$ such that every vector $(x,y,z,w) \in \mathbb{R}^4$ can be written as a linear combination of the given vectors. That is, \begin{align}(x,y,z,w)&=a(1,1,1,1)+b(0,1,1,1)+c(0,0,1,1)+d(0,0,0,1)\\
&=(a,a+b,a+b+c,a+b+c+d).
\end{align}
Equating components, we have... |
1,559,906 | <p>Is there an equivalent in mathematical language to the modulo (or <code>mod</code>) function in computing? </p>
| Archis Welankar | 275,884 | <p>The mathematical symbol in many computer languages and also mathematics is mod it represents modular divsion . Eg $15 mod 2=7$</p>
|
3,367,672 | <p>Assume <span class="math-container">$a_n$</span> is a non-negative sequence, that is, <span class="math-container">$a_n \geq 0$</span> for every <span class="math-container">$n \in \mathbb{N}$</span>. Prove that if <span class="math-container">$a_n$</span> converges then the limit is non-negative. Clue: prove it by ... | Kavi Rama Murthy | 142,385 | <p>You have assumed what you are asked to prove.</p>
<p>If possible let <span class="math-container">$a_n \to L$</span> with <span class="math-container">$L <0$</span>. Take <span class="math-container">$\epsilon =-L$</span>. Then there exits <span class="math-container">$n_0$</span> such that <span class="math-con... |
3,367,672 | <p>Assume <span class="math-container">$a_n$</span> is a non-negative sequence, that is, <span class="math-container">$a_n \geq 0$</span> for every <span class="math-container">$n \in \mathbb{N}$</span>. Prove that if <span class="math-container">$a_n$</span> converges then the limit is non-negative. Clue: prove it by ... | Nicholas Roberts | 245,491 | <p>You seemed to have used the result you are trying to prove in your proof! This is not allowed in mathematics. </p>
<p>The clue suggesting contradiction seems to be a good approach. Indeed, suppose <span class="math-container">$a_n \longrightarrow L$</span> where <span class="math-container">$L < 0$</span>. Then,... |
3,995,272 | <p>I am a master student in mathematical physics. I study soliton and traveling wave solutions of the differential equations.</p>
<p>Let's consider the following ODE:
<span class="math-container">$$Q^{\prime}(\xi)=ln(A)(\alpha+\beta Q(\xi)+\sigma Q^2(\xi))$$</span>
where
<span class="math-container">$A \neq 0,1.$</span... | SteelCubes | 859,262 | <p><span class="math-container">$$\lim _ { x \to \frac { \pi } { 2 } } \frac { \ln \sin x } { \cos ^ { 2 } x }\\
=\lim _ { x \to \frac { \pi } { 2 } } \frac { \ln( \sin x -1 +1)} { 1-\sin ^ { 2 } x }\\
=\lim _ { x \to \frac { \pi } { 2 } } \frac { \ln( 1+ \sin x -1)} { -(\sin x-1)(1+\sin x) }\\
=\frac{-1}{2}$$</span>
A... |
3,995,272 | <p>I am a master student in mathematical physics. I study soliton and traveling wave solutions of the differential equations.</p>
<p>Let's consider the following ODE:
<span class="math-container">$$Q^{\prime}(\xi)=ln(A)(\alpha+\beta Q(\xi)+\sigma Q^2(\xi))$$</span>
where
<span class="math-container">$A \neq 0,1.$</span... | Fred | 380,717 | <p>Let <span class="math-container">$t= \sin x$</span>. Then we have to compute</p>
<p><span class="math-container">$$\lim_{t \to 1} \frac{ \ln t}{1-t^2}.$$</span></p>
<p>We have</p>
<p><span class="math-container">$$\frac{ \ln t}{1-t^2}=\frac{ 1}{1+t} \cdot \frac{ \ln t}{1-t}.$$</span></p>
<p>It is clear that <span cl... |
3,752,770 | <p>I tested this in python using:</p>
<pre><code>import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 10*2*np.pi, 10000)
y = np.sin(x)
plt.plot(y/y)
plt.plot(y)
</code></pre>
<p>Which produces:</p>
<p><a href="https://i.stack.imgur.com/pCwoV.png" rel="nofollow noreferrer"><img src="https://i.stack.img... | Kraigolas | 655,232 | <p>The function</p>
<p><span class="math-container">$$f(x)=\frac{\sin(x)}{\sin(x)}$$</span></p>
<p>has an infinite number of what are called <a href="https://en.wikipedia.org/wiki/Removable_singularity" rel="noreferrer">removable singularities</a>. If we consider the singularity at <span class="math-container">$x=0$</s... |
137,595 | <p>I was using the <code>StreamPlot</code> function to plot the direction field of a system of two first order differential equations. Is there any way I could add solution curves to my direction field with this function? Or is there another function that could do that for me? I looked around, but I couldn't find anyth... | zhk | 8,538 | <p>Here is another way to super impose solution curves on the stream lines. For this I choose a random IVP.</p>
<p><strong>Random example</strong></p>
<pre><code>soln[y0_?NumericQ]:=First@NDSolve[{y'[x] == -1 + Sin[y[x]], y[0] == y0}, {y}, {x, -10,10}];
sp = StreamPlot[{1, (-1 + Sin[y])}, {x, -3, 3}, {y, -3, 3}];
Sho... |
3,374,248 | <p>I haven't worked out all the details yet, but it seems to be true for the following functions:</p>
<ul>
<li><span class="math-container">$f(k) = 1$</span></li>
<li><span class="math-container">$f(k) = 1/k!$</span></li>
<li><span class="math-container">$f(k) = a^k$</span></li>
<li><span class="math-container">$f(k) ... | Ali Ashja' | 437,913 | <p>If you think the error term is produced from approximation of an integral:
<span class="math-container">$$\frac{1}{n} \int_0^n f*f=\frac{1}{f^2(n/2)}$$</span>
That means you should search for functions that mean of their convolution to themselves is equal to inverse of square of them at mid point of interval, that i... |
101,058 | <p>Let $M$ be a module over a ring $R$.</p>
<p>Let $\operatorname{Ass}(M)$ be the set of annihilator ideals $\operatorname{Ann}(x)$, which are prime, so</p>
<p>$$\operatorname{Ass}(M) = \{\operatorname{Ann}(x) \mid \operatorname{Ann}(x)\text{ is prime}, x \in M\}.$$</p>
<p>Recall that $\operatorname{Ann}(x) = \{r \i... | Community | -1 | <p>You could use the fact that for a submodule <span class="math-container">$N\subseteq M$</span> we have <span class="math-container">$\mathrm{Ass}(M)\subseteq \mathrm{Ass}(N)\cup \mathrm{Ass}(M/N)$</span>.<br />
This can be proved directly: Let <span class="math-container">$\mathrm{Ann}(m)=\mathfrak p\in\mathrm{Ass}(... |
101,058 | <p>Let $M$ be a module over a ring $R$.</p>
<p>Let $\operatorname{Ass}(M)$ be the set of annihilator ideals $\operatorname{Ann}(x)$, which are prime, so</p>
<p>$$\operatorname{Ass}(M) = \{\operatorname{Ann}(x) \mid \operatorname{Ann}(x)\text{ is prime}, x \in M\}.$$</p>
<p>Recall that $\operatorname{Ann}(x) = \{r \i... | Zhen Lin | 5,191 | <p>Let <span class="math-container">$\mathfrak{p} \in \textrm{Ass}(M_1 \oplus M_2)$</span>. Suppose <span class="math-container">$\mathfrak{p} = \textrm{Ann}(m_1 + m_2)$</span> where <span class="math-container">$m_1 \in M_1$</span> and <span class="math-container">$m_2 \in M_2$</span>. (I am considering <span class="m... |
101,058 | <p>Let $M$ be a module over a ring $R$.</p>
<p>Let $\operatorname{Ass}(M)$ be the set of annihilator ideals $\operatorname{Ann}(x)$, which are prime, so</p>
<p>$$\operatorname{Ass}(M) = \{\operatorname{Ann}(x) \mid \operatorname{Ann}(x)\text{ is prime}, x \in M\}.$$</p>
<p>Recall that $\operatorname{Ann}(x) = \{r \i... | Community | -1 | <p><em>This is just a restructuring of <a href="https://math.stackexchange.com/a/101099/279515">@ZhenLin's answer</a> in light of the comments below it by <a href="https://math.stackexchange.com/questions/101058/set-of-associated-primes-of-direct-sum#comment237758_101099">@DylanMoreland</a> and <a href="https://math.st... |
3,895,314 | <p>How do I prove <span class="math-container">$x ^ {1 - x}(1 - x) ^ {x} \le \frac{1}{2}$</span>, for every <span class="math-container">$x \in (0, 1)$</span>.</p>
<hr />
<p>For <span class="math-container">$x = \frac {1}{2}$</span> the LHS is equal to one half. I tried studying what happens when <span class="math-cont... | TheSilverDoe | 594,484 | <p>Because <span class="math-container">$1-x+x=1$</span>, and <span class="math-container">$x \in (0,1)$</span>, by concavity of the <span class="math-container">$\ln$</span>, one has
<span class="math-container">$$(1-x)\ln(x)+x \ln(1-x) \leq \ln((1-x)x+x(1-x))=\ln(2x(1-x))$$</span></p>
<p>But it is easy to see that fo... |
1,400,394 | <p>Given that $u(x,y)$ can someone please explain to me how the result as asked in the question is achieved? Steps would be really appreciated, thanks.</p>
| HK Lee | 37,116 | <p>Note that $u_{xx}=0$ implies that $$ u_x= f(y)
$$</p>
<p>(Here fix $y_0$. Then $u_x(x,y_0)$ is a function of variable $x$ only. If $u_{xx}=0$ then $u_x$ is constant : If not, that is, for example $u_x(x_1) < u_x(x_2)$, then note that by mean value theorem, $$ u_x(x_1)-u_x(x_2)= u_{xx} (x_3)(x_1-x_2),\ x_3\in [x... |
4,461,327 | <p>To show that they are equal, I need to show</p>
<p><span class="math-container">$\bigcap_{n=1}^{\infty}[0,1+1/n) \subset [0,1]$</span> and <span class="math-container">$[0,1] \subset \bigcap_{n=1}^{\infty}[0,1+1/n)$</span></p>
<p>My attempt is: let <span class="math-container">$x \in [0,1] \Rightarrow 0 \leq x \leq ... | Hidda Walid | 736,203 | <p>If <span class="math-container">$x$</span> is in the intersection, that means that <span class="math-container">$0\leq x<1+\frac{1}{n}$</span> for all <span class="math-container">$n\geq1$</span>, using the fact that if <span class="math-container">$w_n<u_n<v_n$</span> for all <span class="math-container">$... |
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