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 |
|---|---|---|---|---|
971,997 | <p>How do we find
$$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$</p>
<p>In the shortest and easiest possible manner?</p>
<p>I cannot think of anything good.</p>
| Anastasiya-Romanova 秀 | 133,248 | <p>Here is Feynman's style answer. We will evaluate the general integral using differentiation under integral sign method. Consider
\begin{equation}
I(\alpha)=\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\alpha\theta}}d\theta\right]=\int_0^{\large\frac{\pi}{2}} e^{\alpha\cos\theta}\cos(\alpha\sin\theta)\ d\theta... |
2,477,730 | <p>How would one show (preferably using congruences) that $$37\not\mid n^{9^9}+4 $$ for any $n \in \mathbb Z$?</p>
| Oscar Lanzi | 248,217 | <p>For the divisibility to hold we must have $n$ prime to $37$. Then $n^{36}\equiv 1 \bmod 37$. Now, $4×9^9$ is a multiple of $36$ so $n^{4×9^9}=(n^{9^9})^4\equiv 1$. Alas, $(-4)^4\equiv 256\equiv 34$, and $1\equiv 34$ won't work $\bmod 37$.</p>
|
2,477,730 | <p>How would one show (preferably using congruences) that $$37\not\mid n^{9^9}+4 $$ for any $n \in \mathbb Z$?</p>
| fleablood | 280,126 | <p>If $n\equiv 0 \mod 37$ then that statement isn't true of course.</p>
<p>If $n\not\equiv 0\mod 37... $</p>
<p>37 is prime. So $n^{36}\equiv 1\mod 37$. So if $9^9\equiv k \mod 36$ then ${n^9}^9\equiv n^k \mod 37$.</p>
<p>Hmmm... $4*9=36$ so $9*9\equiv 9+9*4+9*4\equiv 9\mod 36$ so $9^2\equiv 9\mod 36$ and inductive... |
1,255,629 | <p>Show that </p>
<p>$$\sin\left(\frac\pi3(x-2)\right)$$ </p>
<p>is equal to </p>
<p>$$\cos\left(\frac\pi3(x-7/2)\right)$$</p>
<p>I know that $\cos(x + \frac\pi2) = −\sin(x)$ but i'm not sure how i can apply it to this question.</p>
| Oscar | 215,894 | <p>The two angles $\frac{\pi}{3}(x-2)$ and $\frac{\pi}{3}(x-7/2)$ differs by $\pi/2$.
In fact,
$$ \frac{\pi}{3}(x-2) - \frac{\pi}{3}(x-\frac{7}{2}) = \frac{\pi}{3}x- \frac{2}{3}\pi - \frac{\pi}{3}x + \frac{7}{6}\pi = - \frac{7-4}{6}\pi = -\frac{\pi}{2}\, , $$
thus $\sin(\frac{\pi}{3}(x-2)) = \sin(\frac{\pi}{3}(x-7/2) -... |
1,030,327 | <p>Im trying to do an exercise from the book Algebraic Curves of Fulton (Exercise $\:6.26^{*}$).</p>
<p>It says:</p>
<p>Let $f:X\rightarrow Y$ be a morphism of affine varieties. Show that $f(X)$ in dense in $Y$ if and only if the homomorphism $\tilde{f}:\Gamma(Y)\rightarrow\Gamma(X)$ is one-to-one.</p>
| Alex Youcis | 16,497 | <p>This is not true. The statement is that $f:\text{Spec}(B)\to\text{Spec}(A)$ is dominant (has dense image), if and only if $\ker(A\to B)$ is contained in $\text{nil}(A)$, the nilradical of $A$. Indeed, consider the map $\text{Spec}(k)\to\text{Spec}(k[x]/(x^2))$ coming from the quotient $k[x]/(x^2)\to k$.</p>
<p>Here... |
147,441 | <p>I've just been looking through my Linear Algebra notes recently, and while revising the topic of change of basis matrices I've been trying something:</p>
<p>"Suppose that our coordinates are $x$ in the standard basis and $y$ in a different basis, so that $x = Fy$, where $F$ is our change of basis matrix, then any m... | Thomas | 26,188 | <p>Your approach seems correct.</p>
<p>I don't know if the following helps, but anyway: So you have a vector space $V$ (over say the complex numbers) and you have say two basis $E$ and $D$. </p>
<p>That $F$ is a change of basis matrix means that if as column vector $y = (y_i)$ written with respect to the basis $E$, t... |
4,004,157 | <p>A firm wants to know how many of its employees have drug problems. Realizing the sensitivity of this issue, the personnel director decides to use a randomized response survey.</p>
<p>Each employee is asked to flip a fair coin,</p>
<p>If head (H), answer the question “Do you carpool to work?”</p>
<p>If tail (T), answ... | Mark Bennet | 2,906 | <p>Let's look at the concrete example, to see what is going on, and assume we are not in characteristic <span class="math-container">$2$</span>.</p>
<p>Then, using the normal quadratic formula if <span class="math-container">$$x^4-10x^2+1=0$$</span> then <span class="math-container">$$x^2=\frac {10\pm \sqrt{100-4}}{2}=... |
4,179,720 | <p>I'm starting to study triple integrals. In general, I have been doing problems which require me to sketch the projection on the <span class="math-container">$xy$</span> plane so I can figure out the boundaries for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. For example, I had... | quasi | 400,434 | <p>Label the chairs<span class="math-container">$\;1,...,8\;$</span>in clockwise order.</p>
<p>
There are <span class="math-container">$4$</span> pairs of opposite chairs, namely <span class="math-container">$(1,5),(2,6),(3,7),(4,8)$</span>.
<ul>
<li>For the first couple (couple #<span class="math-container">$1$</span>... |
810,110 | <p>How to prove that the following holds? $$\frac{d}{dx}a^x=(\ln a)a^x.$$
Just a hint will do it.</p>
| Hakim | 85,969 | <p><strong>Hint:</strong> $$\dfrac{\mathrm d}{\mathrm dx}a^x=\dfrac{\mathrm d}{\mathrm dx}e^{\ln a\cdot x}.\tag{$a>0$}$$ Now use the <a href="http://quantizd.blogspot.com/p/blog-page_7049.html" rel="nofollow">chain rule</a> knowing that: $$\dfrac{\mathrm d}{\mathrm dx}e^x=e^x.$$</p>
|
810,110 | <p>How to prove that the following holds? $$\frac{d}{dx}a^x=(\ln a)a^x.$$
Just a hint will do it.</p>
| Jika | 143,855 | <p>For $a>0$, we know that $a^x=e^{x\log a}$. Hence to calculate the derivative of $a^x$, you have to calculate the derivative of $e^{g(x)}$ where $g(x)=x\log a$ for $a>0$ which is $g'(x)e^{g(x)}$ using the chain rule.</p>
<p>For $a\le0$, I do not think such a function exists.</p>
|
2,711,189 | <p>Is their anyway you can integrate a wedge of cylinder (picture below)
not by adding up the parts cut in parallel lines but
by adding up parts cut in pivotal lines to the center of the half circle?</p>
<p><a href="https://i.stack.imgur.com/BrucD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Br... | Matthew Leingang | 2,785 | <p>The standard multivariable integration method can integrate along radial lines. If the base disk of the cylinder is in the $(x,y)$-plane, and the wedge runs along the $x$-axis, then the height function for the wedge can be represented as $z=y\tan\alpha$. In cylindrical coordinates, the wedge can be described as
$$... |
2,338,508 | <p>Given this matrix that stretches to infinity to the right and up:
$$
\begin{matrix}
...&...&...\\
\frac{1}{4}& \frac{1}{8}& \frac{1}{16}&... \\
\frac{1}{2} & \frac{1}{4}& \frac{1}{8}&... \\
1 & \frac{1}{2}& \frac{1}{4}&... \\
... | Emilio Novati | 187,568 | <p>The ''triangular array'' is not really an array but a column of values:</p>
<p>$ 2 $</p>
<p>$2-1=1$</p>
<p>$2-1-\frac{1}{2}=\frac{1}{2}$</p>
<p>$2-1-\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$</p>
<p>$2-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}=\frac{1}{8}$</p>
<p>$\cdots$</p>
<p>so <strong>there is not a diagonal</st... |
3,215,553 | <p>Recently I have been reading a lot about <span class="math-container">$\mathbb{Z}_2$</span>-actions on topological spaces. Mainly I was focused on surfaces such as the sphere, torus and Klein bottle and here the existence of a nontrivial <span class="math-container">$\mathbb{Z}_2$</span>-action is rather simple. But... | PR_ | 667,990 | <p>Consider <span class="math-container">$X=[0,1)$</span>. By connectedness arguments it follows that <span class="math-container">$0$</span> is fixed by every homeomorphism and as a consequence every <span class="math-container">$\mathbb{Z}_2$</span>-action on <span class="math-container">$X$</span> is trivial (to ver... |
3,659,942 | <p>I have the following problem:</p>
<p>Find all <span class="math-container">$n$</span> that are natural numbers such that <span class="math-container">$\sqrt{n}+\sqrt{n+m}\in\mathbb{N}$</span>, where <span class="math-container">$m$</span> is a positive natural number that is the product of two primes that are not e... | Brian M. Scott | 12,042 | <p>SKETCH: Let <span class="math-container">$B'$</span> be a subset of <span class="math-container">$A$</span> order-isomorphic to <span class="math-container">$B$</span>. Recursively construct an order-isomorphism from <span class="math-container">$B'$</span> onto an initial segment, not necessarily proper, of <span c... |
3,414,072 | <blockquote>
<p>Let <span class="math-container">$U \subset \mathbb{R}^N$</span> be an open set, let <span class="math-container">$f : U \times [a, b] \to \mathbb{R}$</span> be a continuous function. Consider the function <span class="math-container">$$g(x):= \int_a^b f(x,y) \,dy$$</span>
with <span class="math-con... | Masacroso | 173,262 | <p>First note that continuity is a <strong>local property</strong>. You can exploit this property together with the assumption that <span class="math-container">$U$</span> is open in <span class="math-container">$\Bbb R ^n$</span>. That is: we can simplify the exercise to show that <span class="math-container">$g$</spa... |
758,158 | <p>I am trying to use <span class="math-container">$f(x)=x^3$</span> as a counterexample to the following statement. </p>
<p>If <span class="math-container">$f(x)$</span> is strictly increasing over <span class="math-container">$[a,b]$</span> then for any <span class="math-container">$x\in (a,b), f'(x)>0$</span>. <... | binkyhorse | 18,357 | <p>Here is an intuitive proof that could be made rigorous using Lebesgue measure. </p>
<p>Start with $x>0$. Place a cube with edge length $x$ so that it has three faces on the coordinate planes in three-dimensional space and two corners at $(0,0,0)$ and at $(x,x,x)$. Then $f(x)=x^3$ is the volume of the cube. </p>
... |
725,241 | <p>Where s=circumcenter, H= orthocenter, and A'= midpoint of one side of triangle.
<img src="https://i.stack.imgur.com/Pg6rv.png" alt="enter image description here"></p>
<p>How can can I determine the location of the three vertices of the triangle? </p>
| Marc van Leeuwen | 18,880 | <p>It becomes easier if you first recognise that you may rename $k-1$ to $k'$ and $i-1$ to $i'$, so that you need to prove
$$
\binom{n+k'}{n-1}=\sum_{i'=0}^{k'}\binom{k'}{i'}\binom n{i'+1},
\qquad\text{for $0\leq k'<n$}
$$
then apply symmetry on the last binomial coefficient, drop the primes, and write $n-1=m$
$$
... |
725,241 | <p>Where s=circumcenter, H= orthocenter, and A'= midpoint of one side of triangle.
<img src="https://i.stack.imgur.com/Pg6rv.png" alt="enter image description here"></p>
<p>How can can I determine the location of the three vertices of the triangle? </p>
| mathse | 136,490 | <p>Yes, both answers are perfect, but to add to Marc van Leeuwen's answer - the formula is merely a simple application of the Vandermonde convolution, which states that</p>
<p>$\binom{n+m}{r} = \sum_{i+j=r} \binom{n}{i}\binom{m}{j}$</p>
<p>(with $m=k-1$ and $r=n-1$). So, from this you see that many other 'fancy' form... |
696,900 | <p>This should be trivial but for some reason I cannot think of a formula which enlarges a number in proportion to another number decreasing in the negative direction</p>
<p>Example:</p>
<p>if value 1 = $-0.1$, value 2 should be set to $0.9$<br>
if value 1 = $-0.2$, value 2 should be set to $0.8$<br>
if value 1 = $-0... | Nathan Smith | 690,955 | <p>In your description you mention that one should be increasing while the other is decreasing. In your examples, both value1 and value2 are getting smaller.</p>
|
1,708,072 | <p>This is Exercise 2 of section 1.8 in Humphreys' "Reflection groups and Coxeter groups", p.16.</p>
<p>The longest word $w_0$ in a finite reflection group $W$ acting on a Euclidean space (with a specified basis/system of simple roots $\Delta$) is defined as the unique element of $W$ that maps the fundamental chamber ... | anon | 194,722 | <p>I believe I have found an easy way to proceed. We know that $w_0$ maps every positive root to a negative root; in particular, it maps $\beta$ to a negative root. But by Proposition 1.4 of the book, for any simple root $\alpha \in \Delta$, the simple reflection $s_\alpha$ permutes all the positive roots that are dis... |
1,708,072 | <p>This is Exercise 2 of section 1.8 in Humphreys' "Reflection groups and Coxeter groups", p.16.</p>
<p>The longest word $w_0$ in a finite reflection group $W$ acting on a Euclidean space (with a specified basis/system of simple roots $\Delta$) is defined as the unique element of $W$ that maps the fundamental chamber ... | Ben West | 37,097 | <p>I believe your argument still works. Suppose $w_0=s_{\alpha_1}\cdots s_{\alpha_r}$ is a reduced expression in terms of various simple reflections, and suppose $\beta$ is a simple root distinct from all these $\alpha_i$. </p>
<p>I assume there is some positive definite bilinear form $(-,-)$ on the Euclidean space, s... |
1,506,805 | <p>$\lim _{x\to 0}\left(\frac{\cos\left(x+\frac{\pi }{2}\right)}{x}\right)\:$</p>
| John Joy | 140,156 | <p><a href="https://i.stack.imgur.com/5OBMN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5OBMN.png" alt="enter image description here" /></a></p>
<p>For acute angles <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, use the law of cosines to show that
<s... |
3,758 | <p>When implicitly finding the derivative of: </p>
<blockquote>
<p>$xy^3 - xy^3\sin(x) = 1$</p>
</blockquote>
<p>How do you find the implicit derivative of:</p>
<blockquote>
<p>$xy^3\sin(x)$</p>
</blockquote>
<p>Is it using a <em>triple</em> product rule of sorts?</p>
| Bill Dubuque | 242 | <p><strong>REMARK</strong> $\ $ Pierre's answer is a generating function approach:</p>
<p>$\displaystyle\rm\quad\quad\quad\;\; f(x+t) \;=\; \sum_{n=0}^\infty \;\; f^{(n)}(x) \; \frac{t^n}{n!}$</p>
<p>$\rm\quad (fgh)(x+t) \;=\; f(x+t)\: g(x+t)\: h(x+t)$</p>
<p>$\rm\quad\quad\quad\quad\quad\quad\quad\quad\;\: \;=\;... |
3,218,200 | <p>Let <span class="math-container">$\Bbbk$</span> be a field. A well-known result (see <a href="https://www.jstor.org/stable/2373270?origin=crossref&seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">Larson, Sweedler, <em>An Associative Orthogonal Bilinear Form for Hopf Algebras</em></a> and <a href="http... | Marco Farinati | 664,350 | <p>Here is an example. Take k a field and S the multiplicative semigroup {0,1}. For notational convenience denote it S={1,x}. Consider B=k[S] the semigroup algebra.</p>
<p>B is bialgebra and not Hopf because S is a semigroup and nota a group.</p>
<p><span class="math-container">$B\cong k[x]/(x^2-x)\cong k\times k$</s... |
1,799,559 | <blockquote>
<p>Let $f:[a,b]\to\mathbb{R}$ be a continious function. Show that if $$\int_a^b f(x)g(x)dx=0$$
for all continious functions $g:[a,b]\to\mathbb{R}$ with $g(a)=g(b)=0$, then $f(x)=0$ $\forall x\in[a,b]$</p>
</blockquote>
<p>I have difficulties proving this question. Consider for example $g(x)=0$ $\fora... | levap | 32,262 | <p>Note that if $f$ would satisfy $f(a) = f(b) = 0$, you could take $g = f$ and get $\int_a^b f^2(x) \, dx = 0$ and since $f^2(x)$ is continuous and non-negative, you would immediately get $f(x) \equiv 0$. In general, you can modify $f$ continuously so that it stays the same on $[a + \frac{1}{n}, b - \frac{1}{n}]$ but ... |
84,897 | <pre><code>Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}]
</code></pre>
<p>$$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$</p>
<p>gives $0$</p>
<p>This cannot be true. What is going on?</p>
<p>If I insert a number into <code>a</code>, it gives a reasonable result:</p>
<pre><code>NIntegrate[2/(Sin[t]^2 + 4), {t, 0,... | Dr. Wolfgang Hintze | 16,361 | <pre><code>$Version
(*
Out[228]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)"
*)
</code></pre>
<p>The result provided by <em>Mathematica</em> is correct:</p>
<pre><code>Integrate[a/(a^2 + Sin[t]^2), {t, 0, 2 π}]
(*
Out[213]= (2 π)/(Sqrt[1 + 1/a^2] a)
*)
</code></pre>
<p>Now the same procedure as "always"... |
1,614,899 | <p>Construct Context-free Grammar for integers. Integer can begin with + or - and after that we have non-empty string of digits. Integer must not contain unnecessary leading zeros and zero should not be preceded by + or -.
For example: 0; 123; -15; +9999 are correct, but +0; 01; +-3; +09; + are incorrect.</p>
<p>I hav... | David K | 139,123 | <p>In general, when you take a union of the sets in a collection of sets,
you can easily end up with a new set that is not in the original collection.</p>
<p>I'll assume that when we take $\bigcup_{\gamma\in A} \gamma$, where each
$\gamma$ is a "cut", we are actually referring to a union of the lower sets
in each cut... |
4,490,117 | <blockquote>
<p>(a) Let <span class="math-container">$f : [a, b] \to \mathbb{R}$</span> be continuous and suppose that <span class="math-container">$f(x) \gt 0$</span> for all <span class="math-container">$x$</span>. Show that there is some <span class="math-container">$L\gt 0$</span> such that <span class="math-contai... | user1035015 | 1,035,015 | <p>I would check out Set Theory: The Structure of Arithmetic by Hamilton and Landin. I love this book. They define addition formally and prove the addition tables and all the usual properties of addition. Unfortunately they do not go into algorithm, but I think you will find the content very enlightening and interest... |
668,451 | <p>A wizard has commanded you, the master architect, to build towers by stacking stone blocks.</p>
<p>You have at your disposal five stone masons of limited intellect. Each mason is capable of making only a single type of cylindrical block, the height of which must be a positive integer, and the diameter of which is ... | Bob Brooks | 244,123 | <p>Solve[a n1 + b n2 + c n3 + d n4 + e n5 == RandomInteger[{1, 300}], {a,
b, c, d, e, n1, n2, n3, n4, n5}]</p>
<blockquote>
<p>Just guessing.</p>
</blockquote>
|
668,451 | <p>A wizard has commanded you, the master architect, to build towers by stacking stone blocks.</p>
<p>You have at your disposal five stone masons of limited intellect. Each mason is capable of making only a single type of cylindrical block, the height of which must be a positive integer, and the diameter of which is ... | Mr.Wizard | 9,909 | <p>I don't know how to answer this directly, but I think I can help you approach it. We can show that the upper limit to the best solution is 251 different heights, as that is the maximum number of unique sums that can be produced. Using <em>Mathematica</em>:</p>
<pre><code>p = {a, b, c, d, e};
Array[Tr /@ Tuples[p,... |
2,862,166 | <p>In a paper I've been studying it says:</p>
<blockquote>
<p>Let $x$ in the cone $\mathbb R_+^n$ of all vectors in $\mathbb R^n$ with nonnnegative components ($n\in\mathbb N$)</p>
</blockquote>
<p>Somebody tell me what does it means, please? $\mathbb R_+^n$ should be $[0,\infty)\times\dots\times [0,\infty)$ ($n$ t... | Fred | 380,717 | <p>Yes, we have $\mathbb R_+^n= [0,\infty)\times\dots\times [0,\infty)$.</p>
<p>If $C \ne \emptyset$ is a subset of $\mathbb R^n$, then $C$ is called a cone if $x \in C$ and $t \ge 0$ imply that $tx \in C$.</p>
<p>Hence $\mathbb R_+^n$ is a cone in $\mathbb R^n$</p>
|
1,637,237 | <p>I'm studying differential calculus, but one of the questions involves solving an inequality:</p>
<p>$$(x-2)e^x < 0$$</p>
<p>I intend to go deeper in solving inequalities later, but I just want to understand how the teacher got the following solution in order to advance in these lectures:
$$x-2 < 0$$
$$x <... | BCLC | 140,308 | <p>Recall that for $a > 0$</p>
<p>$$ab < ac \iff b < c$$</p>
<p>Note that $e^x > 0 \ \forall x \in \mathbb R$</p>
|
118,680 | <p>The question is motivated by this one <a href="https://mathoverflow.net/questions/118626/real-symmetric-matrix-has-real-eigenvalues-elementary-proof">real symmetric matrix has real eigenvalues - elementary proof</a>: </p>
<p>Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed field... | Name | 30,062 | <p>This is a generalization of the idea of Will Sawin.
The Stufe of such field should be infinite. In fact if $-1$ is a sums of
squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$, then
$$A:=
\begin{pmatrix}
1 & a_1&\cdots & a_{n-1} \\
a_1 & a_1^2 &\cdots & a_1a_{n-1} \\
\vdots & \vdots&\ddots ... |
3,239,689 | <p><span class="math-container">$$n\in\mathbb{N}^{*}; S_{n}=\sum_{k=1}^{n}k!(k^2+1)$$</span></p>
<p>I need to find <span class="math-container">$S_n$</span></p>
<p>I started like this: <span class="math-container">$S_{n}=\sum_{k=1}^{n}(k+2)!-3(k+1)!+2k!$</span></p>
<p>How to continue?I tried to give the k values but... | AccidentalFourierTransform | 289,977 | <p><span class="math-container">$$
\sum_{k=1}^{n}\bigg((k+2)!-3(k+1)!+2k!\bigg)=\sum_{k=1}^{n}\bigg(\bigg[(k+2)!-(k+1)!\bigg]-2\bigg[(k+1)!-k!\bigg]\bigg)
$$</span>
and telescope.</p>
|
2,996,640 | <p>Maybe trivial for number theorists, but not for me: is the title meaningful to ask for (<span class="math-container">$p$</span> is a prime)? If so, what's the answer? Thanks</p>
| Ekesh Kumar | 614,428 | <p>Note that <span class="math-container">$x > 0$</span> and <span class="math-container">$y < 1$</span> can only satisfy <span class="math-container">$x > y$</span> on the interval <span class="math-container">$(0, 1)$</span>. </p>
<p>We can choose <span class="math-container">$x$</span> arbitrarily on the i... |
256,415 | <p>I need to store values of a function of x and y defined as <code>f[x_, y_] := Sin[Pi*x/3]*Sin[Pi*y/3]</code> at the following points <code>xval = {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14, 14.5, 15};</code>, same for y.</p>
<p>I need... | Henrik Schumacher | 38,178 | <p>Maybe this is what you want:</p>
<pre><code>Map[
X \[Function] {Indexed[X, 1], Indexed[X, 2], f[Indexed[X, 1], Indexed[X, 2]]},
Tuples[{xval, yval}]
]
</code></pre>
|
4,500,550 | <p>I have a very basic question about cohomology of sheaves. Suppose <span class="math-container">$\mathcal{F}$</span> is a sheaf of abelian groups over a topological space <span class="math-container">$X$</span>. Then <span class="math-container">$\mathcal{F}$</span> itself is a topological space with a continuous map... | Qiaochu Yuan | 232 | <p>They have nothing much to do with each other. Take the special case that <span class="math-container">$X$</span> is discrete, so a sheaf <span class="math-container">$F$</span> of abelian groups is just a collection <span class="math-container">$F_x, x \in X$</span> of abelian groups indexed by <span class="math-con... |
3,572,721 | <p>A finite sum of cosine functions weighted with different amplitude and phase, but with a fixed frequency,<span class="math-container">$$f(x) = \sum_{n=1}^{N}A_{n}cos(x+\phi_{n})$$</span>
the question is if I were to fit <span class="math-container">$f(x)$</span> with <span class="math-container">$cos(x)$</span>, wha... | mdnestor | 519,413 | <p>We have:
<span class="math-container">$$ \sum_{n=1}^{N}A_n\cos(x+\phi_n)=A\cos(x+\phi)$$</span>
Use the angle-sum formula <span class="math-container">$\cos(x+\phi)=\cos x\cos\phi-\sin x\sin\phi$</span>:
<span class="math-container">$$ \sum_{n=1}^{N}A_n(\cos x\cos\phi_n-\sin x\sin\phi_n)=A(\cos x\cos\phi-\sin x\sin\... |
459,067 | <p>In set theory, numbers are often constructed, e.g. from nestings of sets which eventually contain the empty set. The operations are defined in term of taking unions etc.etc. </p>
<p>The extended real number line hat plus and minus infinity $\infty,-\infty$ in addition to the reals, and they fulfill <a href="https:/... | The_Sympathizer | 11,172 | <p>One way to do this is using the Dedekind cut construction of the real numbers from the rationals. If one relaxes the requirement that the left and right sets of a Dedekind cut must be nonempty, one has the two cuts $(\emptyset, \mathbb{Q})$ and $(\mathbb{Q}, \emptyset)$. These can be identified with $-\infty$ and $\... |
215,227 | <p>I'm looking at a past homework solution and there is a part of it I don't understand. Specifically I'm talking about number 2 in <a href="http://www.math.cmu.edu/~af1p/Teaching/Combinatorics/F08/hw5a.pdf" rel="nofollow">this problem set</a>:</p>
<blockquote>
<p>Let $m=\lfloor(8/7)^{n/3}\rfloor$. Show that there e... | Brian M. Scott | 12,042 | <p>Suppose that $A_i\cap A_j\subseteq A_k$. Then for each $r\in[n]$ one of the following must be true:</p>
<ol>
<li>$r\in A_i$ and $r\in A_j$ and $r\in A_k$. </li>
<li>$r\in A_i$ and $r\notin A_j$, in which case it doesn’t matter whether $r\in A_k$. </li>
<li>$r\notin A_i$ and $r\in A_j$, in which case it doesn’t ma... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | Joel Cohen | 10,553 | <p>Any polygon (regular or not) can be described by an equation involving only absolute values and polynomials. Here is a small explanation of how to do that.</p>
<p>Let's say that a curve $C$ is given by the equation $f$ if we have $C = \{(x,y) \in \mathbb{R}^2, \, f(x,y) = 0\}$.</p>
<ul>
<li><p>If $C_1$ and $C_2$ a... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | Hans-Peter Stricker | 1,792 | <p>In the context of <a href="https://math.stackexchange.com/questions/3174514/fourier-series-of-regular-polygons">this question</a> you may define for a given number <span class="math-container">$n$</span> some (Fourier) coefficients by these sensible equations (as you have asked for):</p>
<p><span class="math-contai... |
4,370,760 | <p>The two statements are:</p>
<p><span class="math-container">$\forall a\in A, \exists b\in B, M(a)\implies N(b)$</span></p>
<p><span class="math-container">$(\forall a\in A, M(a))\implies (\exists b\in B, N(b))$</span></p>
<p>I am not so good in logics. I feel the second statement implies first one. The existence of ... | Arthur | 15,500 | <p>They are not the same. Consider the following statements:</p>
<ul>
<li><span class="math-container">$M(a)$</span>: Person <span class="math-container">$a$</span> has a dog</li>
<li><span class="math-container">$N(b)$</span>: Person <span class="math-container">$b$</span> has a cat</li>
</ul>
<p>Then the first statem... |
3,910,015 | <p>My impression has been that <span class="math-container">$\mathbb{Z_n}$</span> is the set <span class="math-container">$\{0,1,...,n-1\}$</span> under binary operation addition modulo <span class="math-container">$n$</span>. However I'm also coming across this notion that <span class="math-container">$\mathbb{Z_n}$</... | Bill Dubuque | 242 | <p>Below I explain the general idea, which works not only for rings but also for any algebraic structure definable by <span class="math-container">$\rm\color{#c00}{universal}$</span> equational axioms ("identities") such as <span class="math-container">$\,\color{#c00}{\forall x,y,z\!:}\,x(y+z) = xy+xz\,$</spa... |
79,040 | <p>Let <span class="math-container">$X$</span> be a compact Riemann surface, and <span class="math-container">$f$</span> a meromorphic function on X.
There's a theorem telling us that <span class="math-container">$\deg(\mathrm{div}(f)) = 0$</span>.</p>
<p>But is the inverse statement also true? I mean, is it true that:... | rla | 18,373 | <p>That is not true. Here a counterexample: consider a Riemann surface $X$ of genus $g \geq 1$. Fix $p, q \in X$ distinct points and consider the divisor $D = p - q$. This divisor has degree $0$, but it is not principal, because on the contrary there would be a holomorphic map $f: X \rightarrow \overline{\mathbb{C}}$ o... |
1,352,170 | <p>I have a homework question that I wasn't positive about. This is the first probability course I have taken and the class is only taught using excel so I apologize for the lack of formulas in my reasoning. </p>
<p>The size of a fish in a lake follows a normal distribution with a mean = 1 lb 4 oz and standard deviati... | Ross Millikan | 1,827 | <p>Given the clarification that the teams may split up, it takes some work. Let the heads label the new teams. Assign three people to the first head. They might be all from the same team, so you have a similar problem for four teams (but there are fewer constraints). They might be from three separate teams, or two ... |
1,352,170 | <p>I have a homework question that I wasn't positive about. This is the first probability course I have taken and the class is only taught using excel so I apologize for the lack of formulas in my reasoning. </p>
<p>The size of a fish in a lake follows a normal distribution with a mean = 1 lb 4 oz and standard deviati... | mjqxxxx | 5,546 | <p>The Python code below counts the number of possible assignments for $m$ teams of size $n$. It recursively assigns staff members to new teams, keeping track of how many empty slots and unassigned members there are for each team, and aggressively caching partial results.</p>
<p>For $n=1$, this is just the number of ... |
281,387 | <p>I've been reading Girard et al's 'Proofs and Types', which in Chapter 6 presents a proof of strong normalisation for the simply typed lambda calculus with products and base types. The proof is based on Tait's method of defining a set of 'reducible terms for type $T$' by induction on the type $T$. For example a term ... | Damiano Mazza | 45,027 | <p>Let me add to the above (perfectly correct) answers that there is a more general perspective, which works for all positive connectives, of which sums are a special case.</p>
<p>The fundamental insight is that reducibility actually arises from a duality between terms and contexts, which is itself parametric in a set... |
98 | <p>I am looking for a reference (book or article) that poses a problem that seems to be a classic, in that I've heard it posed many times, but that I've never seen written anywhere: that of the possibility of a man in a circular pen with a lion, each with some maximum speed, avoiding capture by that lion. </p>
<p>Refe... | BBischof | 16 | <p>Here is a book on this type of problem</p>
<ul>
<li>Paul J. Nahin, <a href="http://books.google.com/books?id=HeESjfM2geUC" rel="nofollow noreferrer">Chases and escapes: the mathematics of pursuit and evasion</a>, Princeton University Press, 2007.</li>
</ul>
<p>it is also briefly mentioned in his other book "Eul... |
98 | <p>I am looking for a reference (book or article) that poses a problem that seems to be a classic, in that I've heard it posed many times, but that I've never seen written anywhere: that of the possibility of a man in a circular pen with a lion, each with some maximum speed, avoiding capture by that lion. </p>
<p>Refe... | Aidan | 2,506 | <p>The book is Coffee in Memphis by Bollobas. It's the first problem, and there are loads more :</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0521693950" rel="noreferrer">http://www.amazon.com/Art-Mathematics-Coffee-Time-Memphis/dp/0521693950</a></p>
|
2,480,493 | <p>I have a problem with what looks like a very easy equation to solve $\left|\frac{1 + z}{1- i\overline z}\right| = 1$ . ($z$ is a complex number, $\overline z$ is a conjugate of $z$) I got stuck at the point when after replacing $\overline z = a-bi $ and $z = a +bi$ and getting rid of absolute value I end up with $... | Andreas Blass | 48,510 | <p>You want $|1+z|=|1-i\bar z|$. The left side here is the distance from $z$ to $-1$. The right side equals $|i+\bar z|$ which in turn equals $|-i+z|$, the distance from $z$ to $i$. So $z$ satisfies your equation iff it is equidistant from $-1$ and $i$. These $z$'s form a straight line in the complex plane, whose equ... |
2,480,493 | <p>I have a problem with what looks like a very easy equation to solve $\left|\frac{1 + z}{1- i\overline z}\right| = 1$ . ($z$ is a complex number, $\overline z$ is a conjugate of $z$) I got stuck at the point when after replacing $\overline z = a-bi $ and $z = a +bi$ and getting rid of absolute value I end up with $... | Community | -1 | <p>Hint: Let $z=a+ib$, then $|\frac{1+z}{1-i\overline{z}}|=\frac{|1+z|}{|1-i\overline{z}|}=\frac{|(1+a)+ib|}{|(1-b)-ia|}=\frac{\sqrt{(1+a)^2+b^2}}{\sqrt{(1-b)^2+a^2}}=\frac{\sqrt{1+a^2+2a+b^2}}{\sqrt{1+b^2-2b+a^2}}=1$ </p>
<p>Now this equality is true if and only if $a=-b$ which gives us that if $z=a+ib$ then $iz=ai-b... |
3,515,656 | <p>Given <span class="math-container">$V(x) = -5x_1^2 (x_1^2 - 1) - 4x_1x_2$</span>.
Show that there exist <span class="math-container">$x_0 \in \mathbb{R}^2$</span> arbitrarily close to <span class="math-container">$x = 0$</span> such that <span class="math-container">$V(x_0) > 0$</span>.
I have no idea how to pr... | Siong Thye Goh | 306,553 | <p>Let <span class="math-container">$0<\epsilon < 1$</span> and <span class="math-container">$(x_1, x_2) = (\epsilon, 0)$</span>, then we have </p>
<p><span class="math-container">$$V(x)=-5\epsilon^2(\epsilon^2-1)>0$$</span></p>
|
3,515,656 | <p>Given <span class="math-container">$V(x) = -5x_1^2 (x_1^2 - 1) - 4x_1x_2$</span>.
Show that there exist <span class="math-container">$x_0 \in \mathbb{R}^2$</span> arbitrarily close to <span class="math-container">$x = 0$</span> such that <span class="math-container">$V(x_0) > 0$</span>.
I have no idea how to pr... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$V(\frac 1 n , \frac 1 n)=-\frac 5 {n^{4}} +\frac 5 {n^{2}}-\frac 4 {n^{2}} >0$</span> for all <span class="math-container">$n \geq 3$</span>. Every neighborhood of the origin contains the points <span class="math-container">$(\frac 1 n , \frac 1 n)$</span> for infinitely many value... |
4,435,945 | <p>Near point <span class="math-container">$(0,0)$</span> find taylor formula for <span class="math-container">$f(x,y)=\ln(1+x+y)$</span></p>
<p><span class="math-container">$f(x)=f(x_0)+\sum_{k=1}^n \frac{1}{k!}d^kf(x_0)+o(|x-x_0|^n)$</span></p>
<p><span class="math-container">$$d^kf(0,0)=\sum_{n=1}^k\binom{k}{n}\frac... | trancelocation | 467,003 | <p>A bit late answer but I think worth mentioning it because it doesn't need any Gram-Schmidt:</p>
<p>Note that the direction vector <span class="math-container">$\begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}$</span> of <span class="math-container">$X_2$</span> is also a spanning vector of <span class="math-container">$P$<... |
621 | <p>The OP of this question is asking if <a href="https://mathematica.stackexchange.com/questions/9639">https://mathematica.stackexchange.com/questions/9639</a> could be reopened. I did not want to act unilaterally, so I want to ask the rest of you whether you think the question should be reopened. Plead your case for o... | J. M.'s persistent exhaustion | 50 | <p>For those who'd rather vote than write an answer:</p>
<p>Vote this answer up if you think <a href="https://mathematica.stackexchange.com/questions/9639">https://mathematica.stackexchange.com/questions/9639</a> should be reopened.</p>
|
1,609,854 | <p>If a polyhedron is made only of pentagons and hexagons, how many pentagons can it contain? With the assumption of three polygons per vertex, one can prove there are 12 pentagons.</p>
<p>Let's not make that assumption, and only use pentagons. </p>
<p>12 pentagons: dodecahedron and <a href="https://en.wikipedia.org... | user159517 | 159,517 | <p>Hint: Find the laurent expansion of $1/z$ and differentiate.</p>
|
804,673 | <p>EDIT: Small Typo Fixed now, Thanks to Sir Chen Wang!</p>
<p>Hi I am trying to prove this result without using a series approach
$$
\int_0^1\log(1+x)\frac{1+x^2}{(1+x)^4}dx=-\frac{\log 2}{3}+\frac{23}{72}.
$$I know we can just solve it by writing
$$
\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\left( \int_0^1\frac{x^n}{(... | Ben Grossmann | 81,360 | <p>Not a solution, but a possible step in the right direction (too long for a comment):</p>
<p>Following Cramer's rule, we note that the solution $\vec y = (y_1,\dots,y_{2014})^T$ to
$$X \,\vec y = \pmatrix{
\frac{2^2}{4} & \frac{3^2}{5} & \dfrac{4^2}{6} & \cdots & \dfrac{2014^2}{2016}
}^T$$
Will sati... |
804,673 | <p>EDIT: Small Typo Fixed now, Thanks to Sir Chen Wang!</p>
<p>Hi I am trying to prove this result without using a series approach
$$
\int_0^1\log(1+x)\frac{1+x^2}{(1+x)^4}dx=-\frac{\log 2}{3}+\frac{23}{72}.
$$I know we can just solve it by writing
$$
\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\left( \int_0^1\frac{x^n}{(... | Will Orrick | 3,736 | <p>Consider something a bit more general. Let $$X=\left[ {\begin{array}{cc} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\
1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\
1 & x_3 & x_3^2 & \cdots & x_3^{n-1} \\
\vdots & \vdots & \vdots & \cdots & \vdots \\
1 &am... |
2,713,853 | <p>This is the question:
"Let X and Y be independent random variables representing the lifetime (in 100 hours) of Type A and Type B light bulbs, respectively. Both variables have exponential distributions, and the mean of X is 2 and the mean of Y is 3.
What is the variance of the total lifetime of two Type A bulbs and... | V. Vancak | 230,329 | <p>Note that if $X\sim \mathcal{E}xp(\lambda)$, thus $\mathbb{E}[X] = 1/\lambda$. Hence,
if $A$'s mean life time is $2$, that is $\mathbb{E}[X] = 2 = 1/\lambda \to \lambda =1/2.$ As such, $\operatorname{var}(X) = 1/\lambda^2 = 1/(1/2)^2 = 4$. Same is true for type $B$. </p>
|
679,341 | <p>The 4 bytes used in CDROM sectors is just a usual 32 bit CRC. It uses the polynomial </p>
<pre><code>P(x) = (x^16 + x^15 + x^2 + 1).(x^16 + x^2 + x + 1)
</code></pre>
<p>which expands to </p>
<pre><code>x^32 + x^31 + 2x^18 + 2x^17 + 3x^16 + x^15 + x^4 + x^3 + 2x^2 + x + 1
</code></pre>
<p>The CRC process rever... | Hagen von Eitzen | 39,174 | <p>This boils down to a sequence of <em>shift</em> and <em>xor</em> operations, see for example <a href="http://en.wikipedia.org/wiki/Cyclic_redundancy_check" rel="nofollow">here</a> for an explanation, including an example calculation.</p>
|
427,663 | <p>If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $</p>
<p>i found one link that had a similar problem.
<a href="https://math.stackexchange.com/questions/343982/prove-if-z-1-and-w-1-then-1-zw-neq-0-and-z-w-over-1">Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w}... | Berci | 41,488 | <p><strong>Hint:</strong> For complex numbers $a,b$ we have $a/b\in\Bbb R\iff a\bar b\in\Bbb R$.</p>
|
1,769,596 | <p><strong>Question</strong> Calculate the critical point(s) for the following function $$f =\sin(x)+\cos(y) +\cos(x-y)$$</p>
<p><em>Hint:</em> Use the trigonometric identity $$\sin(\alpha) +\sin(\beta) = 2\sin(\frac{\alpha+\beta}{2})\cos(\frac{\alpha -\beta}{2})$$</p>
<p><strong>My attempt:</strong></p>
<p>$$f_x=\c... | Christian Blatter | 1,303 | <p>From the two equations concerning $f_x$, $f_y$ you obtained it immediately follows that $$\sin y=\sin(x-y)=\cos x\ .$$
Now $\sin y=\cos x$ is equivalent with $${\rm (i)} \quad y={\pi\over2}+x\qquad\vee\qquad{\rm (ii)} \quad y={\pi\over2}-x$$ (all angles are modulo $2\pi$).</p>
<p>In case (i) we then obtain $\cos ... |
2,835,177 | <p>For example 6,</p>
<p>convert to the base 2 = 110</p>
<p>number of 1's is 2(prime), number of zero's is 1(prime - 1)</p>
<p>or 496 = (111110000)2</p>
<p>5(prime) times 1 and 4 times 0</p>
<p>Is this always correct?</p>
| BruceET | 221,800 | <p><strong>Extended Comment:</strong> As indicated in the Comment by @HagenvonEitzen, one way to work the initial problem (on the probability D6 shows a larger value than D10) is to enumerate
cases. In particular, you might make a $10 \times 6$ array of possible pairs
of outcomes and highlight the pairs that satisfy yo... |
3,974,200 | <p>Let <span class="math-container">$X$</span> be an <span class="math-container">$n\times n$</span> negative definite matrix such that <span class="math-container">$X_{ii}<0$</span> for all <span class="math-container">$i$</span> and <span class="math-container">$X_{ij}\geq 0$</span> for <span class="math-container... | thedude | 297,270 | <p>By the <a href="https://en.wikipedia.org/wiki/Stationary_phase_approximation" rel="nofollow noreferrer">stationary phase method</a>, the integral is approximately given by
<span class="math-container">$$ \int_0^1 e^{inp(x)}dx= \sum_{x_0} \sqrt{\frac{2\pi}{np''(x_0)}}e^{inp(x_0)+\mathrm{sign}(p''(x_0))i\pi/4}+o(1/\sq... |
341,946 | <p>Using $n$-th Taylor polynomial for $f_1(x)=\frac{1}{1-x}$ with center in $0$, find $4$-th derivative of $f_2(x)=\frac{1+x+x^2}{1-x+x^2}$ in the point $0$ without calculating it's $1$,$2$ or $3$ derivative. </p>
<p>I'm looking for hints, it's a homework. I've tried using the Taylor expansion with Lagrange rest, but... | Ian Coley | 60,524 | <p>Let $(f,g)=\int_{-1}^1f(x)g(x)\,dx$ denote the $L^2$ inner product.</p>
<p>We seed the process with $P_0(x)=1,P_1(x)=x$. We'll prove this by induction and try to first prove $n=2$. We know $P_2(x)$ satisfies $(P_2,P_0)=\int_{-1}^1 P_2(x)\cdot 1=0$ and $(P_2,P_1)=\int_{-1}^1 P_2(x)\cdot x=0$. We don't actually have ... |
1,623,796 | <p>In the structure $\langle \mathbb{Q}, < \rangle$ which of the following axioms hold? How about when we use the weak versions of the axioms (all $\leftrightarrow$ replaced with $\rightarrow$ )? </p>
<p>Extensionality, Empty Set, Power Set, Infinity</p>
<p>Find an instance of Separation that is true in $\langle \... | Asaf Karagila | 622 | <p>One easy way to see this is false is that the $\sf ZF$ axioms prove that $\in$ is not transitive on every set. Namely, there exists $a,b,c$ such that $a\in b$ and $b\in c$, but $a\notin c$.</p>
<p>For example, $a=\varnothing, b=\{\varnothing\}, c=\{\{\varnothing\}\}$.</p>
<p>On the other hand, $<$ is transitive... |
262,601 | <p>What does the following statement mean, that Nature likes to minimize things (like energy) and this equation describes one particular minimization problem? </p>
<p>What is the Variational Principle?</p>
| Marcel Rubió | 53,712 | <p>The variational principle is, in fact, a physical principle. To get an idea, the easiest example of a variational principle is Fermat's principle in non-relativistic optics (the one taught at school via geometric means).
The idea to use a variational principle is that at the end we get an equation, or even a set of... |
2,755 | <p>As Akhil had great success with his <a href="https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry">question</a>, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enou... | Jon Yard | 434 | <p>My favorite book right now on representation theory is Claudio Procesi's <a href="http://books.google.com/books?id=Sl8OAGYRz_AC&dq=Procesi+Lie+groups&printsec=frontcover&source=bl&ots=8LwROHsMyF&sig=HUTjse0dj6r2K-CcAFjteYyTwD0&hl=en&ei=uabnSuPNF4H2sgOd96ybBQ&sa=X&oi=book_result&am... |
1,856,225 | <p>Right, so as the final step of my project draws near and after having made a bad layout sort of question, I am posting a new one very clear and unambiguous. I need to find this specific definite integral which Mathematica could not solve:</p>
<p>$$ \int_{x=0}^\pi \frac{\cos \frac{x}{2}}{\sqrt{1 + A \sin^2 \frac{x}... | achille hui | 59,379 | <p>Let $\mathcal{I}$ be the integral at hand and $B = \frac{1+A}{\sqrt{A}}\omega$. Introduce variables $y, t, \theta, z$ such that</p>
<p>$$y = \sin\frac{x}{2},\quad t = \tanh\theta = \frac{\cos\frac{x}{2}}{\sqrt{1+A\sin^2\frac{x}{2}}}\quad\text{ and }\quad z = e^\theta$$</p>
<p>Notice
$$t = \sqrt{\frac{1-y^2}{1+Ay^2... |
2,167,598 | <p>I've seen the answer for gcd$(a,b)$ but never for the lcm$(a,b)$?</p>
| Carl Schildkraut | 253,966 | <p>For lcm$(a,b)$, it's a multiple of $a$, so we can take $x=\frac{lcm(a,b)}{a}, y=0$. The gcd case, as you can see, is more interesting. </p>
|
2,412,839 | <p>I figured I could use the binomial formula to solve this problem but then remembered it only works when the order does matter (repeated combinations are included). Is there a formula that does not account for the repeated combinations and that could help me solve this problem?</p>
| Graham Kemp | 135,106 | <p>The trick with "order doesn't matter" is to ask "to whom doesn't it matter and why?" </p>
<blockquote>
<p>A Die is Rolled 3 times, what is the probability that a six is rolled at least once when the order doesn't matter?</p>
</blockquote>
<p>Why doesn't order matter? Because, the observer loses track of ... |
677,574 | <p>Do there exist examples of non-empty, infinite spaces <span class="math-container">$X$</span> not equipped with the discrete topology for which <span class="math-container">$X \cong X \times X$</span>?</p>
| omar | 12,273 | <p>Take for example $X=\cup_{n\ge1}\mathbb R^n$ with the topology defined by $U\subset X$ is open if and only if $U\cap\mathbb R^n $ is open for all $n$. It is then easy to check that the map $\phi:X\times X\to X$ defined by $\phi((x_i),(y_i))=(x_1,y_1,x_2,y_2,x_3,\dots)$ is a homeomorphism here an element of $X$ is re... |
677,574 | <p>Do there exist examples of non-empty, infinite spaces <span class="math-container">$X$</span> not equipped with the discrete topology for which <span class="math-container">$X \cong X \times X$</span>?</p>
| Henno Brandsma | 4,280 | <p>Elementary examples are often zero-dimensional: $\mathbb{Q}$, $\mathbb{P}$ (= the irrationals as a subset of $\mathbb{R}$) and the Cantor set $C \subset [0,1]$ all are homeomorphic to their squares (so even every finite power of itself; even countable power, for the irrationals and $C$.). A trivial example is an inf... |
325,236 | <blockquote>
<p>Is there a bijection between $\mathbb N$ and $\mathbb N^2$?</p>
</blockquote>
<p>If I can show $\mathbb N^2$ is equipotent to $\mathbb N$, I can show that $\mathbb Q$ is countable. Please help. Thanks,</p>
| André Nicolas | 6,312 | <p>It is easier to write an answer than to find one of its several occurrences on this site. </p>
<p>Let $n$ be a positive integer. Then there exist uniquely determined positive integers $u$ and $v$ such that $n=2^{u-1}(2v-1)$. This is because every positive integer $n$ can be written in one and only one way as a powe... |
325,236 | <blockquote>
<p>Is there a bijection between $\mathbb N$ and $\mathbb N^2$?</p>
</blockquote>
<p>If I can show $\mathbb N^2$ is equipotent to $\mathbb N$, I can show that $\mathbb Q$ is countable. Please help. Thanks,</p>
| Julien | 38,053 | <p>Here is a non constructive proof.</p>
<p>Of course, there is an injection of $\mathbb{N}$ into $\mathbb{N}\times\mathbb{N}$.</p>
<p>Now consider the map
$$
(x,y)\longmapsto 2^x\cdot 3^y
$$
By the fundamental theorem of arithmetic, this is an injection of $\mathbb{N}\times\mathbb{N}$ into $\mathbb{N}$.</p>
<p>By <... |
574,220 | <p><strong>Question 1:</strong></p>
<p>let Polynomial <span class="math-container">$f(x)=\displaystyle\sum_{i=0}^{3}a_{i}x^i,$</span> have three real numbers roots,where <span class="math-container">$a_{i}>0,i=1,2,3$</span>.</p>
<blockquote>
<p>show that:
<span class="math-container">$$g(x)=\sum_{i=0}^{3}a^m_{i}x^i$... | zy_ | 27,648 | <p>It is really a tough question. A generalization of this problem follows from a theorem in the famous book by Polya and Szego(Book II, Chapter 5, Problem 155), which states:</p>
<p>If</p>
<p>$$a_0+a_1 x+\cdots+a_n x^n$$</p>
<p>and</p>
<p>$$b_0+b_1 x+\cdots+b_n x^n$$</p>
<p>have only real zeros, while all the zer... |
3,789,429 | <blockquote>
<p><strong>Problem</strong>:
Can an <span class="math-container">$f$</span> function be created where:<span class="math-container">$$f\colon\mathbb Q_{+}^{*}\to \mathbb Q_{+}^{*}$$</span>
The function is defined on the set of fully positive rational numbers and is achieved:
<span class="math-container">$\f... | Marktmeister | 762,010 | <p>There is no such function.</p>
<p><em>Proof:</em>
Specifically for <span class="math-container">$x = 1$</span>, we obtain that <span class="math-container">$f(f(y)) = \frac{f(f(1))}{y}$</span> for all <span class="math-container">$y$</span>.</p>
<p>Now, let <span class="math-container">$a \in \mathbb Q$</span>, <spa... |
2,491,560 | <p>Given a sequence $(s_n)$ in $\mathbb R$ such that
$$\lim \limits_{n \to \infty}( s_{n+1}-s_n)=0,$$
I am asked to prove
$(s_n)$ converges. </p>
<p>I know all Cauchy sequences converge in $\mathbb R^k$. So I want to prove that $(s_n)$ is Cauchy. </p>
<p>I am stuck as to how to show the given sequence is a Cauchy. ... | egreg | 62,967 | <p>It’s not surprising you’re stuck, because the claim is false.</p>
<p>Consider $s_n=\sqrt{n}$. Then
$$
\lim_{n\to\infty}(s_{n+1}-s_n)=
\lim_{n\to\infty}(\sqrt{n+1}-\sqrt{n})=
\lim_{n\to\infty}\frac{1}{\sqrt{n+1}+\sqrt{n}}=0
$$</p>
|
2,071,054 | <p>Let $f(x)$ be continuous in $[a,b]$</p>
<p>Let $A$ be the set defined as:
$$A = \{ x \in [a,b] \mid f(x) = f(a) \}$$</p>
<ol>
<li><p>Does $A$ have a maximum? I guess it it is the max value in $[a,b]$ which $f$ sends to $f(a)$ but I don't know how to prove it.</p></li>
<li><p>Would it still have a maximum if $[a,b]... | Christian Blatter | 1,303 | <p>You can avoid inclusion-exclusion by considering $S$ as a nested sum:
$$S=6\sum_{i=0}^\infty 3^{-i}\sum_{j=i+1}^\infty 3^{-j}\sum_{k=j+1}^\infty 3^{-k}\ ,$$
whereby the factor $6$ compensates for the assumption $i<j<k$. The innermost sum has the value $$3^{-j}\sum_{k'=1}^\infty 3^{-k'}={1\over2} 3^{-j}\ .$$
T... |
1,061,325 | <p>I need to show that for $x>0$, $1+\frac{x}{2} \ge \sqrt{1+x} \ge 1+\frac{x}{2} - \frac{x^2}{8} $.</p>
<p>I used the geometric/arithmetic mean inequality to show that $1+\frac{x}{2} \ge \sqrt{1+x}$ is indeed true. My issue lies in the second part of this.</p>
<p>In trying to show that $\sqrt{1+x} \ge 1+\frac{x}{... | turkeyhundt | 115,823 | <p>Yes. Different structures. Trig/Unit Circle starts with zero at "East" and increases in a counterclockwise direction. Bearings and directions use vertical-up as zero degrees.</p>
<p><img src="https://i.stack.imgur.com/I3zN7.png" alt="enter image description here"></p>
|
1,379,878 | <blockquote>
<p>Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.</p>
</blockquote>
<p>My Try:</p>
<p>I proved that $C(M)$ is ... | Sungjin Kim | 67,070 | <p>Your approach is correct. Let $PMP^{-1}=J$ with invertible $P$ and Jordan form $J$. Then
$$
\begin{align}
AM=MA &\Longleftrightarrow PAMP^{-1}=PMAP^{-1} \\
&\Longleftrightarrow PAP^{-1} PMP^{-1} = PMP^{-1} PAP^{-1}\\
&\Longleftrightarrow PAP^{-1} J= J PAP^{-1}.
\end{align}
$$
Thus, we have
$$
\phi_P ... |
1,685,621 | <p>I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. </p>
<p>So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real vector space), we can say that they are orthogonal iff $\langle \vec x,\vec y\rangle=0$, where $\langle \vec x,\ve... | Theodor Johnson | 320,421 | <p>The unit base vectors $e_1$,$e_2$: form an orthonormal basis.</p>
<p>Let us take a look what this requires:</p>
<ul>
<li>all elements in $M:=\{e_1,e_2,...,e_n \}$ are linearly independent</li>
<li>the vectors $e_1$,$e_2$,...,$e_n$ are pairwise orthogonal, so their dot product $\langle e_i,e_k \rangle=0 $, $\forall... |
5,998 | <p>I have restricted the context of my notebooks to each individual notebook. So variables in each notebook are local and are not seen in another notebook.</p>
<p>But in two of my notebooks I have two functions that I want to plot simultaneously with the command <code>Show</code>. Please note that I want variables of... | celtschk | 129 | <p>You can explicitly define variables in the global context by prefixing their name with <code>Global`</code>, for example, <code>Global`i = 3</code> or <code>Global`f[x_]:=x^2</code>. However if you have set the notebook to have private context, you don't have <code>Global`</code> in your <code>$ContextPath</code> (i... |
2,030,063 | <p>Let $A$ and $B$ be two disjoint subset of $\mathbb{R}^2$ such that $A \cup B$ is open and disconnected in $\mathbb{R}^2$. does it follows both $A$ and $B$ are open. </p>
<p>If $A$ and $B$ are both open, since $A$ and $B$ are disjoint by hypothesis. then by definition $A \cup B$ is disconnected. But for the revers... | Andrew D. Hwang | 86,418 | <p>You've got multiple counterexamples under the hypotheses of the question.</p>
<p>If you assume $A$ and $B$ are <em>connected</em>, however, then "$A \cup B$ open and disconnected" implies $A$ and $B$ are open.</p>
<p>To prove this, note that $A \cap B = \varnothing$: If a family of connected sets has an element in... |
2,030,063 | <p>Let $A$ and $B$ be two disjoint subset of $\mathbb{R}^2$ such that $A \cup B$ is open and disconnected in $\mathbb{R}^2$. does it follows both $A$ and $B$ are open. </p>
<p>If $A$ and $B$ are both open, since $A$ and $B$ are disjoint by hypothesis. then by definition $A \cup B$ is disconnected. But for the revers... | Unknown x | 753,217 | <p>For counterxample , take <span class="math-container">$A=[(0,1)\times (0,1)] \cap [\mathbb Q \times \mathbb R] $</span> and <span class="math-container">$B=[(0,1)\times (0,1)] \cap [\mathbb Q^c \times \mathbb R] $</span></p>
|
2,476,267 | <p>Here's a problem presented in my proofs class that I cannot for the life of me figure out. Apparently the answer is P2 according to the answer book but I have no idea how Help is much appreciated, thanks in advance!</p>
<p>Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper:</p... | zipirovich | 127,842 | <p>You're told that one of these statement — which means <strong>precisely</strong> one of them — is TRUE. So you can simply consider all possible cases in a very straightforward manner:</p>
<ul>
<li>maybe P1 is TRUE and the other two are FALSE;</li>
<li>or maybe P2 is TRUE and the other two are FALSE;</li... |
2,476,267 | <p>Here's a problem presented in my proofs class that I cannot for the life of me figure out. Apparently the answer is P2 according to the answer book but I have no idea how Help is much appreciated, thanks in advance!</p>
<p>Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper:</p... | Marcelo | 487,716 | <p>Try case by case and see the only one who can happen.</p>
<p>If P1 is true than exist just one false, so we have two options P2-T and P3-F; or P2-F and P3-T; note that both are impossible. Then we have P1 must be false (because we showed that in every case of assuming P1 true, contradictions happens).</p>
<p>Now w... |
398,535 | <p>I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am curious why the word "recursive" appears in their name. What does the concept of decidability/recognizability have to do... | Michael Hardy | 11,667 | <p>Quoting from a comment by the original poster: "Is 'recursion' a word in computer science that has two independent meanings ("computable" and "function that calls itself")? Or are those meanings somehow related?"</p>
<p><b>Yes, they're related,</b> but you could reasonably dislike the way in which they're related.<... |
121,453 | <p>I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces when G and K are general topological groups.
Thank you. </p>
| Peter May | 14,447 | <p>The Sullivan conjecture, a beautiful theorem proved by Haynes Miller
(The Sullivan conjecture on maps from classifying spaces) has striking
consequences about maps between classifying spaces when the target is
completed at a prime $p$. Some references are Dwyer and Zabrodsky
(Maps between classifying spaces), Jacko... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Community | -1 | <p>Estimate the result. Multiply the numbers without paying attention to the decimal point. Place the decimal point so that the result is close to the estimate.</p>
|
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Kevin | 13,109 | <p><strong>Do it in slices.</strong></p>
<p>We teach young kids that multiplication is just repeated addition. What's 5x4? It's just 5, added 4 times.</p>
<p>So do the same with the decimal numbers. 3.9 x 7.5. First, let's use 3.9 added 7 times. You can't add it an eighth time, because that's over 7.5 - we only n... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Galactic | 13,851 | <p><span class="math-container">$$\left\lfloor \frac{\left(x+y\right)^2}{4} \right\rfloor - \left\lfloor \frac{\left(x-y\right)^2}{4} \right\rfloor = xy.$$</span></p>
<p>This answer uses the floor function (which essentially means to get rid of the decimal). You can then quickly consult a table to find 0.25*x^2. </p>
... |
2,535,347 | <p>Okay so I'm required to find if the given function is <strong>one-one/many-one and onto/into</strong>.</p>
<p>The function is : $\mathbb R \to \mathbb R$ and $$f(x) = x\left(\frac{2^x - 2^{-x}}{2^x + 2^{-x}}\right)$$</p>
<p>So as this function is even , it can't be one-one .But I'm facing difficulty in deducin... | Burrrrb | 322,248 | <p>Your function is not onto, because it is non-negative. To see this, examine the cases when $x\ge0$ and $x\le0$ separately. The denominator is always positive and the numerator is positive when $x$ is positive and negative when $x$ is negative. So the function is always non-negative.</p>
<p>Also note that $ \lim_{x... |
260,240 | <p><img src="https://i.stack.imgur.com/3WeQw.png" alt="enter image description here"></p>
<p>In the last bullet, it says l must be even and provides an explanation. I don't understand the explanation, however. Why does it have to be even?</p>
| Billy | 39,970 | <p>I don't really follow the argument given. Here's a correct argument...</p>
<p>If $a^2 = 2l$, then $a^2$ is divisible by 2. But 2 is prime, so $a$ must be divisible by 2 too - say $a = 2b$. Then $(2b)^2 = 2l$, i.e. $2b^2 = l$, and so $l$ is divisible by 2.</p>
|
3,700,575 | <p>I have to find Taylor series at <span class="math-container">$a=1$</span> for </p>
<p><span class="math-container">$
f(x)=\begin{cases}
\frac{e^{x}-e}{x-1},\quad &\text{if } x\ne1\\
e,\quad &\text{if } x=1\\
\end{cases}
$</span></p>
<p>I haven't found Taylor series for ... | Community | -1 | <p>With <span class="math-container">$y:=x-1$</span>,<span class="math-container">$$\frac{e^{x-1+1}-e}{x-1}=e\frac{e^y-1}y=e\sum_{k=1}^\infty \frac{y^k}{k!y}=e\sum_{k=1}^\infty \frac{(x-1)^{k-1}}{k!}.$$</span></p>
|
1,735,653 | <p>So I need to use the fact that: $$\cos(4x) + i\sin(4x) = \left(\cos(x) + i\sin(x)\right)^4$$ to derive identities for $\cos(4x)$ and $\sin(4x)$ in terms of $\cos(x)$ and $\sin(x)$. I'm not sure how to go about this, could I please get some help.</p>
| Roman83 | 309,360 | <p>$$(\cos x +i \sin x)^4=\cos 4x+i \sin 4x$$
$$(\cos x +i \sin x)^4=\cos^4x+4\cos ^3x \cdot (i \sin x)+6\cos ^2x \cdot (i \sin x)^2+4\cos x \cdot (i \sin x)^3+(i \sin x)^4=$$
$$=\cos^4x-6 \cos ^2x \sin^2x+\sin^4x+$$
$$+i(4 \cos ^3 x \sin x-4\cos x\sin^3x)$$
Then $$\cos 4x=\cos^4x-6 \cos ^2x \sin^2x+\sin^4x$$
and $$\si... |
3,068,381 | <p>I calculated, using Mathematica, that for <span class="math-container">$4\leq k \leq 100$</span>,
<span class="math-container">$$ \sum_{j=k}^{2k} \sum_{i=j+1-k}^j (-1)^j 2^{j-i} \binom{2k}{j} S(j,i) s(i,j+1-k) = 0,$$</span>
where <span class="math-container">$s(i,j)$</span> and <span class="math-container">$S(i,j)$<... | Marko Riedel | 44,883 | <p>Working with the notation by @GCab we seek to evaluate</p>
<p><span class="math-container">$$S_n = \sum_{j=n}^{2n} \sum_{k=j+1-n}^j
(-1)^j 2^{j-k} {2n\choose j} {j\brace k}
{k\brack j+1-n}.$$</span></p>
<p>With the usual EGFs we get</p>
<p><span class="math-container">$$\sum_{j=n}^{2n} \sum_{k=j+1-n}^j
(-1)^j 2^... |
2,328,677 | <p>I wish to find the area between the curves:</p>
<p>$y=\sqrt{x}$</p>
<p>and </p>
<p>$y=x^{2}-3x-2$</p>
<p>and between $x=1, x=9$</p>
<p>Now, I think that the first thing to do is to find the intersection between the two curves, and then to find which curve is the upper one before and after the intersection point... | Dr. Sonnhard Graubner | 175,066 | <p>you must solve the equation $$\sqrt{x}=x^2-3x-2$$ after squaring and factorizing you will get
$$(x-4) \left(x^3-2 x^2-3 x-1\right)=0$$ the otherone is $$x\approx 3.07959562349143878601$$</p>
|
1,281,950 | <p>Is there a way to express the integral</p>
<p>$I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$,</p>
<p>where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known functions? In particular, I am interested in a form where $t$ appears in the limits and not in the integra... | Daniel | 150,142 | <p>A matrix $M_{n\times n}$ is invertible if (by definition) the exists a matrix $N_{n\times n}$ such that $MN=NM=I$ ($I$: Identity of ${n\times n}$). </p>
<p>Here you have $AB=I$, can you prove $BA=I$? </p>
<p><em>NOTE:</em> This is how I interpreted your question, asking uniqueness of the inverse and that $A$ is th... |
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