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 |
|---|---|---|---|---|
4,133,782 | <p>I am having trouble finding a formula that connects the two and can produce an answer. Anyone know how this is done? I tried y=mx+b, m=3, and b=5-a. But I don't know what to do next or did I even start right.</p>
| Still Learning | 362,881 | <p>As an add on to user169852’s proof, I would note that Strang’s argument that <span class="math-container">$ A^T A $</span> is invertible if A has full column rank is fairly simple.</p>
<p>He shows that for any matrix A, <span class="math-container">$ A^T A $</span> has the same nullspace as A:</p>
<p>(1) Clearly the... |
1,386,343 | <p>Let $P$ be an idempotent $n \times n$ matrix ($P^2 = P$). What is $(I + P)^{-1}$? I've been thinking about this problem for a while, but can't find an answer. I tried a few examples, but I'm not sure what the general pattern is.</p>
| Emilio Novati | 187,568 | <p>Hint:
$$
(I+P)(P-2I)=P-2I+P-2P=-2I
$$</p>
|
301,662 | <p>This is a challenging puzzle I heard from my little brother.</p>
<p>For some $n$ and $x$, $\sum_{k=1}^n \sin^{2k}(x) = 2013$.</p>
<p>Is it possible to deduce
$$\sum_{k=1}^n \cos^{2k}(x) \text{ ?}$$</p>
<p>Edit:
I've just noticed something which now seems obvious to me.<br>
Choose $n = 2013$ and $x = \pi/2$ which ... | Gerry Myerson | 8,269 | <p>As noted, the equation holds if $n=2013$ and $x=\pi/2$. Now let $n=2014$. By continuity, there is a value of $x$ a tiny bit smaller than $\pi/2$ for which the equation will hold, and, for this value of $x$, the cosine sum will not be zero. So one cannot deduce the cosine sum from knowing the first equation holds. </... |
2,349,124 | <p>I keep on hitting a road block in trying to solve this, especially when trying to prove it going from the right hand side to the left hand side. </p>
| Atif Farooq | 451,530 | <p><em>Proof</em>. </p>
<p>$(\Rightarrow).$
Assume that $X=\varnothing$ and that $y\in Y$, evidently $y\not\in X$ therefore $y\in Y\backslash X$ consequently $y\in (Y\backslash X)\cup (X\backslash Y)$, since $y$ was arbitrary we may now conclude that $Y\subseteq (Y\backslash X)\cup (X\backslash Y)$.</p>
<p>Assume now... |
2,349,124 | <p>I keep on hitting a road block in trying to solve this, especially when trying to prove it going from the right hand side to the left hand side. </p>
| Peter Szilas | 408,605 | <p>$ X,Y \subset T$.</p>
<p>$X = \emptyset \iff $</p>
<p>$Y = ( X \cap Y^C ) \cup ( X^C \cap Y )$.</p>
<p>1) $\Rightarrow$ :</p>
<p>Let $X = \emptyset$ .</p>
<p>$( X \cap Y^C ) \cup ( X^C \cap Y)$ =</p>
<p>$\emptyset \cup ( T \cap Y) = Y$.</p>
<p>2) $\Leftarrow$ :</p>
<p>Let $Y = ( X \cap Y^C ) \cup ( X^C \ca... |
716,561 | <p>I want to show that $$\sum\limits_{n=1}^{\infty}\frac{i^n}{\sqrt{n}}$$ is convergent, but not absolutely convergent.</p>
<p>Demonstrating that it is not absolutely convergent is easy since $$\left|\frac{i^n}{\sqrt{n}} \right|=\frac{1}{\sqrt{n}}$$ but $$\sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}}$$ diverges. I'm s... | Community | -1 | <p>We have</p>
<p>$$\left|\sum_{k=1}^n i^k\right|=\left|\frac{1-i^n}{1-i}\right|\le\frac{1+|i^n|}{\sqrt 2}=\sqrt2$$
and the sequence $\left(\frac1{\sqrt n}\right)_n$ is decreasing to $0$ so by the <a href="http://en.wikipedia.org/wiki/Dirichlet%27s_test">Dirichlet's test</a>
the given series is convergent and since it... |
716,561 | <p>I want to show that $$\sum\limits_{n=1}^{\infty}\frac{i^n}{\sqrt{n}}$$ is convergent, but not absolutely convergent.</p>
<p>Demonstrating that it is not absolutely convergent is easy since $$\left|\frac{i^n}{\sqrt{n}} \right|=\frac{1}{\sqrt{n}}$$ but $$\sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}}$$ diverges. I'm s... | Carl Butcher | 85,836 | <p>$\sum_{n=1}^\infty \frac{i^n}{\sqrt{n}} = \sum_{n=1}^\infty \frac{i^{2n}}{\sqrt{2n}} + \sum_{n=1}^\infty \frac{i^{2n-1}}{\sqrt{2n-1}} = \sum_{n=1}^\infty (-1)^{n-1}\frac{1}{\sqrt{2n}} + i \sum_{n=1}^\infty (-1)^{n-1}\frac{1}{\sqrt{2n-1}}$</p>
<p>But the decreasing sequences $\{\frac{1}{\sqrt{2n}}\}_n$ and $\{\frac{... |
112,503 | <p>I am working the a subject guide on involving $L$-Systems and have the alphabet $A = \{a, b, c\}$. The initiator is the string $a$ and the rules of substitution $a \to ba$, $b \to ccb$, $c \to a$. </p>
<p>The study guide gives the first five generations as:</p>
<p>$$[a] \to [ba] \to [ccba] \to [acba] \to [aaba] \t... | Trismegistos | 23,730 | <p>It looks like only one symbol is substituted in one step. The symbol $c$ gets highest priority, followed by $b$ and then $a$. When there are multiple instances of the same symbol, the leftmost is changed. But of course this is guessing, and the example is too short to allow much confidence.</p>
|
2,756,332 | <p>It is <a href="https://math.stackexchange.com/questions/985879/relation-between-trace-and-rank-for-projection-matrices">not difficult</a> to show that if $A \in M_n(k)$ for some field $k$, and $A^2=A$ then $\operatorname{tr}(A) = \dim(\operatorname{Im}(A))$</p>
<p>In <a href="https://mathoverflow.net/questions/1352... | Martin Argerami | 22,857 | <p>Given any $A\in M_n(\mathbb C)$, we have $A=\text{Re}\,A+i\text{Im}\,A$, where
$$
\text{Re}\,A=\frac{A+A^*}2,\ \ \ \text{Im}\,A-\frac{A-A^*}{2i}
$$
are selfadjoint. So it is enough to test the assertion for selfadjoint matrices. In such case we have the spectral theorem available, which tells us that if $A=A^*$, th... |
3,439,626 | <p>I need to proof the following statement:</p>
<p>Let <span class="math-container">$a, b, n \in \Bbb{Z}$</span> with <span class="math-container">$ n≥ 2, gcd(a,n)=1$</span>. Proof that if <span class="math-container">$s_{1},s_{2}$</span> are solutions to <span class="math-container">$ax\equiv b \pmod{n}$</span>, the... | Andrea Mori | 688 | <p>If <span class="math-container">$a\in\Bbb Z$</span> is such <span class="math-container">${\rm gcd}(a,n)=1$</span> then <span class="math-container">$a$</span> has a multiplicative inverse modulo <span class="math-container">$n$</span>, i.e. there exists <span class="math-container">$a^*\in\Bbb Z$</span> such that <... |
4,559,503 | <blockquote>
<p>Let <span class="math-container">$\displaystyle f(x)=\frac{1+\cos(2\pi x)}2$</span> for <span class="math-container">$x\in\mathbb R$</span>, and <span class="math-container">$f^n=\underbrace{ f \circ \cdots \circ f}_{n}$</span>. Is it true that for Lebesgue almost every <span class="math-container">$x$<... | acreativename | 347,666 | <p>It suffices for iteration purposes to restrict <span class="math-container">$f$</span> to <span class="math-container">$[0,1]$</span> as <span class="math-container">$f(.) \in [0,1]$</span>.</p>
<p>Set</p>
<p><span class="math-container">$$A:= \{x: x\in[0,1], \exists n \in \mathbb{N};\text{ } f^{n}(x) \in (l_{2},1]\... |
2,390,077 | <p>is it possible to find a matrix $B$ which fulfills:</p>
<p>$BAA^TB^T=I$, where $I$ is identity matrix and $A$ strictly lower triangular?</p>
<p>Thank you very much in advance!</p>
| tiefi | 415,002 | <p>It is even never possible. Say $A$ is a $n\times n$ matrix. Then, because it is strictly(!) lower triangular, $A$ has rank at most $n-1$. So also $BAA^TB^T$ has rank at most $n-1$. But on the right side of your equation, you have $I$, which has rank $n$. </p>
|
1,342,570 | <p>So, this was my initial proof:</p>
<hr>
<p>Assume $R$ is a ring, and $a,b\in R$</p>
<p>Let $x_1$ and $x_2$ be solutions of $ax=b$</p>
<p>Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$</p>
<p>Thus, we have $x_1-x_2=0_R \Rightarrow x_1=x_2$, and only one solution exists.</p>
<hr>
<p>O... | Brian Fitzpatrick | 56,960 | <p>Let $R$ be a unitial ring with $a,b\in R$ where $a$ is a unit. Then $ax=b$ if and only if $x=a^{-1}b$. Indeed, if $ax=b$, then $$x=1_Rx=a^{-1}ax=a^{-1}b$$
Conversely
$$
a(a^{-1}b)=(aa^{-1})b=1_Rb=b
$$</p>
|
1,342,570 | <p>So, this was my initial proof:</p>
<hr>
<p>Assume $R$ is a ring, and $a,b\in R$</p>
<p>Let $x_1$ and $x_2$ be solutions of $ax=b$</p>
<p>Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$</p>
<p>Thus, we have $x_1-x_2=0_R \Rightarrow x_1=x_2$, and only one solution exists.</p>
<hr>
<p>O... | Alex Mathers | 227,652 | <p>Your solution is close to being great. First off, they told you that $R$ is ring with unity because only those can have units. Second, your proof doesn't require knowledge that it is an integral domain. Towards the end, you have</p>
<p>$$a(x_1-x_2)=0$$</p>
<p>Then, because $a$ is a unit, $a^{-1}$ exists, and</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-$fac... | Zev Chonoles | 264 | <p>It appears to be a conflict between the code formatting and MathJax. For comparison, here is your code, but with all of the dollar signs replaced by S's:</p>
<pre><code> my @cards = qw(BB BR RR);
my Sn_trials = shift || 100;
for (1 .. Sn_trials) {
my Scard = Scards[ int(rand 3) ];
my @faces ... |
138,800 | <h1>Background</h1>
<p>I have a block of code, reproduced at the bottom of this post, consisting of combined \$PreRead and \$PrePrint statements, that automatically formats outputs as 'input = output', and also allows easy inline combination of math and text (allowing the text to be placed either before, after, or bot... | Mr.Wizard | 121 | <h1>Pane</h1>
<p>It appears that someone along the way took out my <a href="https://mathematica.stackexchange.com/a/11987/121">carefully placed</a> <a href="http://reference.wolfram.com/language/ref/Pane.html" rel="nofollow noreferrer"><code>Pane</code></a> which prevented downsizing of graphics. If you just want the... |
3,902,937 | <p>I wish to compute the number of possible partitions <span class="math-container">$S$</span> of a set of cardinal <span class="math-container">$np$</span> into <span class="math-container">$n$</span> subsets of cardinal <span class="math-container">$p$</span>.
It is easy to obtain the formula :
<span class="math-cont... | Wuestenfux | 417,848 | <p>"I wish to compute the number of possible partitions S of a set of cardinal np into n subsets of cardinal p."</p>
<p>Hint: Stirling numbers of 2nd kind: <span class="math-container">$S(n,k)$</span> is the number of partitions of an <span class="math-container">$n$</span> element set into <span class="math-... |
3,902,937 | <p>I wish to compute the number of possible partitions <span class="math-container">$S$</span> of a set of cardinal <span class="math-container">$np$</span> into <span class="math-container">$n$</span> subsets of cardinal <span class="math-container">$p$</span>.
It is easy to obtain the formula :
<span class="math-cont... | Phicar | 78,870 | <p>You are closed, but not quite. Just put your <span class="math-container">$n\cdot p$</span> objects in a line and permute them. Then divide them from left to right taking <span class="math-container">$p$</span> at a time and divide by the order in each group and for the order of the pairings, you will get
<span clas... |
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 ... | Donald Splutterwit | 404,247 | <p>Suppose we have a Young tableaux with shape
\begin{eqnarray*}
(\underbrace{k,\cdots,k}_{a_k \text{times}}, \underbrace{k-1,\cdots,k-1}_{a_{k-1} \text{times}},\cdots,\underbrace{2,\cdots,2}_{a_2 \text{times}}\underbrace{1,\cdots,1}_{a_1 \text{times}})
\end{eqnarray*}
This will give a contribution of $x^n$ to the gen... |
385,537 | <p>How would you go about proving the following?</p>
<p>$${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$</p>
<p>This is what I've done so far:</p>
<p>$$LHS = {1+\cos^2 A -2\cos A + 1 - \cos^2A \over \sin A(1-\cos A)}$$</p>
<p>....no idea how to proceed .... X_X</p>
| user123 | 76,473 | <p>I'll go step by step.</p>
<p>$${1 - \cos A \over \sin A} + {\sin A \over 1 - \cos A}$$</p>
<p>$${(1 - \cos A)^2 \over \sin A (1 - \cos A)} + {\sin^2 A \over \sin A(1 - \cos A)}$$</p>
<p>$$(1 - \cos A)^2 + \sin^2 A \over \sin A(1 - \cos A)$$</p>
<p>Expanding $(1 - \cos A)^2$ yields:</p>
<p>$$1 - 2\cos A + \cos^2... |
385,537 | <p>How would you go about proving the following?</p>
<p>$${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$</p>
<p>This is what I've done so far:</p>
<p>$$LHS = {1+\cos^2 A -2\cos A + 1 - \cos^2A \over \sin A(1-\cos A)}$$</p>
<p>....no idea how to proceed .... X_X</p>
| Inceptio | 63,477 | <p><strong>Hint:</strong></p>
<p>$1-\cos A=1-(1-2 \sin^2\dfrac{A}{2})=2\sin^2 \dfrac{A}{2}$</p>
<p>$\dfrac{2 \sin^2 \dfrac{A}{2}}{2 \sin \dfrac{A}{2} \cos \dfrac{A}{2}}=\tan \dfrac{A}{2}$</p>
<p>The other expression will be $\cot \dfrac{A}{2}$</p>
<p>$(\tan^2 \dfrac{A}{2}+1) /\tan\dfrac{A}{2}= \dfrac{\sec^2 A \cos ... |
385,537 | <p>How would you go about proving the following?</p>
<p>$${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$</p>
<p>This is what I've done so far:</p>
<p>$$LHS = {1+\cos^2 A -2\cos A + 1 - \cos^2A \over \sin A(1-\cos A)}$$</p>
<p>....no idea how to proceed .... X_X</p>
| lab bhattacharjee | 33,337 | <p>LCM is not required.</p>
<p>Observe that the first term already has $\sin A$ in the denominator.</p>
<p>$$\text{Now,}\frac{\sin A}{1-\cos A}=\frac{\sin A(1+\cos A)}{1-\cos^2A}=\frac{\sin A(1+\cos A)}{\sin^2A}=\frac{1+\cos A}{\sin A}$$</p>
<p>$$\text{So,}{1- \cos A \over \sin A } + { \sin A \over 1- \cos A} =\frac... |
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... | Siong Thye Goh | 306,553 | <p>If $x \in \mathbb{R}$,
$$\cos^2x - \sin^2 x= \cos(2x)$$</p>
<p>$$2 \sin x \cos x =\sin(2x)$$ </p>
<p>$$|(\cos x + i \sin x)^2|=\cos^2(2x)+\sin^2(2x)=1$$</p>
|
1,484,838 | <p>Ok, so I just learned trig identities and I come across this problem that had it's answer to it, and I have no idea how they got to that answer.</p>
<p>Here is the problem:</p>
<p>$$
\frac{-\sec\theta}{1-\cos\theta}=\frac{-1-\sec\theta}{\sin^2\theta}
$$</p>
<p>Now the problem calls for the left side to be adjust... | gdm | 196,478 | <p>$1= cos(\theta)^2 + sin(\theta)^2$
~Pythagoras
<a href="https://i.stack.imgur.com/aFKYH.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/aFKYH.jpg" alt="enter image description here"></a></p>
|
12,690 | <p>I understand Lie groups are defined by the structure constants associated with the lie brackets, which are treated as commutators in quantum mechanics, but i dont know of a math theory related to group theory to define or use an anti commutator. If Lie groups theory uses the commutator, what theory uses the anti com... | Raskolnikov | 3,567 | <p>The algebraic structure corresponding most naturally to the anticommutator is that of the <a href="http://en.wikipedia.org/wiki/Clifford_algebra" rel="nofollow">Clifford algebras</a>.</p>
<p>One can construct a Clifford algebra by defining $n$ generating elements $\mathbf{e}_j$ with $j \in \{1,\ldots,n\}$ such that... |
185,112 | <p>Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:</p>
<ul>
<li>The entries of $A$ are $\in \{0, 1\}$.</li>
<li>For all pairs of columns $u, v$ of $A$ the entries of $u - v$ are never either all non-negative or all non-positive (i.e. there is a positive entry and a ... | Joel David Hamkins | 1,946 | <p>Unless I have misunderstood, here is a counterexample. Let $A$ be the $2\times 2$ identity matrix. This has your Pareto property on $x-y$, but if $x=[{a\atop b}]$, then $Ax=x$, and there is no way to have both $a+b=0$ and both $a$ and $b$ non-negative with at least one positive.</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... | Quanto | 686,284 | <p>Substitute <span class="math-container">$x=\sqrt{\frac ab }\frac1t$</span> to get</p>
<p><span class="math-container">$$\int \frac{dx}{x\sqrt{a - bx^2}}=-\frac1{\sqrt a}\int \frac{dt}{\sqrt{t^2-1}}= - \frac1{\sqrt a}\cosh^{-1}t=- \frac1{\sqrt a}\ln\left(t+\sqrt{t^2-1}\right)
$$</span></p>
<p>which is equal to <span ... |
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,... | marty cohen | 13,079 | <p>$\frac{e^x+e^{-x}}{e^x-e^{-x}}
=\frac{e^x-e^{-x}+2e^{-x}}{e^x-e^{-x}}
=1+\frac{2e^{-x}}{e^x-e^{-x}}
=1+\frac{2}{e^{2x}-1}
\to 1
$
as
$x \to \infty
$.</p>
|
4,204,133 | <p>Trying to solve <a href="https://math.stackexchange.com/questions/4201328/how-can-i-arrange-a-group-of-people-at-tables-and-switch-them-around-so-that-no">this problem</a> led me to consider the following generalization.</p>
<p>Let <span class="math-container">$g$</span> and <span class="math-container">$p$</span> b... | Maximilian Janisch | 631,742 | <p>Hint: Is <span class="math-container">$]z-r, z+r[\times\{0\}$</span> an open subset of <span class="math-container">$\mathbb R^2$</span> ?</p>
|
1,511,246 | <blockquote>
<p>What is the value of $0.7\overline{54}$ +$0.69\overline2$?</p>
<p>(a) $\frac{1813}{900}$ (b) $\frac{1783}{910}$ (c) $\frac{14323}{9900} (d) \frac{13243}{9900}$</p>
</blockquote>
<p>I get</p>
<p>@edit</p>
<p>$$754-7/990 + 692-69/900$$=$747$/$990$ + $623$/$900$=$1$/$90$($747$/$11$ + $623$/$10$)<... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Easier to convert to regular fractions:
$$
0.7\overline{54} = \frac{7}{10} + \frac{54}{99 \cdot 10}
$$
Can you convert the other one?</p>
|
112,432 | <p>a) Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then $$|h'(z)|\le \frac{2}{(1-|z|^2)^2}.$$
a') Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then the inequality $$|h'(z)|\le... | fedja | 1,131 | <p>Looks like we are closing the question anyway, so I'll just provide a counterexample quickly before the final vote is cast. </p>
<p>If you think a bit of what is asked and what the natural freedoms and scalings are present here, you'll see that it is enough to get an analytic $f$ in the right half-plane $x>0$ ($... |
3,415,378 | <p>I am looking for an estimation or an approximation of </p>
<p><span class="math-container">$\sum _{k=1}^{n}{\log(k)\binom {n}{k}}$</span></p>
<p>Any hints will be appreciated.
Thank you.</p>
| Jack D'Aurizio | 44,121 | <p>Interesting question. Frullani's integral provides an integral representation for <span class="math-container">$\log(k)$</span>,
<span class="math-container">$$ \log(k) = \int_{0}^{+\infty}\frac{e^{-x}-e^{-kx}}{x}\,dx $$</span>
which in combination with the binomial theorem leads to
<span class="math-container">$$ \... |
428,841 | <p>Let $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$</p>
<p>a) Show that $x_{n} < x_{n+1}$</p>
<p>b) Show that $x_{n+1}^{2} \leq 1+ \sqrt{2} x_{n}$</p>
<p>Hint : Square $x_{n+1}$ and factor a 2 out of the square root</p>
<p>c) Hence Show that $x_{n}$ is bounded above by 2. Deduce that $\lim\limits_{n... | DonAntonio | 31,254 | <p>Hints: Induction on</p>
<p>$$\bullet\;\;x_n<x_{n+1}\iff 1+\sqrt{2+\sqrt{3\ldots+\sqrt n}}<1+\sqrt{2+\sqrt{3+\ldots+\sqrt{n+\sqrt{n+1}}}}\iff$$</p>
<p>$$2+\sqrt{3+\ldots+\sqrt n}<2+\sqrt{3+\ldots\sqrt{n+1}}\iff\ldots$$</p>
<p>$$\bullet\bullet\;x_{n+1}^2=1+\sqrt{2+\sqrt{3+\ldots+\sqrt{n+1}}}\le 1+\sqrt2\le... |
1,606,202 | <p>I'm having trouble figuring out why these two different ways to write this combination give different answers. Here is the scenario:</p>
<p>Q: Choose a group of 10 people from 17 men and 15 women, in how many ways are at most 2 women chosen?</p>
<p>Solution A: From 17 men choose 8, and from 15 women choose 2. Or f... | Thomas Andrews | 7,933 | <p>Consider a simpler problem. How many ways are there to choose ten men from a set of ten men? By your second reasoning, choose eight first, then choose another two. That gives a total of:
$$\binom{10}{8}\binom{2}{2}=45$$
Why is that reasoning problematic?</p>
|
3,985,447 | <p>Which of these is a possible solution for
<span class="math-container">$$\cos^2(x)+\sin^2(x)-1=0$$</span>
in the interval <span class="math-container">$x\in[0,2\pi]$</span></p>
<p>a. <span class="math-container">$x=\frac{\pi}{4}$</span><br />
b. <span class="math-container">$x=\pi$</span><br />
c. <span class="math-... | John Smith | 836,805 | <p>like aryan said it is an identity.
<span class="math-container">$\sin^2(x)+\cos^2(x)=1$</span> for all <span class="math-container">$x$</span> therefore with a simple substitution you have <span class="math-container">$1-1=0$</span>
so it is true for all <span class="math-container">$x$</span></p>
|
3,971,059 | <p><span class="math-container">$x_n$</span> is a sequence. The only thing I can do here is just write the definition of <span class="math-container">$\lim_{n \to \infty} x_n=a$</span>, but that doesn' t seem helpful to me. I tried looking for some inequalities that would help, and the only thing that I found that I t... | Servaes | 30,382 | <p>This follows immediately, by induction on <span class="math-container">$k$</span>, from the product rule for limits; if the limits <span class="math-container">$\lim_{n\to\infty}a_n$</span> and <span class="math-container">$\lim_{n\to\infty}b_n$</span> exist, then
<span class="math-container">$$\lim_{n\to\infty}(a_n... |
3,910,053 | <p>The function
<span class="math-container">$u(x,t) = \frac{2}{\sqrt{\pi}}$$\int_{0}^\frac{x}{\sqrt{t}} e^{-s^2}ds$</span>
satisfies the partial differential equation</p>
<p><span class="math-container">$$\frac{\partial u}{\partial t} = K\frac{\partial^2u}{\partial x^2}$$</span></p>
<p>where <span class="math-containe... | egreg | 62,967 | <p>If <span class="math-container">$\lambda$</span> is an eigenvalue of <span class="math-container">$A$</span>, then <span class="math-container">$Av=\lambda v$</span> for some <span class="math-container">$v\ne0$</span>. Therefore
<span class="math-container">$$
(A+2I)v=Av+2v=\lambda v+2v=(\lambda+2)v
$$</span>
and s... |
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... | Rudy the Reindeer | 5,798 | <p>This is a sum over all entries $M_{ij}$ of $M$, multiplying the diagonal entries $M_{ii}$ by $2$.</p>
|
4,046,356 | <p>Recently, some of the remarkable properties of second-order
Eulerian numbers <span class="math-container">$ \left\langle\!\!\left\langle n\atop k\right\rangle\!\!\right\rangle$</span> <a href="https://oeis.org/A340556" rel="nofollow noreferrer">A340556</a> have been proved on MSE [ <a href="https://math.stackexchan... | Peter Luschny | 406,546 | <p>A sketch in three steps: First, we need a definition for <span class="math-container">$ \operatorname{W}_{n}(x)$</span>. With this, we do not want to assume the
second-order Eulerian numbers. So we take the RHS of the identity.</p>
<p>The next and primary step is to show that in general</p>
<p><span class="math-cont... |
292,651 | <blockquote>
<p>Does an integer $9<n<100$ exist such that the last 2 digits of $n^2$ is $n$? If yes, how to find them? If no, prove it.</p>
</blockquote>
<p>This problem puzzled me for a day, but I'm not making much progress. Please help. Thanks.</p>
| Hanul Jeon | 53,976 | <p>this problem is equivalent to $n^2\equiv n \pmod{100}$. and by <a href="http://www.wolframalpha.com/input/?i=x%5E2-x=0%20mod%20100" rel="nofollow">wolframalpha</a>, solution of this equation is $n=25,76$.</p>
|
292,651 | <blockquote>
<p>Does an integer $9<n<100$ exist such that the last 2 digits of $n^2$ is $n$? If yes, how to find them? If no, prove it.</p>
</blockquote>
<p>This problem puzzled me for a day, but I'm not making much progress. Please help. Thanks.</p>
| lab bhattacharjee | 33,337 | <p>So, we need $n^2\equiv n\equiv{100}\iff 100\mid n(n-1)$</p>
<p>Now, $(n.n-1)=1$ and $100=2^25^2$</p>
<p>So, </p>
<p>either (i) $100\mid n\implies n=100k$ where integer $k\ge0$</p>
<p>So we need $0<100k<100\implies 0<k<1$ which is not possible .</p>
<p>or (ii) $100\mid (n-1)\implies n=100k+1$ wher... |
292,651 | <blockquote>
<p>Does an integer $9<n<100$ exist such that the last 2 digits of $n^2$ is $n$? If yes, how to find them? If no, prove it.</p>
</blockquote>
<p>This problem puzzled me for a day, but I'm not making much progress. Please help. Thanks.</p>
| awllower | 6,792 | <p>Write $n=10a+b$. Then $n² \equiv 20ab+b² \pmod{100}$. So the problem is reduced to solving $20ab+b²\equiv 10a+b \pmod{100}$. Hence $100|b(20a+b-1)-10a$. So $10|b(b-1)$. But $0\leq b<10$, thus either b is even and $b-1$ is divisible by $5$ or $b-1$ is even and $b$ is a multiple of $5$.<br>
In the former case, $b$ ... |
455,230 | <p>I found this proposition and don't see exactly as to why it is true and even more so, why the converse is false:</p>
<p>Proposition 1. The equivalence between the proposition $z \in D$ and the proposition $(\exists x \in D)x = z$ is provable from the definitory equations of the existential quantifier and of the equ... | Matt E | 221 | <p>If $S$ is a simple module over a ring $A$, and $x$ is a non-zero element of
$S$, then the annihilator of $x$ is a maximal left ideal $\mathfrak m$ of $A$. </p>
<p>Thus is $S$ embeds into $A$, then $A$ contains a non-zero element annihilated by $\mathfrak m$.</p>
<p>If $A$ is not a simple module over itself (i.e.... |
3,242,921 | <p>Prove that the equation<span class="math-container">$$x^4+(a-2)x^3+(a^2-2a+4)x^2-x+1=0$$</span>
does not admit <span class="math-container">$$x=-2$$</span> as a triple root.</p>
| JJC94 | 215,893 | <p>I figured it out thanks to Servaes.</p>
<p>Every hyperplane in <span class="math-container">$V=\mathbb{F}_2^n$</span> contains exactly <span class="math-container">$2^{n-1}-1$</span> non-zero vectors; thus, any set of <span class="math-container">$2^{n-1}$</span> distinct, non-zero vectors in <span class="math-cont... |
60,259 | <p>The independence of theorems in some propositional calculus systems seems well studied. For example, if we just have the rules of detachment, substitution, and replacement, and every theorem of this axiom set {((p->q)->((q->r)->(p->r))), ((~p->p)->p), (p->(~p->q))}=<strong>X</strong> as our system <strong>X'</stron... | Kaveh | 468 | <p>In classical propositional logic, we say that a formula $\varphi$ is independent from a theory (i.e. a set of formulas) $T$ if $T \nvDash \varphi$ and $T \nvDash \lnot \varphi$. Alternatively, if $T \cup \{\varphi\}$ and $T \cup \{\lnot \varphi\}$ are both satisfiable. Obviously any classical propositional tautology... |
60,259 | <p>The independence of theorems in some propositional calculus systems seems well studied. For example, if we just have the rules of detachment, substitution, and replacement, and every theorem of this axiom set {((p->q)->((q->r)->(p->r))), ((~p->p)->p), (p->(~p->q))}=<strong>X</strong> as our system <strong>X'</stron... | Doug Spoonwood | 11,300 | <p>Let's call a theorem or axiom A dependent if given a set Z of axioms and primitive inference rules, then a proof of A can get written from Z. An axiom is independent of Z if given Z, then a proof of A cannot get written. A set of axioms S for system Y comes as independent if each subsystem of Y with just one less ... |
2,621,871 | <p>My lecture notes say that</p>
<blockquote>
<p>A topological space is an ordered pair <span class="math-container">$(X, \tau)$</span>, where <span class="math-container">$X$</span> is a set and <span class="math-container">$\tau$</span> is a collection of subsets of <span class="math-container">$X$</span> that satisf... | Cubic Bear | 378,597 | <p>It is an exercise to prove. For $x\in A\cap B$ where $A,B$ are two open set, there exists $\gamma, \delta>0$ such that $\mathbb{B}_\gamma(x)\subseteq A, \mathbb{B}_{\delta }(x)\subseteq B$, then take $\epsilon=\min (\gamma, \delta)$,
$\mathbb{B}_{\delta }(x)\subseteq A \cap B$. This prove $A\cap B$ is open. The... |
3,481,661 | <p>Let <span class="math-container">$X$</span>,<span class="math-container">$Y$</span> be Banach spaces. And <span class="math-container">$T:X\rightarrow Y$</span> a continuous operator when <span class="math-container">$X$</span> is endowed with the weak topology and <span class="math-container">$Y$</span> with the no... | Aphelli | 556,825 | <p>We know that the inverse image of <span class="math-container">$B_Y(0,1)$</span> contains an open subset, so there are linear forms <span class="math-container">$u_1,\ldots,u_n$</span> on <span class="math-container">$X$</span> and some <span class="math-container">$\epsilon > 0$</span> such that for any <span cl... |
2,548,353 | <p><strong>Find the number of $4\times4$ matrices such that $|a_{ij}| = 1 \forall i,j\in[1,4]$ , and sum of every row and column is zero.</strong></p>
<p>I tried 'counting' the number of matrices that satisfy the above conditions, that is, elements are $1$ or $-1$ and sum of every row and column is zero.</p>
<p>In th... | Zach Teitler | 343,280 | <p>The first row has two $1$s and two $-1$s. There are $\binom{4}{2} = 6$ ways they can be arranged in the first row. We'll count the number of matrices with first row $(1,1,-1,-1)$, then multiply by $6$ to account for all the other arrangements of the first row.</p>
<p>The second row can be one of the three following... |
1,528,507 | <p>So I started reading Conjecture and Proof by Miklos Laczkovich and one of the first proofs he provides is that of the irrationality of the square root of two. I am aware there are alternative proofs (one of which is geometric and another that uses the fundamental theorem of arithmetic) but I have a few questions abo... | user247327 | 247,327 | <p>Saying "q is the smallest such number" is the same as saying the fraction is "reduced to lowest terms". Essentially the proof is showing that you <strong>always</strong> have factors of 2 in both numerator and denominator so that a fraction equal to square root of 2 <strong>cannot</strong> be reduced to lowest term... |
1,528,507 | <p>So I started reading Conjecture and Proof by Miklos Laczkovich and one of the first proofs he provides is that of the irrationality of the square root of two. I am aware there are alternative proofs (one of which is geometric and another that uses the fundamental theorem of arithmetic) but I have a few questions abo... | hmakholm left over Monica | 14,366 | <p>The reason for assuming that $q$ is minimal IS that without this assumption we wouldn't reach a contradiction at the end, so the proof wouldn't work.</p>
<p>Writing a proof is not a mechanical process where each step follows with necessity from what comes before it. Instead, very often the reason to do something is... |
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>
| apg | 90,619 | <p>See Mathew Kahle and Elizabeth Meckes <a href="https://arxiv.org/abs/1009.4130" rel="nofollow noreferrer">"Limit theorems for Betti numbers of random simplicial complexes"</a></p>
<p>They say that the distribution is Gaussian in <span class="math-container">$G(n,p)$</span> with <span class="math-container">$n \to \... |
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... | dantopa | 206,581 | <p>As noted by @Emilio Novati, the key idea is to connect to the <em>standard basis</em>. </p>
<p>Vectors will be colored according to the basis membership, and named chromatically:
$$
\color{blue}{\mathbf{B}\ (standard)}, \qquad
\color{red}{\mathbf{R}\ (u)}, \qquad
\color{green}{\mathbf{G}\ (v)}.
$$
$$
\mathbf{R}_... |
2,463,421 | <p>The question is:</p>
<p>Nadir Airways offers three types of tickets on their Boston-New York flights. First-class tickets are \$140, second-class tickets are \$110, and stand-by tickets are \$78. If 69 passengers pay a total of $6548 for their tickets on a particular flight, how many of each type of ticket
were sol... | Donald Splutterwit | 404,247 | <p>i) is fine.</p>
<p>ii) There are $15$ possible values for the first digit, $15$ for the second digit, $14$ for the third and $13$ for the last digit. Giving $ 15 \times 15 \times 14 \times 13$.</p>
<p>iii) There are $15$ possible values for the first digit, $14$ for the second digit, $13$ for the third and $1$ for... |
382,504 | <p>Let <span class="math-container">$n \geq 2$</span> be an integer, and let <span class="math-container">$f(x) = \prod\limits_{k = 1}^n(x - \alpha_k)$</span> be a monic irreducible polynomial in <span class="math-container">$\mathbb Z[x]$</span>, with the property that <span class="math-container">$f(-\alpha_k) \neq 0... | Richard Stanley | 2,807 | <p>We have <span class="math-container">$\mathrm{Res}(f(x),f(-x))=2^n a_n P(\alpha)^2$</span>, where
<span class="math-container">$P(\alpha)=\prod_{1\leq i<j\leq n}(\alpha_i+\alpha_j)$</span>. By
e.g. the case <span class="math-container">$d=2$</span> of Exercise 7.30 in <em>Enumerative Combinatorics</em>,
vol. 2, w... |
2,215,087 | <p>I'm trying to show that $\mathbb{Z}[\sqrt{11}]$ is Euclidean with respect to the function $a+b\sqrt{11} \mapsto|N(a+b\sqrt{11})| = | a^2 -11b^2|$</p>
<p>By multiplicativity, it suffices to show that $\forall x \in \mathbb{Q}(\sqrt{11}) \exists n \in \mathbb{Z}(\sqrt{11}):|N(n-x)| < 1$</p>
<p>For the analogous s... | Gerry Myerson | 8,269 | <p>I believe the result is proved in Oppenheim, Quadratic fields with and without Euclid's algorithm, Math Annalen 109 (1934) 349-352, and I think this paper is freely available <a href="https://eudml.org/doc/159685" rel="noreferrer">here</a>. The proof is essentially the first half of page 350, together with prelimina... |
136,363 | <p>Consider the following case:</p>
<pre><code>(a^3*b) //. {a^2 -> c, a*b -> d}
</code></pre>
<p>instead of <code>c d</code> the output is:</p>
<pre><code>(*a^3*b*)
</code></pre>
<p>How can I get what I want?</p>
| mikado | 36,788 | <p>In general, you want to make the left hand sides of your rules as simple as possible. A simple way of doing what you want is</p>
<pre><code>(a^3*b) //. {a -> Sqrt[c], b -> d/a}
(* c d *)
</code></pre>
|
3,769,843 | <p>This is a multiple choice question from my Text Book</p>
<p>Let <span class="math-container">$A=\{1,2,3\}$</span>. The no. of relations containing <span class="math-container">$(1,2)$</span> and <span class="math-container">$(1,3)$</span> which are reflexive and Symmetric but not transitive is</p>
<p>(A) <span class... | None | 806,517 | <p>Third take. Stop changing the definitions! :)</p>
<p>We have a sequence of cumulative rotation unit quaternions <span class="math-container">$R_i$</span> and cumulative translations <span class="math-container">$T_i$</span> in each local coordinate system, with <span class="math-container">$Q_i$</span> describing t... |
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>
| Alessandro Codenotti | 136,041 | <p>Once you factorize a number as $N=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_n^{a_n}$, $p_i$ prime for every $i$, $a_i>0$ for every $i$ the number of divisors is given by $(a_1+1)(a_2+1)(a_3+1)...(a_n+1)$.</p>
<p>It is easy to see why this formula works from a combinatorial point of view, the divisors of $N$ are also of th... |
4,382,317 | <p>Show that <span class="math-container">$|z+1| = 2\cos(\frac{\theta}{2})$</span>, <span class="math-container">$z = cis(\theta)$</span> and <span class="math-container">$z \in C$</span></p>
<p>Here is what I have managed to do:</p>
<ul>
<li><span class="math-container">$r=1$</span></li>
<li><span class="math-containe... | Samual | 766,626 | <p>note that
<span class="math-container">$z = \text{cis}(\theta)$</span> and <span class="math-container">$1 = \text{cis}(0)$</span>,</p>
<p>you can rewrite the function as
<span class="math-container">$|\text{cis}(\theta)+\text{cis }(0)|$</span></p>
<p>use the identity that
<span class="math-container">$ \text{cis}(x... |
1,963,295 | <p>I have the following equation for a decision boundary line: $-w_0 = w_1x_1 + w_2x_2$ and I want to prove that the distance from the decision boundary to the origin is $l = \frac{w^Tx}{||w||}$. I am having trouble wrapping my mind around how I can just get the distance from a line to a point. Am I supposed to be aver... | dr.ivanova | 192,583 | <p>Let <span class="math-container">$X\sim N(\mu, \sigma)$</span>. "Rectified" Gaussian is then <span class="math-container">$Y = \max(0, X)$</span>. For both the expectation and the variance use the law of total expectation, as suggested by Michael. I'll also make use of the first two moments of the truncated normal d... |
3,339,780 | <p>The popular definition of a vector is </p>
<blockquote>
<p>A vector is an object that has both a magnitude and <strong>a</strong> direction.</p>
</blockquote>
<p>We know that zero vector has no specific <strong>single</strong> direction.</p>
<p>Then how can it be a vector?</p>
| John Bentin | 875 | <p>You are right to question this definition. It suggests that every vector is associated with a unique direction. This is almost true, with the sole exception of the zero vector, which cannot sensibly be said to have any direction. Unfortunately, a more accurate definition</p>
<p>“a vector is an object that has both ... |
3,339,780 | <p>The popular definition of a vector is </p>
<blockquote>
<p>A vector is an object that has both a magnitude and <strong>a</strong> direction.</p>
</blockquote>
<p>We know that zero vector has no specific <strong>single</strong> direction.</p>
<p>Then how can it be a vector?</p>
| Joonas Ilmavirta | 166,535 | <p>What you quote is a reminder, not a definition.
If you are asked to calculate a vector, you know the answer shouldn't be a single number.
The zero vector disagrees with that reminder, and that is a useful caveat to learn.</p>
<p>When vectors are first introduced, they might be treated as arrows and explained with p... |
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">$$2+\sum_{n=1}^{\infty}\frac{1-n}{9n^3-n} =2+\sum_{n=1}^{\infty}\frac{2}{3n... | J.G. | 56,861 | <p>Let's rewrite <span class="math-container">$\frac{1}{n}$</span> as <span class="math-container">$\frac{3}{3n}$</span>, so your sum is <span class="math-container">$S+2$</span> with<span class="math-container">$$S:=\sum_{n\ge 1}\left(\frac{1}{3n-1}-\frac{3}{3n}+\frac{2}{3n+1}\right)\\=\sum_{n\ge 1}\int_0^1 x^{3n-2}\l... |
141,138 | <p>In "Catégories Tannakiennes" by Savedra Rivano (under A. Grothendieck supervision) at pag.78 he define a rigid category $\mathscr{C}$ as a monoidal symmetrical closed such that the natural morphisms $[X, X'] \otimes [Y, Y'] \to [X \otimes Y, X' \otimes Y' ]$ and $\theta: X \mapsto [[X, I], I] $ are isomorphi... | Todd Trimble | 2,926 | <p>Rigid monoidal categories in this sense are equivalent to compact closed categories. Their coherence was studied by Kelly and LaPlaza: </p>
<ul>
<li>G.M. Kelly and M. LaPlaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra, vol. 19, pp. 193-213, 1980. </li>
</ul>
<p>In particular, the... |
141,138 | <p>In "Catégories Tannakiennes" by Savedra Rivano (under A. Grothendieck supervision) at pag.78 he define a rigid category $\mathscr{C}$ as a monoidal symmetrical closed such that the natural morphisms $[X, X'] \otimes [Y, Y'] \to [X \otimes Y, X' \otimes Y' ]$ and $\theta: X \mapsto [[X, I], I] $ are isomorphi... | mszyld | 39,603 | <p>The answer to both your questions is yes. The second one
may be more elementary, but since I'm used to working with a definition
of duality different that the one of Saavedra Rivano, I prefer to answer
both questions in order.</p>
<p>Answer to the first question: I copy from the introduction of my degree
thesis (<a... |
3,012,558 | <p>Let <span class="math-container">$C \subseteq \mathbb{R}^n$</span> be a closed convex set, and <span class="math-container">$x^* \in C^c$</span> (not in <span class="math-container">$C$</span> and its closure). </p>
<p>Define the Euclidean distance from <span class="math-container">$x^*$</span> to <span class="math... | B.Swan | 398,679 | <p>Since <span class="math-container">$C \subseteq D$</span>, the minimum over <span class="math-container">$C$</span> can only be greater or equal to the minimum over <span class="math-container">$D$</span>, so </p>
<p><span class="math-container">$$d_D(x^*)=\min_{z \in D}\|z -x^*\|_2 \leq \min_{z \in C}\|z -x^*\|_2=... |
501,512 | <p>The lines CD and EF are perpendicular with points $C(1,2)$, $D(3,-4)$, $E(-2,5)$, and $F(k,4)$. Find the value of the constant $k$.</p>
| Ted Shifrin | 71,348 | <p>The OP presumably does not know about vectors or dot products. Here's the appropriate hint: What do you know about the <em>slopes</em> of perpendicular lines? Now can you do the problem?</p>
|
2,425,127 | <p>I am learning about general solutions to differential equations and would like to ask whether my solution is mathematically correct. </p>
<p>I was asked to find the general solution to the differential equation </p>
<p>$$\frac{dy}{dx} = 2e^{x-y}$$</p>
<p>So I did the following - </p>
<p>$$\int e^y dy = 2\int e^x... | doraemon | 473,786 | <p>Note this inequality: $$2e^x\neq e^{x^2}$$</p>
|
315,004 | <p>For a knot <span class="math-container">$K$</span>, let <span class="math-container">$\Sigma_K$</span> be the double cyclic branched cover of a knot. </p>
<p>By the classical work of <strong>Casson</strong> and <strong>Gordon</strong>, we know that if <span class="math-container">$K$</span> is smoothly slice, then ... | Oğuz Şavk | 131,172 | <p>Probably, I found some more explicit examples:</p>
<p><strong>Akbulut</strong> and <strong>Larson</strong> recently showed in <a href="https://arxiv.org/pdf/1704.07739.pdf" rel="nofollow noreferrer">AL18</a> that the family of Brieskorn spheres <span class="math-container">$\Sigma(2,4n+1,12n+5)$</span> and <span cl... |
3,984,480 | <p>Show that <span class="math-container">$-\vec{0} = \vec{0}$</span> in any vector space.</p>
<p>I know this is a seemingly obvious statement but is the following justification correct:</p>
<p>Assume <span class="math-container">$-\vec{0} \neq \vec{0}$</span>.</p>
<p><span class="math-container">$$(4): \vec{u} + \vec{... | Teplotaxl | 613,332 | <ol>
<li>A Theorem of Gromov states that: If <span class="math-container">$G$</span> is quasi-isometric to <span class="math-container">$\mathbb{Z}^{n}$</span> then <span class="math-container">$G$</span> has a finite index subgroup isomorphic to <span class="math-container">$\mathbb{Z}^{n}$</span>. (From a geometric i... |
649,570 | <p>How do we show that there is only one solution to,$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+x}}}}$$</p>
<p>I guess it is only $x=2$.
Please help.</p>
| achille hui | 59,379 | <p>Let $\;f(x) = \sqrt{2+x}\;$ and $\;g(x) = \sqrt[3]{6+x}$, they are strictly increasing function in $x$ when $x \ge -2$.
Since $(x+2)^3 - (x+6)^2 = (x-2)(x^2 + 7x + 14)$ and $x^2 + 7x + 14 > 0$ for all $x$,
we have
$$\begin{cases} f(x) > g(x) > 2,& x > 2\\f(x) = g(x) = 2,& x = 2\\f(x) < g(x) &l... |
1,339,875 | <p>Suppose we are given a function
$$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$
and
$$g\left ( k\pi \right )=\lim _{x\rightarrow k\pi}g\left ( x \right )$$
I found that $\lim _{x\rightarrow k\pi}g\left ( x \right )= \sum_{n=1}^{... | Claude Leibovici | 82,404 | <p>I am not sure that I arrive to the same result $$g(k\pi)=\sum_{n=1}^{\infty}\frac{1}{10^{n}}=\frac{1}{9}$$ </p>
<p>Using complex representation of $\sin(x)$, what I got is that $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )}=-\frac{10 e^{i x}}{\left(-10+e^{i ... |
1,339,875 | <p>Suppose we are given a function
$$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$
and
$$g\left ( k\pi \right )=\lim _{x\rightarrow k\pi}g\left ( x \right )$$
I found that $\lim _{x\rightarrow k\pi}g\left ( x \right )= \sum_{n=1}^{... | Christian Blatter | 1,303 | <p>It is a basic precalculus fact that for arbitrary complex $z$ with $|z|<1$ one has
$$\sum_{k=0}^\infty z^k={1\over 1-z}\ .$$
Furthermore, by Euler's equation, for arbitrary real $x$ we may write $\sin x={\rm Im}\>e^{ix}$ . Putting these two together we obtain
$$\eqalign{\sum_{n=1}^\infty{\sin (nx)\over 10^n}&a... |
1,915,366 | <p>Suppose we have two Gaussian distributed random variable $X$~$N(0,\sigma^2)$ and $Y$~$N(0,\sigma^2)$. These variables are not independent. What will be the expected value of product of square of this random variables</p>
<p>$E[X^2Y^2]$ = ??</p>
<p>Edit 1: They are jointly Gaussian distributed with correlation coef... | Frank | 87,112 | <p>See <a href="https://stats.stackexchange.com/a/182822/">Variance of product of dependent variables</a> at CrossValidated, specifically the second answer, for the outline of a derivation. In your case, because $X$ and $Y$ have zero mean and the same variance, the formula is simpler.</p>
<p>As explained there, the co... |
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>
| murray | 148 | <p>Let me try with a few simple ("obvious"?) style guidelines I try to follow:</p>
<ul>
<li>Use meaningful names that are spelled out or that use widely-adopted abbreviations from the field of application.</li>
<li>Begin names with lower-case letters (except when they're going into a Package for others' use) and then ... |
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>It is more easy to prove this inequality without induction. Really $$0 < i\cdot (n + 1 - i) = \left(\frac{n+1}2 + \frac{2i - n - 1}2\right)\left(\frac{n+1}2 - \frac{2i - n - 1}2\right) = \left(\frac{n+1}2\right)^2 - \left(\frac{2i - n - 1}2\right)^2 \le \left(\frac{n+1}2\right)^2.$$
Multiply this inequalities for... |
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>
| R. J. Mathar | 185,604 | <p>The n-th partial sum of the triangular numbers as listed in <a href="http://oeis.org/A000292" rel="nofollow">http://oeis.org/A000292</a> .</p>
|
1,027,807 | <p>So I have this question that looks like</p>
<p>$$ \frac{x^3 + 3x^2 - x - 8}{x^2 + x - 6} $$</p>
<p>and first I got the partial fraction so getting </p>
<p>$$ x + 2 + \frac{3x + 4}{x^2 + x -6} $$</p>
<p>but now I'm trying to integrate it and I cannot remember for the life of me how I should integrate the fraction... | rae306 | 168,956 | <p><strong>Hint</strong>:</p>
<p>Remember that $x^2+x-6=(x-2)(x+3)$.</p>
<p>Now apply the partial fraction decomposition again:</p>
<p>$$\frac{3x+4}{x^2+x-6}=\frac{3x+4}{(x-2)(x+3)}=\frac{A}{x-2}+\frac{B}{x+3}$$</p>
<p>Also, it seems as if your division is not correct:</p>
<p>$$\require{cancel}\frac{x^3 + 3x^2 - x... |
3,275,892 | <p>Im having problem graphing the following inequality:
<span class="math-container">$(x+y) \div (x-y) \ge 0$</span>.
I know what the graph looks like, but I can't grasp the thought process behind solving the problem.</p>
<p>Thanks in advance!</p>
| Dr. Sonnhard Graubner | 175,066 | <p>First of all it must be <span class="math-container">$$x\neq y$$</span>- So, now we have two cases:
<span class="math-container">$$x>y$$</span> then we get <span class="math-container">$$x\geq -y$$</span>
Second case:
<span class="math-container">$$x<y$$</span> then we get <span class="math-container">$$x\le-y... |
4,185,658 | <p>In the proof of Proposition. 1.3. page 100 Functional Analysis book of Conway the following claim (<span class="math-container">$X$</span> is a TVS and <span class="math-container">$p$</span> is a seminorm.)</p>
<p>If <span class="math-container">$0 \in \operatorname{Int}{\{x \in X : p(x) \le 1}\}$</span> then <span... | Ramiro | 190,563 | <p>Suppose <span class="math-container">$0 \in \operatorname{Int}{\{x \in X : p(x) \le 1}\}$</span>.
Given <span class="math-container">$\varepsilon >0$</span>, let <span class="math-container">$f: X \rightarrow X$</span> be defined by <span class="math-container">$f(x) = \frac{1}{\varepsilon} x$</span>. We have th... |
138,698 | <p>I want to evaluate the following double summation</p>
<pre><code>Sum[(-1)^(i + j + i*j)*Exp[-Pi/2*( i^2 + j^2)], {i, -Infinity,
Infinity}, {j, -Infinity, Infinity}]
</code></pre>
<p>I am really new both in using Mathematica and in doing mathematics using computer. I don't know if there is some special technics ... | mikado | 36,788 | <p>As noted by Szabolcs, there is an inefficiency in generating the table of real elements one at a time. </p>
<pre><code>v = RandomReal[{0, 1}, 1000000];
Table[Timing[Length[Select[v, # < 0.5 &]]][[1]], {10}]
(* {0.384, 0.38, 0.376, 0.38, 0.38, 0.38, 0.504, 0.48, 0.384, 0.38} *)
</code></pre>
<p>However, a mo... |
4,478,109 | <p>Find the value of <span class="math-container">$$\int\frac{1+\ln x}{4+x\ln x^2}\mathrm{d}x$$</span> I have a very bad understanding of integrals where some function of a variable is in the denominator. I know I have to do some kind of substitution and I even tried that but can't get any help from it. Please forgive ... | insipidintegrator | 1,062,486 | <p><span class="math-container">$$x\ln(x^2)=2\cdot x\ln x. $$</span>
Now, try to differentiate the function <span class="math-container">$f(x)= x\ln x.$</span> So by using the product rule, <span class="math-container">$$\frac{d}{dx}(x\ln x)= x\cdot\frac 1x + 1\cdot \ln x=1+\ln x.$$</span> This is the numerator of your... |
4,478,109 | <p>Find the value of <span class="math-container">$$\int\frac{1+\ln x}{4+x\ln x^2}\mathrm{d}x$$</span> I have a very bad understanding of integrals where some function of a variable is in the denominator. I know I have to do some kind of substitution and I even tried that but can't get any help from it. Please forgive ... | Davius | 910,130 | <p>First note that <span class="math-container">$\ln(x^2)$</span> is just <span class="math-container">$2\ln x$</span>. Then, consider a funciton <span class="math-container">$f(x) = ln(a+bx\ln(x))$</span> then you have:</p>
<p><span class="math-container">$$f'(x) = \frac{b+b\ln(x)}{a+bx\ln x}$$</span></p>
<p>Now, you ... |
638,566 | <p>I came across a question yesterday about combinations, and I wanted to know what the correct answer was. The question states as follows:</p>
<p>There are 8 spaces that are alternately black and white. There is one king, one queen, two identical rooks, two identical bishops, and two identical knights. The king needs... | bof | 111,012 | <p>Assuming that the king and rooks do <strong>not</strong> have to be on consecutive squares, the question is about the number of legal arrays in <a href="http://www.chessvariants.org/diffsetup.dir/fischer.html" rel="nofollow">Fischer Random Chess</a> aka <a href="http://en.wikipedia.org/wiki/Chess960" rel="nofollow">... |
3,066,530 | <p><span class="math-container">$$\lim_{x\to 0} \frac {(\sin(2x)-2\sin(x))^4}{(3+\cos(2x)-4\cos(x))^3}$$</span> </p>
<p>without L'Hôpital.</p>
<p>I've tried using equivalences with <span class="math-container">${(\sin(2x)-2\sin(x))^4}$</span> and arrived at <span class="math-container">$-x^{12}$</span> but I don't kn... | MSDG | 447,520 | <p><strong>Hint:</strong> Note that
<span class="math-container">$$ 3+\cos(2x)-4\cos(x) = 3 + 2\cos^2(x) - 1 - 4\cos(x) = 2(\cos(x)-1)^2, $$</span>
and that
<span class="math-container">$$ \sin(2x) - 2\sin(x) = 2\sin(x)\cos(x)-2\sin(x) = 2\sin(x)(\cos(x)-1). $$</span></p>
|
2,479,290 | <p>Came across this question in my textbook:</p>
<p>$f(x) = (1+2x)^{10}$. Determine $f^{(5)}(0)$ using the binomial theorem.</p>
<p>If I am correct, the author of the book want me not to use the power rule. How else do I compute this? </p>
| Graham Kemp | 135,106 | <p><strong>Strong Induction</strong> :</p>
<p>We let <span class="math-container">$n$</span> refer to some arbitrary natural number, and by assuming <span class="math-container">$P(0),..., P(n-1)$</span>, we make <em>a valid argument</em> that this infers, <span class="math-container">$P(n)$</span>. Thus we conclude ... |
2,724,744 | <p>I have a basic math question.</p>
<p>If I have the following inequality:
$$-a-b > -1$$
and I want to flip (or reverse) the sign. What is the correct way of the following? And why?</p>
<p>i) $a+b \le 1$<br>
ii) $a+b < 1$</p>
<p>Many thanks! (:</p>
| CY Aries | 268,334 | <p>$-a-b>-1$ $\implies$ $-a-b+(a+b+1)>-1+(a+b+1)$ $\implies$ $1>a+b$</p>
|
4,609,001 | <p>The following definition of pmf is on page51 from Probability and statistical inference by Robert V. Hogg, etc.</p>
<p>The pmf <span class="math-container">$f(x)$</span> of a discrete random variable X is a function that satisfies the following properties:<br />
(a)<span class="math-container">$f(x)\gt 0, x\in S;$</... | Abolfazl Chaman motlagh | 938,462 | <p>for any <span class="math-container">$x \in (x_1,x_2)$</span> we can write : <span class="math-container">$x=tx_1+(1-t)x_2$</span> which <span class="math-container">$t \in (0,1)$</span>. if you solve it for <span class="math-container">$t$</span> you get :
<span class="math-container">$$
t = \frac{x_2-x}{x_2-x_1}
... |
444,448 | <p>Let $M$ be a set of prime numbers of $\mathbb{Q}$ . The limit
$$d(M)= \lim_{s\rightarrow 1^+} \frac{ \sum_{p \in M} p^{-s} }{ - \log(s-1)}$$
Where $p$ is a prime of $\mathbb{Q}$ is called <strong>Dirichlet Density</strong> of $M$. Also, the <strong>Natural density</strong> of $M$ is the limit
$$ \delta(M)= \lim_{x\... | Kunnysan | 84,764 | <p>This is a non-trivial result, I may give you an idea of proof but first you should look at <a href="http://ac.els-cdn.com/0022314X84900611/1-s2.0-0022314X84900611-main.pdf?_tid=40706d58-ed8a-11e2-921a-00000aab0f26&acdnat=1373919061_d93a04eb74763c51d5c52d7dc3395665" rel="nofollow">this</a>.</p>
|
3,265,243 | <p>The number of ways of placing <span class="math-container">$n$</span> objects not in position is given by the inclusion-exclusion number <span class="math-container">$D_n$</span>: </p>
<p><span class="math-container">$n! \left( 1-\dfrac{1}{1!}+\dfrac{1}{2!}+....+(-1)^n\dfrac{1}{n!} \right)$</span></p>
<p>which can... | XbarJim | 682,244 | <p>Let <span class="math-container">$LHS(n)$</span> denote the expression <span class="math-container">$n!\sum_{i=1}^{n-1} (-1)^{i+1}/(i+1)!$</span></p>
<p>Let <span class="math-container">$RHS(n)$</span> denote the expression <span class="math-container">$(n-1)!\sum_{i=1}^{n-1} (-1)^{i+1}(n-i)/(i-1)!$</span></p>
<p>... |
224,019 | <p>I am trying to compute the volume of intersection of the following two regions:</p>
<pre><code>a = 0.857597;
b = 1.653926;
hexagon = Polygon[{{0, (b - a)/2, 1/2}, {(b - a)/2, 0, 1/2},
{1/2, 0, (b - 1)/(2 a)}, {1/2, (b - 1)/2, 0}, {(b - 1)/2, 1/2, 0},
{0, 1/2, (b - 1)/(2 a)}}];
octahedron = ImplicitRegion[Ab... | flinty | 72,682 | <p>This is not ideal, but it gives an approximate resulting region. I first generate random points on the hexagon and add a random vector on the unit sphere. I take the convex hull of the points which is acceptable because the blob must be convex. Finally I discretize the octahedron and intersect with <code>crudehexago... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | Heike | 46 | <p>This answer uses <code>RegionPlot</code> to plot the rounded rectangle. In <code>roundedRect</code>, <code>{{xmin, xmax}, {ymin, ymax}}</code> is the range of the rectangle and <code>rad</code> the rounding radius. <code>roundedRect</code> accepts any option of <code>RegionPlot</code>, in particular <code>ColorFunc... |
3,940,447 | <p>Disclaimer: I believe this proof is wrong, but I'm asking because I can't find what's wrong with it, which means I must have some basic misunderstanding of the concepts involved.</p>
<p>First, some definitions. Recursively, we define an ordinal <span class="math-container">$\alpha$</span> to be <span class="math-con... | lulu | 252,071 | <p>The distribution is correct but the expected value is not.</p>
<p>I think it is simpler to approach this by noting that you expect it to take <span class="math-container">$\frac 1p$</span> tries for a girl and <span class="math-container">$\frac 1{1-p}$</span> for a boy so the expected number of times to get one of ... |
2,492,206 | <p>Let $(X, \mathcal{X})$ and $(Y,\mathcal{Y})$ be measurable spaces, and $f: X \to Y$ a measurable function. Let $A \in \mathcal{X}$ be a measurable subset of $X$.</p>
<p>Is it guaranteed that $f_{\mid A}: A \to Y$ is measurable?</p>
<p>The measurable space on $A$ is $(X, \mathcal{X})$ restricted to $A$. Formally, t... | Lucio | 495,116 | <p>The proof, as pointed out, is correct.</p>
<p>Regarding the notions of measurability: no, they are not the same. First of all, the last notion, preimage of open set is open, is the notion of continuity. Continuity is in general much stronger than measurability. Secondly, observe that you haven't defined any topolog... |
2,391,624 | <p>This question pertains to Mosteller's classic book <em>Fifty Challenging Problems in Probability</em>. Specifically, this in regards to an algebraic operation Mosteller performs in the solution to the first question, entitled "The Sock Drawer."</p>
<p>Mosteller writes:</p>
<blockquote>
<p>Then we require the pro... | Angina Seng | 436,618 | <p>You have two positive numbers, which I'll call $x$ and $y$, with
$xy=\frac12$ and $x>y$.
Then
$$x^2>xy>y^2$$
which is the same as
$$x^2>\frac12>y^2.$$</p>
|
4,030,296 | <p>I'm reading Carothers' Real Analysis, and I'm currently looking at <strong>homeomorphisms</strong>. The author says "two intervals that look different, are different" - i.e. they are not homeomorphic. The proof is done for the case <span class="math-container">$(0,1]$</span> and <span class="math-container... | DanielWainfleet | 254,665 | <p>In <span class="math-container">$\Bbb R^2$</span> let <span class="math-container">$C$</span> be the circle centered at <span class="math-container">$(0,2)$</span> with radius <span class="math-container">$1.$</span> Let <span class="math-container">$D=\{(x,y)\in C: y<2\}$</span>...Now <span class="math-container... |
1,266,210 | <p>Hello everybody my query is regarding the number of positive integral solution.</p>
<blockquote>
<p>In the sport of cricket, find the number of ways in which a batsman can score $14$ runs in $6$ balls not scoring more than $4$ runs in any ball.</p>
</blockquote>
| Sachin Gupta | 490,560 | <p>my solution is based on non-negative intergral solution..</p>
<p>$a+b+c+d+e+f=14$ provided the condition $a,b,c,d,e,f \leq 4$</p>
<p>let $a=4-a_1,b=4-a_2,....f=4-a_6$ and replace in given equation</p>
<p>we get ,</p>
<p>$(4-a_1)+(4-a_2)+(4-a_3)+(4-a_4)+(4-a_5)+(4-a_6)=14$</p>
<p>$24-(a_1+a_2+a_3+a_4+a_5+a_6)=14... |
3,988,084 | <p>I have 3 points:</p>
<p><span class="math-container">$$A = (0,4) \\
B=(-5,0) \\
C=(5,0)$$</span></p>
<p>I need to find a polynomial that goes through B and C, and is tangent to <span class="math-container">$f(x) = (2/3)x+4$</span> at A.</p>
<p>I know that tangent means it must be equal to the derivative of f(x) at t... | Hagen von Eitzen | 39,174 | <p>You have four conditions
<span class="math-container">$$\begin{align}f(0)&=4\\f(-5)&=0\\f(5)&=0\\f'(0)&=\frac23\end{align} $$</span>
and so we should be looking for four unknowns, i.e., <span class="math-container">$f$</span> is of degree <span class="math-container">$3$</span>, or
<span class="math-... |
1,679,615 | <p>From what I have read about a transitive relation is that if xRy and yRz are both true then xRz has to be true. </p>
<p>I'm doing some practice problems and I'm a little confused with identifying a transitive relation. </p>
<p>My first example is a "equivalence relation"
$S=\{1,2,3\}$ and $R = \{(1,1),(1,3),(2,2),... | Ove Ahlman | 222,450 | <p>$xRy$ and $yRz$ should imply $xRz$ for all choises of $x,y,z$.</p>
<p>In your case you have just noticed that $1R3$ hold in both examples. This is however not sufficient for transifitivty, you have to check all possible cases of $x,y$ and $z$ in order to show that a relation is transitive. </p>
<p>In the case of $... |
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