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 |
|---|---|---|---|---|
1,621,019 | <blockquote>
<p>Let $a,b,$ and $c$ be the lengths of the sides of a triangle, prove that $$a(b^2+c^2-a^2)+b(a^2+c^2-b^2)+c(a^2+b^2-c^2) \le 3abc.$$</p>
</blockquote>
<p>I can't really factor this into something nice. Also using AM-GM or Cauchy-Schwarz doesn't look like it will help. I am thinking we need to bound th... | yurnero | 178,464 | <p><strong>Hint</strong>: Expand the following, divide by 2, and rearrange:
$$
(a+b-c)(a-b)^2+(b+c-a)(b-c)^2+(c+a-b)(c-a)^2\geq 0.
$$</p>
<blockquote class="spoiler">
<p>How I thought of this: I was thinking of a multinomial with degree $3$ that is symmetrical in $a,b,c$, is nonnegative, uses the triangle inequality... |
2,310,828 | <blockquote>
<p>Let $f:\Bbb{R}^2 \to \Bbb{R}^2$ be continuously differentiable. Suppose $f'(0)$ has non-zero determinant. Let $U = \left\{x \in \Bbb{R}^2 : ||f'(x)-f'(0)||<\frac{1}{2||f'(0)||}\right\}$. Show that $f(U)$ is open.</p>
</blockquote>
<p>I have tried doing this using generic properties of a norm but t... | José Carlos Santos | 446,262 | <p>This is not true. Take $f(x,y)=\left(\left(x-\frac14\right)^2,\left(y-\frac14\right)^2\right)$. Then $f'\left(0,0\right)=-\frac12\operatorname{Id}$ and so $\left\|f'\left(0,0\right)\right\|=\frac12$. On the other hand, $f'\left(\frac14,\frac14\right)=0$ and so $\left(\frac14,\frac14\right)\in U$. But then $f(U)$ is ... |
2,310,828 | <blockquote>
<p>Let $f:\Bbb{R}^2 \to \Bbb{R}^2$ be continuously differentiable. Suppose $f'(0)$ has non-zero determinant. Let $U = \left\{x \in \Bbb{R}^2 : ||f'(x)-f'(0)||<\frac{1}{2||f'(0)||}\right\}$. Show that $f(U)$ is open.</p>
</blockquote>
<p>I have tried doing this using generic properties of a norm but t... | Severin Schraven | 331,816 | <p>Your statement is in general not true. Take</p>
<p>$$ f_c: \mathbb{R}^2 \rightarrow \mathbb{R}^2, \ f_c(x,y)=\left(\sin \left(\frac{x}{c}\right), \sin\left(\frac{y}{c}\right)\right).$$</p>
<p>One computes</p>
<p>$$ f_c'(x,y)= \frac{1}{c}\begin{pmatrix} \cos\left(\frac{x}{c}\right) & 0 \\ 0 & \cos\left(\fr... |
773,880 | <p>What approach would be ideal in finding the integral $\int4^{-x}dx$?</p>
| DeepSea | 101,504 | <p>Hint: let $u = -x$, then you can go from here.</p>
|
202,040 | <p>I'd like to get separate plots for the functions in a list, and I'm trying the following, which doesn't work. What is the correct way to do that?</p>
<pre><code>Table[ContourPlot3D[f, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}], {f, {x + y + z + x y z == 0, x + y + z^2 + x y z^2 == 0, x + y^2 + z + x y^2 z == 0}}]
</code><... | yarchik | 9,469 | <pre><code>Table[ContourPlot3D[
Evaluate[f], {x, -2, 2}, {y, -2, 2}, {z, -2,
2}], {f, {x + y + z + x y z == 0, x + y + z^2 + x y z^2 == 0,
x + y^2 + z + x y^2 z == 0}}]
</code></pre>
<p>Just add <code>Evaluate</code></p>
|
4,234 | <p>I recently pasted the following code:</p>
<pre><code> my @cards = qw(BB BR RR);
my $n_trials = shift || 100;
for (1 .. $n_trials) {
my $card = $cards[ int(rand 3) ];
my @faces = split //, $card;
my $face_choice = int(rand 2);
my ($face, $other_face) = @faces[$face_choice, 1-$fac... | Davide Cervone | 7,798 | <p>The handling of MathJax in Markdown is rather awkward, since TeX and Markdown have different interpretations for things like underscores. You need to keep Mardown from processing those character inside of mathematics, and the way it works on StackExchange is that the mathematics is removed before Markdown runs, and... |
69,272 | <p>By the way, does anyone know how to prove in an elementary way (i.e. expanding) that $\prod_1^n (1+a_i r)$ tends to $e^r=\sum \frac{r^k}{k!}$ as you let $\max|a_i|\to 0$ with $0\leq a_i \leq 1$ and $\sum a_i = 1$? An easy solution goes by writing the product with the exponential function so that you get the exponent... | Geoff Robinson | 14,450 | <p>I record this answer because I think that Pietro Majer's comment can be made into a solution which
meets the proposer's criterion of potentially being able to be used to define $e^r$ ( I had been thinking along the same lines, although I am not sure there would be an advantage over the usual $\lim_{n \to \infty} (... |
2,018,239 | <p>I have to show, using induction, that $2^{4^n}+5$ is divisible by $21$. It is supposed to be a standard exercise, but no matter what I try, I get to a point where I have to use two more inductions.</p>
<p>For example, here is one of the things I tried:</p>
<p>Assuming that $21 |2^{4^k}+5$, we have to show that $21... | Denis Korzhenkov | 367,345 | <ol>
<li><p>$2^{4^1} \equiv -5 \bmod 21$</p></li>
<li><p>$2^{4^{n+1}} = 2^{4^n \cdot 4} = (2^{4^n})^4 \equiv (-5)^4 = 625 \equiv -5 \bmod 21$</p></li>
</ol>
|
2,018,239 | <p>I have to show, using induction, that $2^{4^n}+5$ is divisible by $21$. It is supposed to be a standard exercise, but no matter what I try, I get to a point where I have to use two more inductions.</p>
<p>For example, here is one of the things I tried:</p>
<p>Assuming that $21 |2^{4^k}+5$, we have to show that $21... | Steven Alexis Gregory | 75,410 | <p>$2^{4^k}+5 \equiv (-1)^{4^k}+5 \equiv 1+5 \equiv 0 \pmod 3$</p>
<p>Proof by induction that $2^{4^k}+5 \equiv 0 \pmod 7$ for all non negative ingeters $k$.</p>
<p>If $k=0$, then $2^{4^k}+5 \equiv 2+5 \equiv 0 \pmod 7$</p>
<p>Suppose that $2^{4^m}+5 \equiv 0$ for some non negative integer $m$.</p>
<p>Then
$2^{4^{... |
2,372,762 | <p>So I know in order to prove a function is bijective, you need to prove that it is both injective and surjective. I know that to prove it is an injection, I need to make $f(x) = f(y)$, and try to get $x=y$ from that, but I can't seem to manipulate the equations to do so. </p>
<p>Also, how would I prove that this is ... | Tai | 290,413 | <p>Let $p(n,k)$ denote the number of partitions of $n$ with largest part at most $k$. Then, of the $p(n,k)$ such partitions, there are some that have a partition with part $k$ and some that don't. Those that don't have part $k$ are all precisely the partitions of $n$ with largest part at most $k-1$, so there are $p(n,k... |
3,063,577 | <p>Please suggest a book on applications of Diophantine equations in physics, chemistry, and biology. This book should be suitable to introduce this subject to students who are not mathematics specialists. </p>
| José Carlos Santos | 446,262 | <p>I am unaware of any such book, but you may find <a href="http://ijmaa.in/v5n2-b/217-222.pdf" rel="nofollow noreferrer">this article</a> interesting.</p>
|
723,633 | <p>My book asserts that for fixed $w$ where $w\neq 0$ that $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$</p>
<p>My book has a general corralary that $v\to P(v)$ is a bounded linear transformation and the fact that $P^2=P$ implies it is a projection. I'm not sure how they made the assertation. Any ideas?</p>
| vadim123 | 73,324 | <p>Hint: $$P^2(v)=P(P(v))=\frac{\langle P(v),w\rangle}{\|w\|^2}w$$</p>
|
3,290,095 | <p>Now first something that I already know;
<span class="math-container">\begin{eqnarray}
∞/ ∞ = undetermined ( ≠1 ) \\
∞- ∞ = undetermined (≠0)\\
\end{eqnarray}</span></p>
<p>So basically one reason for this is that the <span class="math-container">$∞$</span> I assume is not as same as the <span class="math-contain... | mlchristians | 681,917 | <p>Strange things can happen when considering the infinite.</p>
<p>Consider the infinite set <span class="math-container">$\lbrace 1, 2, 3, \ldots \rbrace$</span>. If I multiply all the elements of this set by <span class="math-container">$2$</span>, I get <span class="math-container">$\lbrace 2, 4, 6, \ldots \rbrace$... |
2,464,756 | <p>When I was trying to prove a relation from solid state physics, I reached this mathematical problem. In the equation</p>
<p>$$\sum_{i=1}^Nm_ix_i=n$$</p>
<p>$m_i$ and $n$ are known integers, $N=3$, and $x_i$ are unknown integers. Also we know that the greatest common factor of $\left\{m_i\right\}$ is 1. I don't nee... | Bernard | 202,857 | <p>It's a standard result from <em>Arithmetic</em>:</p>
<p>The ideal $\;(x_1,\dots , x_N)\subset\mathbf Z$ generated by $x_1, \dots, x_n$, i.e. the set of linear combinations of $x_1,\dots, x_N$ with integer coefficients is the principal ideal generated by $\gcd(x_1,\dots,x_N)$.</p>
<p>Hence, if the generators are co... |
2,870,729 | <blockquote>
<p>Why does $|e^{ix}|^2 = 1$?</p>
</blockquote>
<p>The book said $e^{ix} = \cos x + i\sin x$, and square it, then $|e^{ix}|^2 = \cos^2x + \sin^2x = 1$.</p>
<p>But, when I calculated it, $ |e^{ix}|^2 = \left|\cos x + i\sin x\right|^2 = \cos^2x - \sin^2x + 2i\sin x\cos x$.</p>
<p>I can't make it to be e... | Mason | 552,184 | <p>$|z|^2=z\bar{z}$</p>
<p>But the complex conjugate of ${e^{xi}}$ is $e^{-xi}$.</p>
<p>$|e^{xi}|^2=e^{xi}e^{-xi}=1$</p>
|
748,815 | <blockquote>
<p>$\displaystyle\sum\limits_{k=1}^nk^2(k-1){n\choose k}^2 = n^2(n-1)
{2n-3\choose n-2}$ considering $n\ge2$</p>
</blockquote>
<p>Can somebody help with this combinatorial proof?
I'm struggling a lot.
Thanks.</p>
<p><strong>EDIT:</strong> Ok. I could figure it out, if we had $\displaystyle\sum\limits_{... | DirkGently | 88,378 | <p>There's a factor of $n-1$ missing from the right side of the equation. Let us write the equation as
$$\sum_{k=1}^n k (k-1){\binom{n}{k}}\cdot k{\binom{n}{n-k}}=n^2(n-1)\binom{2n-3}{n-2}.$$
Now count the number of sequences of length $2n$ on the alphabet $\{a_0,a_1,a_2,b_0,b_1\}$ with $n$ $a$'s (with any subscript) ... |
748,815 | <blockquote>
<p>$\displaystyle\sum\limits_{k=1}^nk^2(k-1){n\choose k}^2 = n^2(n-1)
{2n-3\choose n-2}$ considering $n\ge2$</p>
</blockquote>
<p>Can somebody help with this combinatorial proof?
I'm struggling a lot.
Thanks.</p>
<p><strong>EDIT:</strong> Ok. I could figure it out, if we had $\displaystyle\sum\limits_{... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
933,604 | <p>Hi can anyone solve these two questions using logs and indices</p>
<p>a.
$$4^{2x}-2^{x+1}=48$$</p>
<p>b.
$$6^{2x+1}-17*{6^x}+12=0$$</p>
<p>Thanks.</p>
| Brass2010 | 173,503 | <p>Thanks for the answers, after a night of thoughts, I came up with the idea of using quadratic equations and logarithms to solve the answers</p>
<p>Let 2x=a</p>
<p>a^2-2a-48=0</p>
<p>a=8 or a=-6(rejected)</p>
<p>2^x=3</p>
<p>log(2)8=3</p>
<h2>x=3</h2>
<p>Let 6x=a</p>
<p>6a^2-17a+12=0</p>
<p>a=1.5 or4/3</p>
... |
971,617 | <p>Suppose $p,q$ are two distinct prime numbers, $q \geq 3$ and $p \not\equiv 1 \pmod q$. Then I have the following problem: Prove that there is no integer $x \in \mathbb{Z}$ such that $1+x+x^2+...+x^{q-1} \equiv 0 \pmod p$. </p>
<p>It is obvious that $x$ cannot be $0 \pmod p$, and I also found that when $p$ is even, ... | Xoff | 36,246 | <p>You are looking for an element $x$ of the finite field with $p$ elements such that $x^q=1$ and $x\neq 1$. But for any finite field, the multiplicative group is cyclic, so $q$ (as prime) must be a divisor of $p-1$, so $p=1 (\mod q)$</p>
|
189,650 | <p>let $S=\{s_1, s_2, s_3 \}$, if $s_1$ can be represented as a linear combination of $s_2$ and $s_3$, $s_2$ can be represented as a linear combination of $s_1$ and $s_3$ but $s_3$ can not be represented as a linear combination of $s_1$ or $s_2$ or $s_1$ and $s_2$, can we call $S$ a linearly dependent set? </p>
| Community | -1 | <p>Yes, the elements of $S$ are linearly dependent. To be linearly dependent means that there exist scalars $a, b, c$, not all zero such that
$$
as_1 + bs_2 + cs_3 = 0.
$$
This is true of your elements because we know
$$
s_1 = ms_2 + ns_3
$$
for some scalars $m, n$ and we can rearrange this equation to get $a = 1$, $b ... |
2,979,226 | <p>Consider you are given following </p>
<blockquote>
<p><span class="math-container">$$\biggr (x-\dfrac{2}{x^2}\biggr )^6$$</span></p>
</blockquote>
<p>I'm trying to evaluate the constant term. What I've done so far is given below</p>
<p><span class="math-container">$$\sum^{6}_{n = 0} \binom{6}{r}x^{6-r}\times (... | Sujit Bhattacharyya | 524,692 | <p><strong>EDIT:</strong></p>
<p>Here is a simple observation:</p>
<p>Constant term will have no <span class="math-container">$x$</span>. So in order to omit <span class="math-container">$x$</span> we must have, <span class="math-container">$\displaystyle\frac{x^r}{x^{2(6-r)}}=1\implies x^r=x^{12-2r}\implies r=4$</sp... |
4,045,238 | <p>I was working on the problems in Mathematical Methods for Physics and Engineering by Riley,Hobson & Bence.
In Problem 2.34 (d) I'm supposed to find this integral: <span class="math-container">$$J=\int\frac{dx}{x(x^n+a^n)}.$$</span>
I used partial fractions and arrived at the form
<span class="math-container">$$J... | Kenta S | 404,616 | <p><span class="math-container">\begin{align*}
J&=\int\frac{dx}{x(x^n+a^n)}\\
&=\frac1{a^n}\int\left(\frac1{x}-\frac{x^{n-1}}{x^n+a^n}\right)dx\\
&=\frac1{a^n}\ln|x|-\frac1{na^n}\int\frac{nx^{n-1}}{x^n+a^n}dx\\
&=\frac1{a^n}\ln|x|-\frac1{na^n}\ln|x^n+a^n|+C,\\
\end{align*}</span>
where <span class="math... |
1,180,437 | <p>I am trying to understand this proof. Rather an important part of the proof. I have already shown this is true for $n=2$ and am assuming the $a_n$ case is true.</p>
<p>$$(a_1^2+a_2^2+...+a_n^2) \le (a_1+a_2+...+a_n)^2$$
Want to show that
$$(a_1^2+a_2^2+...+a_n^2 + a_{n+1}^2) \le (a_1+a_2+...+a_n+a_{n+1})^2$$
$=$... | Math-fun | 195,344 | <p><em>inductive step</em>: </p>
<p>the claim being correct for
$$(a_1^2+a_2^2+...+a_n^2) \le (a_1+a_2+...+a_n)^2$$
implies
$$(a_1^2+a_2^2+...+a_n^2+a_{n+1}^2) \le (a_1+a_2+...+a_n+a_{n+1})^2$$
<strong>Proof</strong>
\begin{align}
a_1^2+a_2^2+...+a_n^2+a_{n+1}^2 &=(a_1^2+a_2^2+...+a_n^2)+a_{n+1}^2\\
& \leq... |
192,334 | <p>I want to partition string into longest substrings that each contain only specific characters, beginning from left to right with no overlaps, always choosing the longest one possible at current position. In my example only substrings that contain only characters <code>d,f,g</code> or <code>d,e,h</code> or <code>a,b,... | Carl Woll | 45,431 | <p>You could do:</p>
<pre><code>StringCases[
"ABCDEFGH",
Longest[p__] /; StringMatchQ[p,("D"|"F"|"G")..|("D"|"E"|"H")..|("A"|"B"|"C"|"G")..]
]
</code></pre>
<blockquote>
<p>{"ABC", "DE", "FG", "H"}</p>
</blockquote>
|
3,111,985 | <p><span class="math-container">$f_n(x)= \frac{x}{(1+x)^n}\quad f_n(0)=0$</span></p>
<p>pointwise convergence: <span class="math-container">$\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}=x \sum_{n=1}^{\infty} \frac{1}{(1+x)^n}$</span> and the series is a geometric series convergent if <span class="math-container">$|x+1|>1... | MathFail | 978,020 | <p>The series is convergent on <span class="math-container">$x\in E, ~\text{where} ~E=(-\infty, -2)\cup [0,\infty)$</span></p>
<p>Define the parital sum as <span class="math-container">$S_n(x)=\sum_{k=0}^{n} f_k(x)$</span></p>
<p><span class="math-container">$$S_n(x)=x\left(\frac{1}{1+x}+\cdots+\frac{1}{(1+x)^n}\right)... |
4,573,566 | <p>So I have to find the bifurcation points of the system: <span class="math-container">$\dot{x}=(ax-x^3+x^5)(x-a+2)$</span>, where <span class="math-container">$a\in\mathbb{R}$</span> is a parameter.</p>
<p>Attempt:<br />
I know that a bifurcation point is the point, where there is a change in stability or number of f... | Gregory | 197,701 | <p>Lets take this step-by-step:</p>
<ul>
<li>Find the fixed points as a function of our parameters (e.g. <span class="math-container">$a$</span>).</li>
<li>Investigate the changes of the locations as we vary the parameters</li>
<li>Determine the stability of the fixed points.</li>
<li>Investigate the nature of these fi... |
3,362,000 | <p>From listing the first few terms, I suspect that the sequence is increasing, so I wanted to use mathematical induction to verify my suspicion.</p>
<p>I have assumed that <span class="math-container">$a_k<a_{k+1}$</span>, I don't see how I can obtain <span class="math-container">$a_{k+1}<a_{k+2}$</span> becaus... | mathcounterexamples.net | 187,663 | <p>The sequence is positive. Easy proof by induction.</p>
<p>Then <span class="math-container">$a_n - a_{n-1} = 1/a_{n-1} >0$</span> proving that the sequence is increasing.</p>
|
3,953,681 | <p>I have a basic question but I have failed in solving it. I have the equation of a cylinder which is <span class="math-container">$y^2 + z^2 = r^2$</span> (centered in the x-axis). The parametric equation (dependent on <span class="math-container">$L$</span> and <span class="math-container">$s$</span>) is <span class... | Z Ahmed | 671,540 | <p><span class="math-container">$$I=\int \frac{dx}{x\sqrt{a-bx^2}}$$</span>
Let <span class="math-container">$x=\sqrt\frac{a}{b}\sin t$</span>, then
<span class="math-container">$$I=\int \frac{\sqrt{a/b} \cos t!dt}{\sqrt{a/b} \sin t \sqrt{a} \cos t}.$$</span>
<span class="math-container">$$\implies I=\frac{1}{\sqrt{a}}... |
2,690,433 | <p>If $V$ is a vector space that has closure properties and satisfies the axioms and $S$ is a subset of $V$, why wouldn't $S$ always have closure under addition and scalar multiplication (which are required to show $S$ is a subspace) because since $S$ is a subset of $V$, doesn't that mean $S$ would have the same proper... | Matteo Casarosa | 539,948 | <p>No, because it could be that $v, w \in S $ but $v+w \not\in S $ or $\lambda v \not\in S$.</p>
<p>Note that you have to prove <em>both</em> closure properties: one does not entail the other.</p>
<p>Here are some examples:</p>
<p>Take $V = \mathbb{R}^2 $ . Now, $ \{(x,y)\in \mathbb{R}^2 \vert x\in \mathbb{Z}, y\i... |
1,574,663 | <p>I'm a first time Calc I student with a professor who loves using $e^x$ and logarithims in questions. So, loosely I know L'Hopital's rule states that when you have a limit that is indeterminate, you can differentiate the function to then solve the problem. But what do you do when no matter how much you differentiate,... | Matematleta | 138,929 | <p>Factor out $e^x$: </p>
<p>$\frac{\left(e^x+e^{-x}\right)}{\left(e^x-e^{-x}\right)}=\frac{1+e^{-2x}}{1-e^{-2x}}\rightarrow 1$ as $x\rightarrow \infty $.</p>
<p>In other situations, Taylor expansions and algebraic operations usually are sufficient. There are also many standard inequalities which can help with these ... |
3,734,216 | <p>Say <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are operators on Hilbert spaces <span class="math-container">$H_A,H_B$</span> respectively. If the Hilbert spaces are finite dimensional, then I know the tensor <span class="math-container">$A\otimes B$</span> can be represented ... | Ben Grossmann | 81,360 | <p>We can make the Kronecker-product formula work in the following way. If <span class="math-container">$\{u_j\}_{j \in \Bbb N}$</span> is a basis of <span class="math-container">$H_A$</span>, then we have
<span class="math-container">$$
H_{A} \otimes H_B \cong \bigoplus_{j \in \Bbb N} H_B,
$$</span>
with an isomorphis... |
1,602,271 | <p>Someone can help me to solve this differential equation with method of undetermined coefficient.
$$ y''-2y'+y=x\sin x$$
Thanks</p>
| Cesareo | 397,348 | <p>As the DE is linear we have</p>
<p><span class="math-container">$$
y = y_h + y_p
$$</span></p>
<p>with</p>
<p><span class="math-container">$$
\cases{
y''_h-2y'_h+y_h = 0\\
y''_p-2y'_p+y_p = x\sin x
}
$$</span></p>
<p>The homogeneous has the solution</p>
<p><span class="math-container">$$
y_h(x) = c_1 e^x+c_2 x ... |
2,648,626 | <p>Is the set $(e_n)_{n>0}$ a (vector space) basis for the sequence Hilbert space $l^2$? It is a Hilbert space basis anyway.</p>
<p>I would say no, because the sequence $\left(\frac{1}{n}\right)_{n>0}$ is in $l^2$ but it can't be written as a finite linear combination of $e_i$'s.</p>
<p>Is that right?</p>
| Henno Brandsma | 4,280 | <p>Yes, that's true. Any actual vector space basis for $\ell^2$ has to have the same size as $\mathbb{R}$ and cannot be explicitly written down. </p>
|
3,041,656 | <p>I need some help in a proof:
Prove that for any integer <span class="math-container">$n>6$</span> can be written as a sum of two co-prime integers <span class="math-container">$a,b$</span> s.t. <span class="math-container">$\gcd(a,b)=1$</span>.</p>
<p>I tried to go around with "Dirichlet's theorem on arithmetic ... | fleablood | 280,126 | <p>Well if <span class="math-container">$n$</span> is odd you can always do <span class="math-container">$n-2$</span> and <span class="math-container">$2$</span>. Or you can do <span class="math-container">$\frac {n-1}2$</span> and <span class="math-container">$\frac {n+1}2$</span>.</p>
<p>If <span class="math-contain... |
2,516,942 | <p>Trying to find all solutions on (-infinity,+infinity) for :
$y''+4y = 0$</p>
<p>I know that the discriminant of the characteristic equation is -16 so the roots are complex. so $k=0.5 \cdot \sqrt{-16} = 2i$</p>
<p>$f_1(x) = e^{(2ix)} = \cos(2x) + i\sin(2x)$</p>
<p>$f_2(x) = e^{(-2ix)} = \cos(2x) - i\sin(2x)$</p>... | Nirvanacs | 70,592 | <p>It can be solved directly by induction.</p>
<p>Note that
$$
(n!)^22^{2n}=(2\cdot 4\cdot 6 \cdot\cdots \cdot 2n)^2,
$$
the middle expression can be rewritten as
$$
\prod_{k=1}^n\left(1-\frac{1}{2k}\right).
$$
By indcution, for the right inequality, it suffices to show that
$$
\frac{1}{\sqrt{2n+1}}\frac{2n+1}{2n+2}&l... |
2,516,942 | <p>Trying to find all solutions on (-infinity,+infinity) for :
$y''+4y = 0$</p>
<p>I know that the discriminant of the characteristic equation is -16 so the roots are complex. so $k=0.5 \cdot \sqrt{-16} = 2i$</p>
<p>$f_1(x) = e^{(2ix)} = \cos(2x) + i\sin(2x)$</p>
<p>$f_2(x) = e^{(-2ix)} = \cos(2x) - i\sin(2x)$</p>... | Jack D'Aurizio | 44,121 | <p>$$\begin{eqnarray*}\frac{1}{4^n}\binom{2n}{n}=\prod_{k=1}^{n}\left(1-\frac{1}{2k}\right)&=&\sqrt{\frac{1}{4}\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)\prod_{k=2}^{n}\left(1+\frac{1}{4k(k-1)}\right)}\\&=&\frac{1}{2\sqrt{n}}\sqrt{\prod_{k=2}^{n}\left(1-\frac{1}{(2k-1)^2}\right)^{-1}}\\(\text{Wallis prod... |
3,971,059 | <p><span class="math-container">$x_n$</span> is a sequence. The only thing I can do here is just write the definition of <span class="math-container">$\lim_{n \to \infty} x_n=a$</span>, but that doesn' t seem helpful to me. I tried looking for some inequalities that would help, and the only thing that I found that I t... | mathcounterexamples.net | 187,663 | <p>This is an immediate consequence of the continuity of the map <span class="math-container">$f_n(x)=x^{n}$</span>.</p>
|
3,971,059 | <p><span class="math-container">$x_n$</span> is a sequence. The only thing I can do here is just write the definition of <span class="math-container">$\lim_{n \to \infty} x_n=a$</span>, but that doesn' t seem helpful to me. I tried looking for some inequalities that would help, and the only thing that I found that I t... | mathcounterexamples.net | 187,663 | <p>Use the equality</p>
<p><span class="math-container">$$x^n-a^n =(x-a)(x^{n-1} +x^{n-2}a + \dots + a^{n-1})$$</span></p>
<p>To prove the inequality</p>
<p><span class="math-container">$$\vert x^n -a^n \vert \le n \vert x -a\vert \vert \vert 2a \vert^{n-1}$$</span></p>
<p>valid for <span class="math-container">$\vert ... |
1,832,177 | <p><em>(see edits below with attempts made in the meanwhile after posting the question)</em></p>
<h1>Problem</h1>
<p>I need to modify a sigmoid function for an AI application, but cannot figure out the correct math. Given a variable <span class="math-container">$x \in [0,1]$</span>, a function <span class="math-contain... | Ron | 680,211 | <p>I was just working on finding a similar function. In case anyone else lands here in the future, here is a sigmoidal function which satisfies:</p>
<p><span class="math-container">$$f(0)=0$$</span>
<span class="math-container">$$f(1)=1$$</span></p>
<p>and has a single parameter, <span class="math-container">$k$</spa... |
634,127 | <p>How to prove this (true or not)?</p>
<blockquote>
<p>$f(a,b) = f(a,c)$ must hold if $b = c$</p>
</blockquote>
<p><b>Note:</b> <i><b>f(a,b)</b> is a function with <b>a</b> & <b>b</b></i> parameters</p>
<p>thanks</p>
| angryavian | 43,949 | <p>Hint: by definition, a function takes in some inputs, and produces a <em>unique</em> output.</p>
|
227,096 | <p>Q:
Let $A$ be an $n\times n$ matrix defined by $A_{ij}=1$ for all $i,j$.
Find the characteristic polynomial of $A$.</p>
<p>There is probably a way to calculate the characteristic polynomial $(\det(A-tI))$ directly but I've spent a while not getting anywhere and it seems cumbersome. Something tells me there is a mor... | josh314 | 42,904 | <p>The Kronecker-delta factor does a "trace," meaning a summation over the diagonal components. Remember that $\delta_{ij}=0$ if $i\ne j$. So, then for any function $f(i,j)$, you'd have
\begin{equation}
\sum_{i,j} \delta_{ij} f(i,j) = \sum_{i=j} f(i,j) = \sum_i f(i,i)
\end{equation}</p>
<p>In your example, then,
\beg... |
227,096 | <p>Q:
Let $A$ be an $n\times n$ matrix defined by $A_{ij}=1$ for all $i,j$.
Find the characteristic polynomial of $A$.</p>
<p>There is probably a way to calculate the characteristic polynomial $(\det(A-tI))$ directly but I've spent a while not getting anywhere and it seems cumbersome. Something tells me there is a mor... | Ernesto Lopez Fune | 115,376 | <p>This transformation can be decomposed in the sum of two transformations (supposing of course that the index <span class="math-container">$i$</span> and <span class="math-container">$j$</span> run over a finite ordered set, otherwise you have to check convergence piece by piece first): <span class="math-container">$M... |
203,456 | <p>Please help me proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, where $a,b,c$ positive number different for 1.</p>
| Karolis Juodelė | 30,701 | <p><strike>By definition</strike> $\log_a b = \frac{\log b}{\log a}$.</p>
|
500,632 | <p>Find all such lines that are tangent to the following curves:</p>
<p>$$y=x^2$$ and $$y=-x^2+2x-2$$</p>
<p>I have been pounding my head against the wall on this. I used the derivatives and assumed that their derivatives must be equal at those tangent point but could not figure out the equations. An explanation will... | Mark Bennet | 2,906 | <p>If $y=x^2$ we have that $\frac {dy}{dx} = 2x$</p>
<p>This gives the gradient of the tangent at the point $(x, y)=(a, a^2)$</p>
<p>If the tangent line is $y=mx + c$ we therefore have $a^2=(2a)\cdot a+c$ whence $c=-a^2$ and the general tangent line to $y=x^2$ is $$y=2ax-a^2$$</p>
<p>If $y=-x^2+2x-2$ we have $\frac ... |
14,552 | <p>What are good examples of proofs by induction that are relatively low on algebra? Examples might include simple results about graphs.</p>
<p>My aim is to help students get a sense of the logical form of an induction proof (in particular proving a statement of the form 'if $P(k)$ then $P(k+1)$'), independent of the ... | John Coleman | 6,891 | <p>A couple of simple examples come to mind:</p>
<p>1) Prove that there are $2^n$ subsets of an $n$-element set.</p>
<p>2) Prove the power rule of derivatives for non-negative integer powers using the product rule.</p>
|
14,552 | <p>What are good examples of proofs by induction that are relatively low on algebra? Examples might include simple results about graphs.</p>
<p>My aim is to help students get a sense of the logical form of an induction proof (in particular proving a statement of the form 'if $P(k)$ then $P(k+1)$'), independent of the ... | Bill Dubuque | 163 | <p>Tiling problems might meet your constraints. A nice simple example is Golomb's Theorem that a chessboard of side $2^n$ with <em>any</em> square omitted can be tiled by trominoes ("L" shapes of 3 squares).</p>
<p>In fact we can modify it to give an example of how <em>strengthening the induction hypothesis</em> is of... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| Martin Brandenburg | 1,650 | <p>$\sqrt{ab} = \sqrt{a} \sqrt{b}$ is correct in the following sense: In an arbitrary field (here it is the field of complex numbers) the root $\sqrt{a}$ is <em>an</em> element in some field extension such that $\sqrt{a}^2=a$. It is not uniquely determined, for if $b$ is a root, then also $-b$ is a root (and these only... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| Husrat Mehmood | 116,634 | <p>Here is the Simple Answer to this question</p>
<p>as $i^2=-1$, if you break $i^3$ into $i^2\cdot i$ then you insert $-1$ into the place of $i^2$ you will get $(-1)\cdot i =-i$</p>
<p>this is the simple trick behind it.</p>
|
2,734,374 | <p>I don't think it is possible because that entails that only the <span class="math-container">$\mathbf 0$</span>-vector is in the eigenspace, but <span class="math-container">$\mathbf 0$</span> is not an eigenvector by definition. </p>
<p>However, my textbook says:</p>
<blockquote>
<p>For an <span class="math-con... | user | 505,767 | <p>By definition to any eigenvalues correspond at least one eigenvector thus for a n-by-n matrix for each eigenvalue $\lambda_i$ we have $1\le$ dim(eigenspace)$\le n$.</p>
|
451,131 | <blockquote>
<p><strong>Problem</strong>: If <span class="math-container">$\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log f(x) +c $</span>. Find <span class="math-container">$f(x)$</span></p>
<p><strong>Solution</strong>: <span class="math-container">$\int f(x) \sin{x} \cos{x}\,\mathrm dx = \f... | André Nicolas | 6,312 | <p>It is a good start. For simplicity write $y$ for $f(x)$. We can rewrite the result you got as
$$\frac{y'}{y^2}=2(b^2-a^2)\sin x\cos x.$$
Integrate both sides. It may be handy to note that $2\sin x\cos x=\sin(2x)$. Or not, since it is clear that $2\sin x\cos x$ is the derivative of $\sin^2 x$. </p>
|
2,856,373 | <blockquote>
<p>If <span class="math-container">$z_{1},z_{2}$</span> are two complex numbers and <span class="math-container">$c>0.$</span> Then prove that</p>
<p><span class="math-container">$\displaystyle |z_{1}+z_{2}|^2\leq (1+c)|z_{1}|^2+\bigg(1+\frac{1}{c}\bigg)|z_{2}|^2$</span></p>
</blockquote>
<p>Try: put <s... | Kavi Rama Murthy | 142,385 | <p>Let $a=|z_1|,b=|z_2|$. Then $2ab=2(a\sqrt c) \frac b {\sqrt c} \leq ca^{2}+\frac {b^{2}} c$ (because $2ts \leq t^{2}+s^{2}$ for any two real numbers $t$ and $s$). Hence $|z_1+z_2|^{2} \leq (a+b)^{2}=a^{2}+b^{2}+2ab\leq (1+c)a^{2}+(1+\frac 1 c )b^{2}$.</p>
|
2,872,492 | <p>My work starts with a supposition of $N$, so that for $n > N$ we have $\vert b \vert ^n < \epsilon$.</p>
<p>Since $0 < \vert b \vert < 1$, we see the logarithm with base $\vert b \vert$ is a decrescent function meaning it will invert the inequality once taken.
$$\vert b \vert ^n < \epsilon $$
$$n >... | marty cohen | 13,079 | <p>You can make janmarqz's result
more explicit.</p>
<p>Since
$|b^n| = |b|^n$,
we can assume that
$0 < b < 1$.</p>
<p>Then
$b = \dfrac1{1+a}$
with $a > 0$,
so that,
from Bernoulli's inequality,
$(1+a)^n \ge 1+an
$.</p>
<p>Therefore,
since
$a = \dfrac1{b}-1$,</p>
<p>$\begin{array}\\
b^n
&=\dfrac1{(1+a)^... |
1,957,084 | <p>Where $n$ is any positive integer.</p>
<p>I'm honestly completely at a loss at how to prove this.<br>
Tested by brute forcing it up to large numbers, and it keeps increasing, although very slowly.</p>
<p>This is actually part of a bigger problem containing harmonic numbers, but I've solved the rest, so that's why ... | user133281 | 133,281 | <p>$$\sum_{i=2^n+1}^{2^{n+1}} \frac1i \geq \sum_{i=2^n+1}^{2^{n+1}} \frac{1}{2^{n+1}} = \frac12.$$ </p>
|
113,725 | <p>Is there a closed form for $\prod_{1 \leq i < j \leq k} (j - i)$? It looks like something like a determinant of a Vandermonde matrix, but I can't seem to get it to fit.</p>
| Brett Frankel | 22,405 | <p>Indeed, the square of this quantity is the discriminant of the polynomial whose roots are the integers from 1 to $k$, so your observation that this is the determinant of a Vandermonde matrix is correct.
None of the below are close forms, but here are two alternative formulas that may (or may not) be helpful: $$\prod... |
443,736 | <p>Find the point on the parabola $3x^2+4x-8$ that is closest to the point $(-2,-3)$.</p>
<p>My plan for this problem was to use the distance formula and then that the derivative to get my answer. I'm having a little trouble along the way.</p>
<p>$$ d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.$$</p>
| Gamma Function | 54,750 | <p>$$ d = \sqrt{(x - (-2))^2 + ((3x^2+4x-8) - (-3))^2} = \sqrt{\left(x+2\right) ^2 + \left(3x^2+4x-5 \right)^2}$$</p>
<p>Instead of minimizing $d$, lets minimize $d^2$. It should be noted that the minimum of $d$ and $d^2$ are exactly the same as $d \geq 0$. This gets rid of the square root, thereby simplifying the pro... |
43,226 | <p>I realize the probability of the following two events are equal. I am curious: is there a reason, besides coincidence, that the probabilities are equal?</p>
<p>Suppose there are five balls in a bucket. 3 of the balls are labelled A, and 2 of the balls are labelled B. There is no way to distinguish between balls lab... | Aryabhata | 1,102 | <p>The claim indeed seems true.</p>
<p>For a way to see this.</p>
<p>Consider the first probability</p>
<p>$P(\{AAABB\}) = \frac{3}{5} \times \frac{2}{4} \times \frac{1}{3} \times 1 = 0.1$</p>
<p>This can actually be written as</p>
<p>$P(\{AAABB\}) = \frac{3}{5} \times \frac{2}{4} \times \frac{1}{3} \times \frac{2... |
172,119 | <p>For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value?
I'm specially interested in what happens for small values of p, before the giant component emerges.</p>
| ericf | 39,187 | <p>In case this isn't obvious, a simple insight used in many papers is that for small p, the network is (whp) simply a random tree, which allows for simple estimates of these and many other properties.</p>
<p>Added in response to comments:
oops, I should have said "tree-like" and one can use "tree counting" techniques... |
172,119 | <p>For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value?
I'm specially interested in what happens for small values of p, before the giant component emerges.</p>
| David White | 11,540 | <p>Have a look at Chapter 11 of Alon and Spencer's <em>The Probabilistic Method</em>. They focus entirely on the case $p = \Theta(1/n)$, and the smallest p they consider is of the form $c/n$, which perhaps counts as "small p" as in your question. In 11.6, they study the smallest case $c < 1$ in detail (they refer to... |
172,119 | <p>For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value?
I'm specially interested in what happens for small values of p, before the giant component emerges.</p>
| Gruddium Fretzke | 159,884 | <p>For <span class="math-container">$p= \frac c N$</span>, the mean number of components per node is <span class="math-container">$(1-s_0) (1- \frac c 2 (1-s_0))$</span>, where <span class="math-container">$s_0$</span> is the fraction of nodes in the largest component. (So, trivially, in the non-percolating phase for <... |
1,645,130 | <p>Is there any known explicit bijection between these two sets? </p>
<p>I know it can be proved that such bijection exists using two injections and Schröder–Bernstein theorem, but I wanted to know whether some explicit bijection is known. I failed to find any except ones constructed awkwardly from the Schröder–Bernst... | Asaf Karagila | 622 | <p>You can do this by steps:</p>
<ol>
<li><p>There are reasonably nice bijections between <span class="math-container">$[0,1]$</span> and <span class="math-container">$(0,1)$</span>, and between <span class="math-container">$(0,1)$</span> and <span class="math-container">$\Bbb R$</span>. So there is a reasonably nice b... |
2,232,952 | <p>Could you please help me solve these quesitons??</p>
<p><a href="https://i.stack.imgur.com/iRgCE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iRgCE.png" alt="enter image description here" /></a></p>
<p>Consider the bases S={<span class="math-container">$u_1, u_2, u_3$</span>}, and T={<span clas... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>If $U$ is the transition matrix from the standard basis to the basis $S$ ( i.e. the matrix that has as columns the vectors $u_i$), and $V$ is the transition matrix from the standard basis to the basis $T$, than the transition from $S$ to $T$ is given by the matrix $V^{-1}U$. </p>
|
142,819 | <p>I am currently studying Serge Lang's book "Algebra", on page 25 it is proved that if $G$ is a cyclic group of order $n$, and if $d$ is a divisor of $n$, then there exists a unique subgroup $H$ of $G$ of order $d$.</p>
<p>I have trouble seeing why the proof (as explained below) settles the uniqueness part.</p>
<p>T... | D_S | 28,556 | <p>Another way is to use the fact that a cyclic group is the same thing as a quotient of $\mathbf{Z}$. If $G$ is a cyclic group, let $a$ be a generator, and define a homomorphism $\mathbf{Z} \rightarrow G$ by $n \mapsto a^n$. This is surjective, so the first isomorphism applies. Now you can classify all subgroups of... |
4,504,768 | <p>We've to prove that
<span class="math-container">$$
\lim_{(x,y)\to(0,0)} \frac{x^3+y^4}{x^2+y^2} =0
$$</span></p>
<p>Kindly check if my proof below is correct.</p>
<p><b>Proof</b></p>
<p>We need to show there exists <span class="math-container">$\delta>0$</span> for an <span class="math-container">$\varepsilon>... | David Sheard | 373,149 | <p><strong>The more general statement is not true</strong>, even if you assume that <span class="math-container">$T$</span> is a set of elements of order <span class="math-container">$2$</span> like transpositions. For an explicit counter example, consider the dihedral group <span class="math-container">$D_8$</span> of... |
637,897 | <p>I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. </p>
<p>My first thought was to take a free subgroup $A_k$ of rank $k$ of the isometries of the hyperbolic plane $H$, which acts fix point free and discontinuous... | Andrea Mori | 688 | <p>I think we are saying the same thing but in a slightly different order. Isn't it true that a sphere with at least 3 points removed is homeomorphic to an open set in a compact Riemann surface of genus $\geq2$?</p>
<p>If so, the latter can be endowed with a constant negative curvature because it is uniformized by the... |
452,653 | <p>If $f:X\rightarrow Y$ is initial in category <strong>Top</strong> then
it is easy to proof that </p>
<blockquote>
<p>(!) the topology on $X$ is the set of preimages of open sets in $Y$. </p>
</blockquote>
<p>Just construct topology $Z$ having
the same underlying subset as $X$ and let the set of these preimages
s... | user37238 | 87,392 | <p>In fact, here your series converges absolutely since we know that
$$ \sum_{n=2}^{+\infty} \frac{1}{n^\alpha (\ln n)^{\beta}}<+\infty \Leftrightarrow \left[(\alpha>1)\text{ or }(\alpha=1\text{ and }\beta >1)\right]$$
You can prove this with integral test. And conclude since the absolute convergence implies t... |
3,849,311 | <blockquote>
<p>What is the remainder when dividing the polynomial
<span class="math-container">$$P(x)=x^n+x^{n-1}+\cdots+x+1$$</span> with the polynomial
<span class="math-container">$$x^3-x$$</span> if <span class="math-container">$n$</span> is a natural odd number?</p>
</blockquote>
<p>So, what I know so far is:</p>... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Experimenting the first few values of <span class="math-container">$n$</span>: <span class="math-container">$\:n= 3,5,7,9,11$</span>, you may conjecture the remainder for the general polynomial <span class="math-container">$P_n(x)=x^n+x^{n-1}+\dots+x^3+x^2+x+1\:$</span> is
<span class=... |
3,849,311 | <blockquote>
<p>What is the remainder when dividing the polynomial
<span class="math-container">$$P(x)=x^n+x^{n-1}+\cdots+x+1$$</span> with the polynomial
<span class="math-container">$$x^3-x$$</span> if <span class="math-container">$n$</span> is a natural odd number?</p>
</blockquote>
<p>So, what I know so far is:</p>... | lhf | 589 | <p>Your approach is fine but you've missed the root <span class="math-container">$x=0$</span> of <span class="math-container">$x^3-x$</span>.</p>
<p>We get <span class="math-container">$R(0)=P(0)=1$</span>, <span class="math-container">$R(1)=P(1)=n+1$</span>, <span class="math-container">$R(-1)=P(-1)=0$</span>.</p>
<p>... |
16,831 | <p>As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presu... | Matthew Towers | 5,316 | <p>Just for fun, here's a proof in non-standard analysis (Nelson-style IST). I write $a \approx b$ to mean that $a-b$ is infinitesimal. </p>
<p>It's enough to prove the result when $f$ is standard. The Weierstrass approximation theorem gives a polynomial $p$ such that $p(x) \approx x f(x)$ for all $x \in [0,1]$. No... |
16,831 | <p>As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presu... | Hans Lundmark | 1,242 | <p>As an aside, the answer is <em>yes</em> if the interval is $(0,\infty)$ instead of $(0,1)$.
For example the "Stieltjes ghost function"
$f(x) = \exp(-x^{1/4}) \sin x^{1/4}$ satisfies
$\int_0^{\infty} f(x) x^n dx = 0$
for all integers $n \ge 0$. </p>
<p>Stieltjes gave this as an example of a case where the
<a href="... |
3,628,277 | <p>I am solving the equation <span class="math-container">$e^z = 1 $</span> in <span class="math-container">$\mathbb{C}$</span>. The book says, other than <span class="math-container">$z = 0$</span>, <span class="math-container">$z = 2 \pi k i$</span> for <span class="math-container">$ k \in \mathbb{Z}$</span> is also ... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>By differentiating the equality, we should prove <span class="math-container">$$\frac{1}{(a+b\cos x)^n}={d\over dx}\frac{A\sin x}{(a+b\cos x)^{n-1}}+{B \over (a+b\cos x)^{n-1}}+{C \over (a+b\cos x)^{n-2}}, \hspace{1cm}$$</span> with <span class="math-container">$|a|\neq |b|$</span>. Als... |
3,628,277 | <p>I am solving the equation <span class="math-container">$e^z = 1 $</span> in <span class="math-container">$\mathbb{C}$</span>. The book says, other than <span class="math-container">$z = 0$</span>, <span class="math-container">$z = 2 \pi k i$</span> for <span class="math-container">$ k \in \mathbb{Z}$</span> is also ... | Quanto | 686,284 | <p>Note,</p>
<p><span class="math-container">$$\frac b{\sin x} \left(\frac{\sin^2x}{(a+b\cos x)^{n-1}}\right)'
=\frac{(n-1)(b^2-a^2)}{(a+b\cos x)^n}+\frac{2a(n-2)}{(a+b\cos x)^{n-1}}
-\frac{n-3}{(a+b\cos x)^{n-2}}$$</span></p>
<p>Integrate both sides and denote <span class="math-container">$I_n= \int \frac{dx}{(a+b\c... |
23,674 | <p>Let $v$ be the 3-adic valuation on $\mathbb{Q}$ and consider the subring $\mathbb{Z}_{(3)}$ of $\mathbb{Q}$ defined by
$$
\mathbb{Z}_{(3)} = \{ x \in \mathbb{Q} : v(x) \geq 0 \}.
$$
That is, $\mathbb{Z}_{(3)}$ is the ring of rational numbers that are integral with respect to $v$. $\mathbb{Z}_{(3)}$ is also the l... | Andrea Mori | 688 | <p>The integral closure of $\Bbb Z$ in $K={\Bbb Q}(\sqrt{-5})$ is the ring of integers ${\cal O}_K={\Bbb Z}[\sqrt{-5}]$, the latter equality because $-5\equiv 3\bmod 4$.</p>
<p>As you observed, ${\Bbb Z}_{(3)}$ is the localization of $\Bbb Z$ at the ideal $(3)$. Then, it follows from general properties (see Atiyah-Mac... |
2,444,196 | <p>The center of a group $G$ is defined as $Z(G):=\{ z\in G : gz = zg, \; \forall g \in G\}$.</p>
<p>The goal is to show that if $\vert G\vert = pq$, where $p$ and $q$ are not necessarily distinct primes then either $G$ is abelian or $Z(G) = \{ e\}$.</p>
<p>I want to suppose that $Z(G) \neq \{ e\}$ and then use the f... | Nicky Hekster | 9,605 | <p>Hint: assume $Z(G) \neq \{1\}$. Then look at $|G:Z(G)| \in \{1,p,q\}$</p>
|
2,444,196 | <p>The center of a group $G$ is defined as $Z(G):=\{ z\in G : gz = zg, \; \forall g \in G\}$.</p>
<p>The goal is to show that if $\vert G\vert = pq$, where $p$ and $q$ are not necessarily distinct primes then either $G$ is abelian or $Z(G) = \{ e\}$.</p>
<p>I want to suppose that $Z(G) \neq \{ e\}$ and then use the f... | GAVD | 255,061 | <p>You already suppose that $Z(G)\neq 1$. Then the order of the quotient group $G/Z(G)$ is one of 1,p,q. </p>
<p>You can follow <a href="https://math.stackexchange.com/questions/106163/show-that-every-group-of-prime-order-is-cyclic">this question</a> to see that all group of prime order is cyclic. So, the group $G/Z(G... |
239,863 | <p>Im trying to reproduce the following graph:
<a href="https://i.stack.imgur.com/tHQX3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tHQX3.png" alt="enter image description here" /></a></p>
<p>I have the parabolas graphed, but I want to indicate the intercepts with the x axis by some point or cros... | MassDefect | 42,264 | <p>Maybe something like this?</p>
<pre><code>eqs = {(x + 5)^2 - 130, (x - 2)^2 - 130};
intercepts = NumericalSort[x /. Solve[Or @@ Thread[eqs == 0], x]];
Plot[
eqs,
{x, -21, 16},
AxesStyle -> White,
Background -> Black,
Frame -> True,
FrameLabel -> {n, \[Epsilon]},
FrameStyle -> White,
FrameTicks... |
2,583,454 | <p>Consider for instance the linear system:</p>
<p>$$\left(
\begin{array}{cc}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)=\left(
\begin{array}{c}
1 \\
2 \\
4 \\
\end{array}
\right)$$</p>
<p>This is over determined and thus has no solution. Y... | Jean Marie | 305,862 | <p>A general principle (some would say "tautology") connecting sets and their defining properties is as follows:</p>
<p>Let $E_1 := \{x | p_1(x)\} $ (with the meaning "$x$ has property $p_1$") and $E_2 := \{x | p_2(x)\} $.</p>
<p>$$\text{If} \ \forall x, (\ p_1(x) \implies p_2(x)), \ \ \text{then} \ \ E_1 \subse... |
8,107 | <p>Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on resear... | Mike Spivey | 2,370 | <p>Since your question is about seeing how your different options play out, and you have a small number of them (the point of my previous question), you can use a <a href="http://en.wikipedia.org/wiki/Decision_tree" rel="nofollow">decision tree</a>. (From Wikipedia: "In decision analysis, a 'decision tree'... is used ... |
8,107 | <p>Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on resear... | Emre | 9,901 | <p>This is a decision-theoretic problem. I suggest you look into <a href="http://en.wikipedia.org/wiki/Multi-armed_bandit" rel="nofollow">multi-armed bandits</a>, which</p>
<blockquote>
<p>have been used to model the problem of managing research projects in a large organization, like a science foundation or a pharma... |
721,449 | <p>I need to determine all the positive divisors of 7!. I got 360 as the total number of positive divisors for 7!. Can someone confirm, or give the real answer?</p>
| Joe K | 64,292 | <p>Just to generalize what others have said, it's a neat little fact that the number of distinct factors of $n!$ is given by:</p>
<p>$$
\prod_{p \in primes}\left( 1 + \sum_{k=1}^{\infty}\left \lfloor \frac{n}{p^k} \right \rfloor \right)
$$</p>
<p>Note that the sum is simply a shortcut to calculating the exponent for ... |
3,237,476 | <p>I have a trouble with calculating the sum of this series:</p>
<p><span class="math-container">$$2+\sum_{n=1}^{\infty}\frac{1-n}{9n^3-n}$$</span></p>
<p>I tried to split it into three separate series like this:
<span class="math-container">$$2+\sum_{n=1}^{\infty}\frac{1-n}{9n^3-n} =2+\sum_{n=1}^{\infty}\frac{2}{3n... | Claude Leibovici | 82,404 | <p>For the direct evaluation of the limit, you have received the good solution from J.G.</p>
<p>You could also consider the partial sum using, as you did, partial fraction decomposition
<span class="math-container">$$\frac{1-n}{9n^3-n}=\frac{1}{3 n-1}+\frac{2}{3 n+1}-\frac{1}{n}$$</span> which makes
<span class="math-... |
926,168 | <p>In how many ways can 3 teachers and 4 pupils be arranged in a line if the pupils and teachers must alternate?
.
how to get the answer?
the ans :144</p>
| duci9y | 165,386 | <p>Thanks to @labbhattacharjee. Here's a step by step version of their solution:</p>
<p>$$\sin 2a + \sin 2b + \sin 2c - \sin 2(a+b+c) = 4 \sin (a+b) \sin (b+c) \sin (c+a)$$</p>
<p>LHS:</p>
<p>$ = 2 \sin (a+b) \cos (a-b) + \sin 2c - \sin((2a+2b)+2c) $
$ = 2 \sin (a+b) \cos (a-b) + 2 \cos (a+b+2c) \sin (-a-b) $
$ = 2 ... |
2,323,845 | <p>I would like to create a 4 on 4 tournament with 8 players (4 players on a team where two teams play against each other each game), where every player plays with every other player an equal number of times. A simple example of this would be if you had a 2 on 2 tournament with 4 players then:</p>
<p>12 v 34</p>
<p>... | AmagicalFishy | 126,921 | <p>I started thinking of it this way: </p>
<p>We have $28$ pairs of players that need to be satisfied three times. Each game can accommodate six pairs. If we can find an optimal configuration of pairings for a certain number of games such that each pair is satisfied, we can just play that configuration three times.</p... |
2,425,127 | <p>I am learning about general solutions to differential equations and would like to ask whether my solution is mathematically correct. </p>
<p>I was asked to find the general solution to the differential equation </p>
<p>$$\frac{dy}{dx} = 2e^{x-y}$$</p>
<p>So I did the following - </p>
<p>$$\int e^y dy = 2\int e^x... | Ross Millikan | 1,827 | <p>You went wrong in two places. First you replaced $2e^x$ with $e^{x^2}$. It seems you were thinking about $2 \ln a= \ln (a^2)$, but note that the $2$ is outside the log here. Second, you are thinking that $e^{x^2}=(e^x)^2$, but the convention is that $e^{x^2}=e^{(x^2)}$ because you can replace $(e^x)^2$ by $e^{2x}... |
425,663 | <p>Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that
$$ \forall x \in X, \|x\|_1 \leq K\|x\|_2 .$$
show then that these two norms are equivalent</p>
| Romeo | 28,746 | <p>Use the open mapping principle: the identity map $F: (X, \Vert \cdot \Vert_2) \to (X, \Vert \cdot \Vert_1)$ is linear and continuous (thanks to hypothesis). Its rank is of the second category, since $X$ is Banach (in particular, a complete metric space under the metric induced by $\Vert \cdot \Vert_1$). </p>
<p>By ... |
2,283,123 | <p>Let $(\mathbb{R}, +, 0)$ be the additive group of reals. Is this structure $\aleph_0$-saturated? </p>
<p>I don't really see how to go about showing this. To show it is not saturated, it is enough to exhibit a type omitted in $(\mathbb{R}, +, 0)$. The interesting statements we can make about groups are usually to do... | Alex Kruckman | 7,062 | <p>Note, as in Zarathustra's answer, that $(\mathbb{R},0,+)$ carries the structure of a $\mathbb{Q}$-vector space. </p>
<p>Let $A$ be a finite subset of $\mathbb{R}$. What are the possible types over $A$? </p>
<p>For every $\mathbb{Q}$-linear combination of elements of $A$, there is a type $p(x)$ which is completely ... |
2,916,685 | <p>What are some common, preferably uncomplicated functions $ f: [a,b] \rightarrow \mathbb{R} $ that are Riemann integrable on $ [c,b] $ for all $ c \in (a,b) $ but not integrable on $ [a,b] $. </p>
<p>I know $ f = 1/x $ is one such function for the interval $ [0,1] $. Are there any other examples? </p>
| zhw. | 228,045 | <p>Any unbounded function $f$ on the closed interval $[a,b]$ such that $f$ is continuous on $(a,b]$ is an example.</p>
|
649,570 | <p>How do we show that there is only one solution to,$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+x}}}}$$</p>
<p>I guess it is only $x=2$.
Please help.</p>
| nbubis | 28,743 | <p>A proof by induction.
Let:
$$f_n(x) = \sqrt[3]{6+\sqrt[3]{6+\ldots+\sqrt[3]{6+x}}},\ g_n(x) = \sqrt{2+\sqrt{2+\ldots+\sqrt{2+x}}}$$
With $n$ terms. Then for $n=1$ you can easily solve the cubic equation to show that $f_1=g_1 $ only at $x=2$ (over the reals).</p>
<p>Now assume our claim is true for $n$, i.e. that $... |
2,797,329 | <p>The function $y_1 = x^2$ is a solution of
$x^2y'' − 3xy' + 4y = 0$.
Find the general solution of the nonhomogeneous linear differential equation
$x^2y'' − 3xy' + 4y = x^2$</p>
<p>I know the equation $x^2y'' − 3xy' + 4y = 0$ is a Euler-Cauchy equation but I'm not sure how to proceed with this question; any help is ... | user577215664 | 475,762 | <p>$x^2$ is a known solution to the homogeneous equation</p>
<p>Let's consider $y_p=x^2v$ as a general solution to the original equation. Then</p>
<p>$$
y_p=x^2v \implies
\begin{cases}
y'_p=2xv+x^2v' \\
y_p''=2v+4xv'+x^2v''
\end{cases}
$$
The equation becomes
$$x^2v''+xv'=1 \implies xv''+v'=\frac 1x$$
$$(xv')'=\frac... |
2,528,306 | <p>The answer is 648 but I tried to solve this problem in reverse, so I ended up with 630. Theee are 10 ways to pick the third digit, 9 ways to pick the second digit, and 7 ways to pick the first digit. So why do these answers differ. Please do not close this question as I am trying to learn mathematics and I have stum... | N. F. Taussig | 173,070 | <p>The main restriction that we need to consider is that the hundreds digit cannot be zero. By choosing the hundreds digit first, we handle this restriction immediately, after which our choices depend only on which digits have already been selected.</p>
<p>The restriction that the hundreds digit may not be zero means... |
80,899 | <p>This is related to a previous post of mine (<a href="https://math.stackexchange.com/questions/78669/limit-superior-of-a-sequence-showing-an-alternate-definition">link</a>) regarding how to show that for any sequence $\{x_{n}\}$, the limit superior of the sequence, which is defined as $\text{inf}_{n\geq 1}\text{sup }... | Brian M. Scott | 12,042 | <p>The problem is that you’ve misunderstood the notion of <em>limit point of a sequence</em>. I prefer the term <em>cluster point</em> in this context, but by either name it’s a point $x$ such for any nbhd $U$ of $x$ and any $n\in\mathbb{N}$ there is an $m\ge n$ such that $x_n \in U$. Thus $1$ and $-1$ are both cluster... |
18,879 | <p>A first-order sentence is (logically) valid iff it's true in every interpretation. And it's valid iff it can be deduced from the FO axioms alone.</p>
<p>One normal case of showing that a FO sentence is true is deducing it (syntactically).</p>
<p>I guess that indirect proofs have to be interpreted more "semanticall... | Aram Kasner | 6,215 | <p>As Moron says, Cardano works. <a href="http://books.google.com/books?id=w47cyQLVQowC&pg=PA228">You merely need to be a bit more careful than usual.</a></p>
|
72,669 | <p>I encountered this site today <a href="https://code.google.com/p/google-styleguide/">https://code.google.com/p/google-styleguide/</a> regarding the programming style in some languages. What would be best programming practices in Mathematica, for small and large projects ?</p>
| Kevin Groenke | 62,350 | <p>Not a general Answer, but I haven't seen such a thing in any language else. What I have found using a 1080p Monitor with the Mathematica Benchmark is: Formatting every function by indending every argument and place the commata on the first level, e.g.</p>
<pre><code>If[ Head@list[[1]] == Symbol,
foo[]
,
If[... |
72,669 | <p>I encountered this site today <a href="https://code.google.com/p/google-styleguide/">https://code.google.com/p/google-styleguide/</a> regarding the programming style in some languages. What would be best programming practices in Mathematica, for small and large projects ?</p>
| Jules Manson | 37,721 | <h1>Separate Styles and Options from Functional Code</h1>
<p><em>Almost nothing was said about</em> <code>Styles</code> <em>and</em> <code>Options</code> <em>so I think I will do that now.</em></p>
<h2>Basis</h2>
<p>In all modern languages it is a <em>best practice</em> to separate presentation and behavior of interfac... |
523,529 | <p>I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this:<br>
$V(1): 1≤1 \text{ true}$ <br>
$V(n): n!≤((n+1)/2)^n$ <br>
$V(n+1): (n+1)!≤((n+2)/2)^{(n+1)}$<br><br></p>
<p>and I've got : <br>$(((n+1)/2)^n)\cdot(n+1)≤((n+2)/2)^{(n+1)}$ <br>$((n+1)^n)n(n+1)... | Smylic | 100,361 | <p>If you really need induction let it be.
Base is $n = 0$: $0! = 1 \le 1 = \left(\frac12\right)^0$.
By induction hypothesis the inequality holds for $n = k$. Let proove it for $n = k+1$.
$$k! \le \left(\frac{k+1}2\right)^k,\\
k!(k+1) \le \left(\frac{k+1}2\right)^k(k+1),$$
$$(k+1)! \le \frac{(k+1)^{k+1}}{2^k}.\tag{*}$$... |
980,941 | <p>How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?</p>
| Timbuc | 118,527 | <p>$$\sum_{k=1}^n(1+\ldots+k)=\sum_{k=1}^n\frac{k(k+1)}2=\frac12\left(\sum_{k=1}^nk^2+\sum_{k=1}^nk\right)$$</p>
<p>and now</p>
<p>$$\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6$$</p>
|
2,292,511 | <p>I need some help to solve this problem:</p>
<blockquote>
<p>Evaluate A such that the exponential distribution with parameter $\alpha, P(X = x) = Ae^{−\alpha x}$, is normalized.
Here, $\alpha > 0$ and $\Omega = \mathbb{R}_{+}$.</p>
</blockquote>
<p>I've been trying to evaluate the following Integral </p>
<p... | Jack D'Aurizio | 44,121 | <p>With Mathematica notation we have that $\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{a+\sin^2\theta}}$ is an elliptic integral, namely
$$ \frac{1}{\sqrt{1+a}}\,K\left(\frac{1}{1+a}\right) $$
hence by differentiating with respect to $a$ we get that
$$ \int_{0}^{\pi/2}\frac{d\theta}{(a+\sin^2\theta)^{3/2}} = \frac{1}{a\sqrt{a... |
3,256,767 | <p>So I'm trying to understand a solution made by my teacher for a question that asks me to determine whether the following is true. I'm having trouble understanding where some values in the steps are coming from.</p>
<p>Like for the first part, I don't really get where n≥5 came from. My guess is getting 16n^2 + 25 to... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Let the matrix of <span class="math-container">$\varphi$</span> be the matrix
<span class="math-container">$$A=\begin{pmatrix}
a_1&b_1&c_1&d_1\\
a_2&b_2&c_2&d_2 \\
a_3&b_3&c_3&d_3
\end{pmatrix}.$$</span></p>
<p>The conditions on <span class="math-co... |
3,342,094 | <p>I am asked to prove following proposition:</p>
<blockquote>
<p><strong>Proposition 1.</strong> If an invertible matrix <span class="math-container">$\mathbf A$</span> has a left inverse <span class="math-container">$\mathbf{B}$</span> and a right inverse <span class="math-container">$\mathbf{C}$</span>, then <spa... | Henry | 6,460 | <p>Assuming that <span class="math-container">$\mathbf A^{-1}$</span> is meaningful seems premature until you have proved the assertion that <span class="math-container">$\mathbf B = \mathbf C$</span></p>
<p>Instead use something like <span class="math-container">$\mathbf{B(AC)} = \mathbf{(BA)C}$</span> since matrix m... |
2,777,631 | <p>Angle bisectors of traingle $ABC$ meet its circum-circle ( after passing through in-center) at opposite points $P, Q$, and $R$ respectively on the circumcircle. </p>
<p>Find $\angle RQP.$ </p>
<p>Is there any way of getting the answer through its in-center properties?</p>
<p>Ans = $90-\frac{B}{2}$ </p>
| Alex D | 477,539 | <ol>
<li><p>Suppose $1<0$. Since $x^{2}\ge0$ holds for all $x$, we also have $$(1)\cdot(1)=1^{2}=1\ge0,$$
hence a contradiction.$$\\ \\$$</p></li>
<li><p>Your second part is correct, for suppose $a>0$ and $a^{-1}<0$. Multiplying through by $a$ doesn't change the order, so
$$a^{-1}<0\implies a^{-1}\cdot a&l... |
270,624 | <p>For a polynomial $f(x) = \sum_{i=0}^dc_ix^i \in \mathbb Z[x]$ of degree $d$, let</p>
<p>$$
H(f):=\max\limits_{i=0,1,\ldots, d}\{|c_i|\}
$$</p>
<p>denote the naive height. Further, define</p>
<p>$$
R(M, r, d) := \#\{f(x) \colon \text{$H(f) \leq M$, $\deg f = d$ and $f(x)$ has extactly $r$ real roots}\}.
$$</p>
<p... | Liviu Nicolaescu | 20,302 | <p>This is a bit too long to be a comment. Let me phrase this probabilistically by saying that the coefficients $c_i$ are independent (discrete) random variables uniformly distributed on the finite set
$$
\big\{\, -M,-(M-1),\dotsc, -1,0, 1, \dotsc, (M-1), M\,\big\}.
$$
Denote by $N_d(f)$ the number of real zeros ... |
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