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 |
|---|---|---|---|---|
320,118 | <p>Suppose $$d = un + vs$$ where $d$ is the $\gcd(n,s)$</p>
<p>Dividing $d$ both sides</p>
<p>$$1 = u(n/d) + v(s/d)$$</p>
<p>So $(n/d)$ and $(s/d)$ are integers that are relatively prime. Why does this show that for any integer dividing both of them must also divide $1$?</p>
<p>This is a part of a proof I am readin... | Julien | 38,053 | <p>This is (sometimes called) Bezout: $\gcd(a,b)=1$ if and only if there exist $u,v$ such that $au+bv=1$.</p>
<p>You are asking about $\Leftarrow$.</p>
<p>If $n$ divides $a$ and $b$, then $n$ divides $au$ and $bv$, so $n$ divides $au+bv$.</p>
<p>Thus $n$ divides $1=au+bv$.</p>
<p>The only divisors of $1$ are $1$ an... |
31,274 | <p>Many image processing libraries like <a href="http://docs.opencv.org/doc/tutorials/imgproc/imgtrans/remap/remap.html" rel="noreferrer">OpenCV</a>, <a href="https://software.intel.com/en-us/node/504403" rel="noreferrer">Intel Performance Primitives</a> or <a href="http://octave.sourceforge.net/image/function/imremap.... | ssch | 1,517 | <p>Straightforward method without interpolation, takes 0.1s but it feels like there is lots of room for improvement. I used the definition with <code>dst_pixel</code> so the orientation is different from result in question.</p>
<p>I don't know of a quick way to have <code>Extract</code> or <code>Part</code> return a d... |
1,455,736 | <p>For example, if I had a set <code>A = {2,4,6}</code> is <code>2 ⊂ A</code> true?</p>
| Belgi | 21,335 | <p>A subset is also a set, so the answer is no (although there are definitions of the number two as a set but that doesn't seem to be the case here) </p>
|
4,386,744 | <blockquote>
<p>Show that if <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are connected, then <span class="math-container">$X \times Y$</span> is also connected.</p>
</blockquote>
<p>Since <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are both... | Community | -1 | <p>The simplest way to solve such questions is to use the known results for <span class="math-container">$\sum i, \sum i^2, ...$</span>.</p>
<p>So, for your second problem you have to find <span class="math-container">$3\sum i^2-\sum i.$</span></p>
<p>In your comments you say that you wanted to use the method of differ... |
3,624,267 | <p>Let <span class="math-container">$f$</span> be an entire function which is not a polynomial, and <span class="math-container">$a \neq b$</span>. Can <span class="math-container">$f^{-1}(\{a,b\})$</span> be a finite set? </p>
| Kavi Rama Murthy | 142,385 | <p><span class="math-container">$f(z)=z$</span> has this property since the inverse image has at most two points. </p>
<p>If <span class="math-container">$f$</span> is not a polynomial then <span class="math-container">$f$</span> has an essential singularity at <span class="math-container">$\infty$</span>. By Picard's... |
856,282 | <p>Can someone tell me how to add parenthesis to this expression </p>
<blockquote>
<p>$a + b \cdot c \cdot d + e$. </p>
</blockquote>
<p>where $a,b,c,d,e$ are algebraic variables, where addition and multiplication is as usual there. :-)</p>
<p>If you have any other complicated examples please explain them to me ... | Mary Star | 80,708 | <p>The operator $"+"$ has a lower priority than the multiplication operator, so you first do the multiplication and the the sum, and the experssion in the parenthesis has high priority so you have to write in the parenthesis the expression with the multiplication operator:
$$a+(b \cdot c \cdot d) +e$$
or
$$a+((b \cdot ... |
856,282 | <p>Can someone tell me how to add parenthesis to this expression </p>
<blockquote>
<p>$a + b \cdot c \cdot d + e$. </p>
</blockquote>
<p>where $a,b,c,d,e$ are algebraic variables, where addition and multiplication is as usual there. :-)</p>
<p>If you have any other complicated examples please explain them to me ... | gnometorule | 21,386 | <p>If @mvw 's comment is correct, then, assuming LR, you'll parse this as $$((a+( ( b \cdot c) \cdot d) ) + e)$$ And a proper tag, if it existed, would be "recursive descent parsing" because this isn't really RegEx analysis. </p>
<p>If this is just standalone evaluation of simple expressions like this, you're fine us... |
335,383 | <p>Hello I am currently stuck at integration by parts. And I was wondering why at the second $u$ substitution I should choose $u=\cos 3x$ instead of the $\sin 2x$. Is there a specific rule which applies?</p>
<p><img src="https://i.stack.imgur.com/mH7JL.png" alt="enter image description here"></p>
<p>Best,
Sebastian</... | Ross Millikan | 1,827 | <p>If you take a $u$ that was $dv$ before you will get back where you started. The idea is to exploit the fact that taking two derivatives of $\cos$ get you back something that looks like the original but has a minus sign. If you do $$\int u dv=uv-\int v du$$ then integrate by parts again you get $$\int u dv=uv-\int ... |
3,981,000 | <p>I have a set of points forming a polygon. However, any 3 points in this polygon can also be represented as an arc (starting at point 1, through point 2, to point 3).</p>
<p><a href="https://i.stack.imgur.com/Tz9L6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Tz9L6.png" alt="Polygon with Arcs" /... | saulspatz | 235,128 | <p>Can't you dissect it into convex pieces as indicated below?</p>
<p><strong>EDIT</strong></p>
<p>In response to the OP's comment, if you had an inward bulge, you could connect the two extreme points, and compute the area of a convex shape minus the area of another convex shape.</p>
<p>If you are trying to write a com... |
835,111 | <blockquote>
<p>How many solutions are there for $a^{2014} +2015\cdot b! = 2014^{2015}$, with $a,b$ positive integers?</p>
</blockquote>
<p>This is another contest problem that I got from my friend.</p>
<p>Can anybody help me find the answer? Or give me a hint to solve this problem?</p>
<p>Thanks</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ If $\,n\,$ is odd, and $\ a^{n-1}\! + n b = (n-1)^{2j+1} $ then every prime $\,p\,$ dividing $\,n\,$ is $\,\equiv 1\pmod 4 $
since $\ n\!-\!1 = 2k,\,$ so $\, {\rm mod}\ p\!:\ (a^k)^2 \equiv\, -1,\ $ so by <a href="http://en.wikipedia.org/wiki/Euler%27s_criterion" rel="nofollow">Euler</a>... |
1,471,023 | <p>I'm trying to get the hang of permutations and combinations for a discrete math class. One of my questions is as follows:</p>
<p>$$\text { You have a group of 30 people, and want to divide the group into two groups of 15 people each. }$$
$$\text { How many ways can this be done? } $$</p>
<p>My intuition is tellin... | Scientifica | 164,983 | <p>A first remark: it should be multiplication by the basic principle of counting, not addition, i.e, it should be $$\dfrac{30!}{15!}\times\dfrac{30!}{15!}$$</p>
<p>Now the way you count doesn't seem correct. In a group of people, we don't care about their order there, so you're counting each outcome in each group $15... |
317,842 | <p>A particular professor is known for his arbitrary grading policies. Each paper receives a grade from the set {A, A-, B+, B, B-, C+}, with equal probability, independently of other papers. How many papers do you expect to hand in before you receive each possible grade at least once?</p>
| Louis | 21,585 | <p>This called the "coupon collectors problem". You can search for it.</p>
<p>The way to see the expectation is the following. Break up the process into a sequence of rounds that end each time you get a new paper. Let $N$ be the total number of grades. The length of the $i$th round has geometric distribution with ... |
38,629 | <p>I've asked this question on <a href="https://math.stackexchange.com/questions/4533/sobolev-space-norm-and-beppo-levi-space-norm">math.stackexchange.com</a> but I'm not satisfied by the answers I got, so I've decided to ask here instead. As always I apologize if my notation is not precise enough. I am a computer scie... | Urg | 9,242 | <p>To attempt an answer to your first question:</p>
<p>For $p=2$, the Sobolev space you've defined assigns an infinite norm to the function $f(x_1,x_2)=1$, while the Beppo-Levi SEMI-norm assigns a quantity of zero to this function (as alext87 already noted in his comment). In the way you've defined them, this means t... |
2,436,775 | <p>Question : </p>
<p>How much work is done in pulling an object constrained to move along the portion of the curve y = $x^2$; z = $x^3$ from (0; 0; 0) to (1; 1; 1) (positions in meters), if the rope pulling it is always in the direction < 1;-3;-4 > and the tension in the rope is constant at 100 Newtons?</p>
<p>At... | neonpokharkar | 477,567 | <p>You could take a more easier route</p>
<blockquote>
<p>As the magnitude and direction of the force of constant it is just like gravitational pull.</p>
</blockquote>
<p>So you could easily learn from the potential energy equation of gravity <em>$mgh$</em> </p>
<p>For your variable like <em>$h$</em> , consider th... |
1,710,244 | <p>So, I have this problem on my analysis homework. We've been looking at sequences of functions. In particular, we are looking at $x^n$ and the question asks us to show that there is a sequence of values of $x$ on $[0,1]$, call it $(x_n)$, such that $\lim_{n\to\infty} (x_{n}^n) \neq 0 = lim_{n\to\infty} x^n$ on $[0,1... | robjohn | 13,854 | <p>The sequence $x_n=1-\frac an$ gives
$$
\lim_{n\to\infty}x_n^n=e^{-a}
$$
For $a\in(0,1)$, the sequence is in $(0,1)$</p>
<hr>
<p>On the other hand, if
$$
0\le\lim_{n\to\infty}x_n=x\lt1
$$
Then for some $N$, if $n\ge N$,
$$
0\le x_n\le\frac{x+1}2=1-\frac{1-x}2
$$
and therefore,
$$
\begin{align}
\lim_{n\to\infty}x_n^... |
2,295,313 | <p>I was learning how to solve $3\sin(t+\pi )=2$. Step one was isolating, then using inverse trig func., when it came to $t+\pi$ there was written that $t+\pi =0.730 \text{ radians}$, why is that? Could someone explain?</p>
| Bobson Dugnutt | 259,085 | <p>Probably just because they didn't want to write out $\arcsin\left(\frac{2}{3}\right)\approx 0.73$ every time. It is not really good practice to do so, however, since $0.73$ is not the exact value. </p>
|
3,275,356 | <blockquote>
<p>Prove that <span class="math-container">$$\frac{9}{1!}+\frac{19}{2!}+\frac{35}{3!}+\frac{57}{4!}+\frac{85}{5!}+......=12e-5$$</span></p>
</blockquote>
<p><span class="math-container">$$
e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+......
$$</span></p>
<p>I have no clue of whe... | TurlocTheRed | 397,318 | <p>You noticed a pattern about the denominators, they are straight forward. How do you figure out what's going on with the numerators? </p>
<p>It's good to check ratios between consecutive terms if they are increasing, but that doesn't get you anywhere here. Anther good rule of thumb, take consecutive differences, per... |
2,279,937 | <p>How to prove that $$\sum_{n=1}^{15} \frac{1}{n^3}\lt\frac 65$$ </p>
<p>I tried to compare this sum to the infinite sum, but <em>Apery</em>'s constant is just above $1.2$ so this approach doesn't work.</p>
<p>Then I typed this into wolfy and the sum seems as if it is just under $6/5$. </p>
<p>However, I was wonder... | marty cohen | 13,079 | <p>This is just
a numerical coincidence,
as far as I am concerned.</p>
<p>As others have noted,
according to Wolfy,
the sum to 15 terms is
$$\dfrac{56154295334575853}{46796108014656000}
=1.199977898097614804322644896619336727642969815345853855758...
$$
and the sum to 16 terms is
$$\dfrac{449325761325072949}{374368864... |
795,469 | <p><img src="https://i.stack.imgur.com/U4ZBh.png" alt="enter image description here"></p>
<p>Can someone please explain this answer to me with a graphical representation or just a have a better explanation than this?
I would really appreciate it, please,
Thanks </p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\,\ {\rm mod}\ 101\!:\,\ \color{#c00}{x^2\equiv 10}\:\Rightarrow\, 1\! \overset{\rm Fermat}\equiv\! (\color{#c00}{x^2})^{50}\equiv \color{#c00}{10}^{50}\equiv (10^2)^{25}\equiv (-1)^{25}\equiv -1\,\Rightarrow\!\Leftarrow$</p>
<p><strong>Remark</strong> $\ $ i.e. $\,{\rm mod}\ p\!:\ x^4 \equiv... |
3,584,718 | <p>My uni is closed because of the pandemic and I'm home learning calculus. There is one problem I am really not sure how is supposed to be solved:</p>
<p><span class="math-container">$$\log(x)+\log(\sqrt[3]{x})+\log(\sqrt[9]{x})+\log(\sqrt[27]{x})+\ldots=6$$</span></p>
<p>I know that I am supposed to show my attempt... | Marine Galantin | 497,474 | <p><strong>Hint :</strong></p>
<p><span class="math-container">$$log (x)+log(\sqrt[3]{x})+log(\sqrt[9]{x})+log(\sqrt[27]{x})+...= log (x)+ \frac 1 3 log(x)+ \frac 1 9log(x)+ \frac 1 {27} log(x)+... $$</span> </p>
<hr>
<p><strong>Full answer :</strong> </p>
<p>Let's massage the expression. This equality is given by ... |
103,133 | <p>I am trying to send message to my LinkedIn contact using Mathematica 10.3, but there is no enough information (documented) on how to do that. I am able to login in LinkedIn and see my user data with:</p>
<pre><code>lk = ServiceConnect["LinkedIn"] (* works fine *)
ud=ServiceExecute[lk, "UserData"]; (* works fine... | dsingh | 26,603 | <p>Currently Mathematica supports two LinkedIn requests.<br />
1."UserData"
2. "Share".</p>
<p>Earlier more requests used to be supported but change in linkedin policies caused others to take back seat. Also currently it is not possible to share a message to other members in your linkedin network.</... |
988,028 | <p><em>Disclaimer: This is just meant as record of a proof. For more details see: <a href="http://blog.stackoverflow.com/2011/07/its-ok-to-ask-and-answer-your-own-questions/">Answer own Question</a></em></p>
<hr>
<p>How to prove that the Lebesgue measure has no atoms:
$$\lambda:\mathbb{R}^n\to\mathbb{R}_+$$
<em>(Reca... | C-star-W-star | 79,762 | <p>Assuming, the Lebesgue measure has an atom $A\in\mathcal{B}(\mathbb{R}^n)$.</p>
<p>First consider growing boxes:
$$Q_k:=[-k,k]^n$$
By definition one of the following always applies:
$$\lambda(A\cap Q_k)=0\lor\lambda(A\cap Q_k^c)=0$$
If always the former case applies then there's a contradiction as $0=\mu(A\cap Q_k)... |
599,651 | <p>I'm hoping for link to some resource which can explain why the following is true.</p>
<p>$$ x^2 + 104x - 896 = 0 $$</p>
<p>Using the quadratic formula we pull a = 1, b = 104, c = 896. Putting that into the formula for the discriminant we get $ 104^2 - 4.1.896 $ which is 10816 - 3584 = 7232.</p>
<p>Using the quadr... | zrbecker | 19,536 | <p>The quadratic formula is $$x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}.$$
Thus in your problem, $$x = \frac{-104\pm\sqrt{104^2 - 4(-896)}}{2}
=\frac{-104 \pm\sqrt{14400}}{2} = \frac{-104 \pm 120}{2}.$$
This gives $x = 8$ or $-112$.</p>
<p>It seems the problem in your computation was you forgot the negative on $896$ when you... |
3,203,090 | <p><a href="https://i.stack.imgur.com/RdEok.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RdEok.png" alt="enter image description here"></a></p>
<p>Could someone help me? I'm not sure how to start</p>
<p>My professor also gave me this hint: </p>
<p><a href="https://i.stack.imgur.com/91Oy2.png" r... | Graham Kemp | 135,106 | <p><span class="math-container">$A$</span> will win the game if:</p>
<ul>
<li><span class="math-container">$A$</span> wins on the first throw (with probability of <span class="math-container">$c_1$</span>), or</li>
<li>All three players lose on their first throw (with probability of <span class="math-container">$c_2$<... |
1,606,727 | <p>If you have a set of $N$ items, how many subsets can you make?
For example, for the set existing of $3$ items (Item1, Item2 and Item3) the subsets are Item1, Item2, Item3, Item2+3, Item1+2, Item1+3 and Item1+2+3. Is there a general formula for this? </p>
<p>I think it might be $2^N-1$, but I'm not sure. </p>
<p>T... | Archis Welankar | 275,884 | <p>You are right. Take it like this. You have $n$ items of course. You can't select $0$ items so now we have to choose $$(1,2,...n)$$ elements so we have $$\sum_{k = 0}^n {n\choose k}$$ Now we know $$\sum_{k = 0}^n {n\choose k}=2^n$$ but as stated above we can't choose $0$ elements so it becomes $$\sum_{k = 1}^{n} {n\... |
753,997 | <p>Someone told me that math has a lot of contradictions. </p>
<p>He said that a lot of things are not well defined.</p>
<p>He told me two things that I do not know.</p>
<ul>
<li>$1+2+3+4+...=-1/12$</li>
<li>what is infinity $\infty$?</li>
</ul>
<p>Since I am not a math specialist and little. How to disprove the pr... | ex0du5 | 31,155 | <p>Mathematics is <em>obviously</em> contradictory. Predicativist systems disagree with classical systems on the allowable scope of quantification in definitions. Constructivists disagree on the use of excluded middle. Ultrafinitists disagree on induction principles.</p>
<p>Not only that, there are mathematical sys... |
1,773,543 | <p>I am trying to understand whether or not the statement $X′ ∩ Y′ ⊆ (X ∩ Y)′$ is true or false.</p>
<p>In trying to develop a counterexample I started asking myself what sets might make it false. That led to the following question. </p>
<p>If $(A ∩ B)$ = Empty Set, what is $(A ∩ B)′$ ?</p>
| fleablood | 280,126 | <p>A counter example to make the statement untrue would be X = the irrationals and Y= the rationals.</p>
<p>Then X' = Y' = the reals and $X' \cap Y' = \mathbb R $. But $X \cap Y =\emptyset $.</p>
<p>So what is $\emptyset'$? Well, every neighborhood of a limit point of the empty set must have a point of the empty se... |
890,770 | <p>In a proof I've come across the following identity:</p>
<blockquote>
<p>$$ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $$</p>
</blockquote>
<p>I see that it's right, when plugging in numbers, but I don't see the algebraic or combinatorial proof behind it. Can anyone help me with that?</... | Quang Hoang | 91,708 | <p>Of course one only needs to consider only $M\le j$. Rewrite the identity as</p>
<p>$$\sum^{n-j}_{l=0}\binom{M-1+l}{M-1}\binom{n-M-l}{j-M}=\binom{n}{j}.$$</p>
<p>The RHS suggests that one should look at how many sequence $a_1<a_2<\cdots <a_j$ can be chosen from $\{1,2,\cdots,n\}$ differently.</p>
<p>$a_M$... |
890,770 | <p>In a proof I've come across the following identity:</p>
<blockquote>
<p>$$ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $$</p>
</blockquote>
<p>I see that it's right, when plugging in numbers, but I don't see the algebraic or combinatorial proof behind it. Can anyone help me with that?</... | Marko Riedel | 44,883 | <p>A similar technique to what has already been presented uses basic complex variables.
<P></p>
<p>Suppose we seek to evaluate
$$\sum_{q=0}^{n-j}
{q+M-1\choose M-1} {n-M-q\choose n-j-q}$$
where $M\le j.$</p>
<p>Start from
$${n-M-q\choose n-j-q}
= \frac{1}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{n-j-q+1}} (1+z)^{n-M-... |
107,546 | <p>My matrix is</p>
<p>$\qquad A= \begin{pmatrix}
{1} & {2} & {3}\\
{4} & {1} & {0}\\
{0} & {5} & {4}
\end{pmatrix} $</p>
<p>I need </p>
<p>$\qquad A^n$</p>
<p>I tried</p>
<pre><code>MatrixPower[A, n]
</code></pre>
<p>I do not understand the result. Should it not go according to <code>... | m_goldberg | 3,066 | <p><em>Mathematica</em> has given you what you asked for. You can see that this is true by looking at the values it takes for values of <code>n = 0, 1, 2, 3</code>.</p>
<pre><code>a = {{1, 2, 3}, {4, 1, 0}, {0, 5, 4}};
b[n_Integer] := Chop[N[MatrixPower[a, n]]]
Column[Table[MatrixForm[b[i]], {i, 0, 3}]]
</code></pre>
... |
160,323 | <p>I am working on a subject of geometric group theory closely related to 3-manifolds, and in order to understand these links, I am seeking a good reference about 3-manifolds, as self-contained as possible, and in particular dealing with: loop and sphere theorems, Heegaard diagrams, Haken manifolds.</p>
<p>I browsed H... | yanqing | 18,496 | <p>Jennifer Schultens has written a notes on 3-manifolds,
see <a href="https://www.math.ucdavis.edu/~jcs/pubs/notes.pdf" rel="noreferrer">https://www.math.ucdavis.edu/~jcs/pubs/notes.pdf</a>.</p>
|
742,093 | <p>Let's define the sequence $a_n$, $n \geq 0$ by making $a_0 = 0$ and $a_{n+1} = 2a_n + n$ for $n \geq 0$. Show that if $F(x) = \sum_{n=0}^{\infty} a_nx^n$ is the generating function of the sequence, then $$F(x) = \frac{x^2}{(1-x)^2(1-2x)}$$</p>
| Imanol Pérez Arribas | 127,622 | <p>$F(x) = \displaystyle{\sum_{n=1}^\infty a_n x^n} = \displaystyle{\sum_{n=1}^\infty (2a_n + n) x^n} = 2\displaystyle{\sum_{n=1}^\infty a_n x^n} + \displaystyle{\sum_{n=1}^\infty n x^n} = 2xF(x) + \displaystyle{\sum_{n=1}^\infty n x^n}$</p>
<p>Now, we know that $\dfrac{1}{1-x} = \displaystyle{\sum_{n=0}^\infty x^n}$,... |
28,361 | <p>My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be compact (with respect to the $p$-adic topology)?</p>
<p>I more or less understand that if $G=SL_1(D)$ where $D$ is a ... | Kevin Buzzard | 1,384 | <p>I'm no expert, but I think the theorem is that (for $G$ reductive over a local field $F$) $G(F)$ is compact iff $G$ is $F$-anisotropic, that is, every $F$-torus in $G$ (or equivalently every maximal $F$-torus) has the property that its only $F$-character is the trivial character.
This will be somewhere in Platonov-R... |
3,256,475 | <p>I'm sorry that I don't know how to upload the context, I'll just write the example:</p>
<blockquote>
<p>Example 7. The product of two quotient maps need not be a quotient map.</p>
<p>Proof: Let <span class="math-container">$X=\mathbb{R}$</span> and <span class="math-container">$\mathbb Q$</span> is set of rational n... | Lê Thành Đạt | 654,749 | <p>Not suprisingly,</p>
<p><span class="math-container">$$x^2y^2 - 4x^2y + y^3 + 4x^2 - 3y^2 + 1 = 0$$</span></p>
<p><span class="math-container">$$\iff (x^2y^2 + y^3 + y^2) - (4x^2y + 4y^2 + 4y) + (4x^2 + 4y + 4) = 3$$</span></p>
<p><span class="math-container">$$\iff y^2(x^2 + y + 1) - 4y(x^2 + y + 1) + 4(x^2 + y ... |
1,914 | <p><a href="https://matheducators.stackexchange.com/a/1528/42">This answer</a> describes an analogy between finite state machines and mazes. This allows for some playful exercises, like </p>
<blockquote>
<p>Draw a representation of a word accepted by the following automaton...</p>
</blockquote>
<p>which sums up to ... | Joe Z. | 856 | <p>Try this special variation on the game of Nim:</p>
<p>Put a pile of 42 stones on the table (although if you don't have 42, any multiple of six will do). Explain to your student that each turn, each of you will remove either 1, 3, or 5 stones, and the person who removes the last stone from a pile wins. Then, explain... |
1,249,726 | <p>I am wondering whether it is possible to construct an experiment, where the probability of occurrence of an event comes out to be an irrational number. </p>
| DRF | 176,997 | <p>This very much depends on whether or not you want a real life experiment and its evaluation in which case this is a physics question and depends a lot on what theory you exactly subscribe to. To me it seems that due to the fact that it seems that space is discretized it should not be possible.</p>
<p>On the other h... |
1,631 | <p>Using <code>MouseAppearance</code> one can change the cursor image when passing over an expression.</p>
<p>Is it possible to change the cursor image for the entire notebook front end (not just one expression within it)?</p>
| acl | 16 | <p>Certainly. For instance, here's how to reduce the number of colours to 10 (randomly chosen in RGB space):</p>
<pre><code>i = Import["ExampleData/lena.tif"]
</code></pre>
<p><img src="https://i.stack.imgur.com/4uKUl.png" alt="Mathematica graphics"></p>
<p>You can try <code>ImageData[i]</code> to see the actual RGB... |
302,519 | <p>My teacher gave us an homework. I solved it, but I don't think I have the right answer.</p>
<p><strong>PROBLEM</strong></p>
<p>We have three coins identical in appearance.</p>
<ul>
<li>Coin A falls on tails and heads with equal probability</li>
<li>Coin B falls twice as much on tails as heads</li>
<li>Coin C alwa... | broccoli | 50,577 | <p>The key point to consider here is that you now have new evidence that favours a certain hypothesis, and less others.
<a href="http://bayesianthink.blogspot.com/2012/12/the-magicians-coin-tossing-experiment.html" rel="nofollow">Here</a> is a write up on dealing with exactly such scenarios.</p>
|
4,169,953 | <p>Suppose we have a function <span class="math-container">$f$</span> on the real line and <span class="math-container">$f > 0$</span> on some interval <span class="math-container">$[a,b)$</span> and <span class="math-container">$f(b) = 0$</span>. Assume also that <span class="math-container">$f'<0$</span> on <sp... | Community | -1 | <p><span class="math-container">$$f(b)=0\land \forall x>b:f'(x)>0$$</span> gives by integration</p>
<p><span class="math-container">$$f(c)=f(b)+\int_b^c f'(x)dx>f(b)>0.$$</span></p>
|
1,781,957 | <p>How would you go about expressing the following as a limit?
$$\int_0^1 \ln(x) dx$$
I know how to express limits on simple equations, but have no clue how to go about expressing an integral as a limit.</p>
<p>Do I need to expand it as if I was performing a normal integration?</p>
| User8128 | 307,205 | <p>Since Riemann integration is only defined for bounded functions, technically the integral $$\int_0^1 \ln(x) dx$$ is not properly defined from the outset; we need to come up with a new definition for it. We see that $\ln(x)$ is bounded on any interval of the form $[\epsilon, 1]$ for $0 < \epsilon < 1$ so $\int_... |
1,781,957 | <p>How would you go about expressing the following as a limit?
$$\int_0^1 \ln(x) dx$$
I know how to express limits on simple equations, but have no clue how to go about expressing an integral as a limit.</p>
<p>Do I need to expand it as if I was performing a normal integration?</p>
| RRL | 148,510 | <p>This integral with unbounded integrand is improper at $x = 0,$ but it converges.</p>
<p>So</p>
<p>$$\int_0^1 \ln x \, dx = \lim_{c \to 0} \int_c^1 \ln x \, dx = \lim_{c \to 0}( x \ln x - x)|_c^1 = -1.$$</p>
<p>It also can be computed as the limit of a right-hand Riemann sum even though it is not properly Riemann ... |
2,112,161 | <p>This is a question out of "Precalculus: A Prelude to Calculus" second edition by Sheldon Axler. on page 19 problem number 54.</p>
<p>The problem is Explain why $(a−b)^2 = a^2 −b^2 $ if and only if $b = 0$ or $b = a$.</p>
<p>So I started by expanding $(a−b)^2$ to $(a−b)^2 = (a-b)(a-b) = a^2 -2ab +b^2$. To Prove tha... | dxiv | 291,201 | <p>Hint: $(a−b)^2 = a^2 −b^2 \iff (a−b)^2 - a^2 +b^2=0 \iff 2b(b-a)=0\,$.</p>
|
2,910,860 | <p>Question: [For which positive real numbers $a$ and $b$ is</p>
<p>$u(x,y) = \cosh(ax)\sin(by)$</p>
<p>harmonic? When $a$ and $b$ satisfy this condition find a holomorphic function $f(z)$ such that $\Re f = u$]</p>
<hr>
<p>I got $a=b$ for the condition so that</p>
<p>$u(x,y) = \cosh(ax)\sin(ay)$ and $v(x,y) = -\s... | Narlin | 490,058 | <p>Here is a general solution that is easy enough to understand. As others have done, let A=(0,0,0) and C=(7,0,0). From your example, $a=5$; $b=6$; $c=7$; Use the law of cosines to get angle $\alpha$ at point A in the xy plane. $a^2+c^2-2ac\,\cos(\alpha)=b^2$. Doing that, point $B=A + a\cdot(cos(\alpha), sin(\alpha),... |
57,489 | <p>If I enter <code>x/x</code>, I get <code>1</code>. Such behavior leads to this:</p>
<pre><code>Simplify[D[Sqrt[x^2], x, x]]
</code></pre>
<blockquote>
<p>0</p>
</blockquote>
<p>The same would be even if I use <code>Together</code> instead of <code>Simplify</code>.</p>
<p>One could then think that $\sqrt{x^2}$ ... | eldo | 14,254 | <pre><code>fun = Sqrt[x^2];
dd = D[fun, x, x];
</code></pre>
<p>With <strong>V10</strong> we can explicitly define:</p>
<pre><code>{dd, FunctionDomain[dd, x]}
</code></pre>
<p><img src="https://i.stack.imgur.com/rc8PA.jpg" alt="enter image description here"></p>
<p>Or</p>
<pre><code>{Simplify @ dd, FunctionDomain... |
3,629,734 | <p>Suppose <span class="math-container">$\{x_n\}$</span> is a sequence of real numbers such that <span class="math-container">$\lim_{n\to +\infty}x_n=0$</span>. If the product sequence <span class="math-container">$\{nx_n\}$</span> is bounded (<span class="math-container">$\exists M>0$</span> such that <span class="... | sirous | 346,566 | <p><a href="https://i.stack.imgur.com/vTx6E.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vTx6E.jpg" alt="enter image description here"></a></p>
<p>We assume point A and parallel lines L1 and L2 are given.We use the fact that in every triangle if we draw the diameter of circumscribed circle ,Here... |
3,145,550 | <blockquote>
<p>Prove if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are real numbers with <span class="math-container">$x \lt y$</span>, then there are infinitely many rational numbers in the interval <span class="math-container">$[x,y]$</span>.</p>
</blockquote>
<p>What I go... | TonyK | 1,508 | <p>Just find two distinct rational numbers <span class="math-container">$p,q\in[x,y]$</span>. Now the rational number <span class="math-container">$p+(q-p)/n$</span> is in <span class="math-container">$[x,y]$</span> for all <span class="math-container">$n\in \Bbb N_{>0}$</span>.</p>
<p>(And you can find such <span ... |
513,822 | <p>Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists.</p>
<p>Sorry if this seems elementary, I have just started my degree...</p>
<p>Thanks in advance.</p>
| drhab | 75,923 | <p>Let $f$ be a function taking values between $1$ and (for instance) $2$ such that $\lim_{x\rightarrow c}f\left(x\right)$ does not exist. Then $g(x)=1/f(x)$ is well defined and $\lim_{x\rightarrow c}g\left(x\right)$ does not exist. This while $f(x)g(x)=1$ for every $x$.</p>
|
513,822 | <p>Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists.</p>
<p>Sorry if this seems elementary, I have just started my degree...</p>
<p>Thanks in advance.</p>
| Peter LeFanu Lumsdaine | 2,439 | <p>Looking at the bigger picture a little, the idea behind most answers given works not just for product but also for sum, difference, quotient, exponent, and most other binary operations.</p>
<p>The common idea is that for many ways that <em>f</em> can vary, <em>g</em> can also vary in some way that cancels it out, m... |
513,822 | <p>Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists.</p>
<p>Sorry if this seems elementary, I have just started my degree...</p>
<p>Thanks in advance.</p>
| john | 266,200 | <p>I would like to reiterate a solution given above. Neither of the limits $$\lim_{x\to0}\left(\sin{\frac1x}\right), ~~~\lim_{x\to0}\left(\frac{1}{\sin{\frac1x}}\right)$$ exist, but $$\lim_{x\to0}\left(\sin{\frac1x}\right)\left(\frac{1}{\sin{\frac1x}}\right)=1.$$
I am aware that $ \large f(x) = \frac{1}{\sin{\frac1x}}$... |
4,022,658 | <p>I am interested in understanding the general construction of important subgroups in reductive groups, and how they are parametrized (Borel, Levi, parabolic, etc.). But for simplicity I take the example of <span class="math-container">$SL(3)$</span> over a field <span class="math-container">$k$</span>.</p>
<p>There a... | Andreas Cap | 202,204 | <p>I should say right away that I am only familiar with the case that <span class="math-container">$k=\mathbb C$</span>, I still hope that the answer is helpful.</p>
<p>Parabolics do not correspond to subsets of positive roots but to subsets of simple roots. There is a description in terms of <em>saturated</em> subsets... |
249,307 | <p>By an <em>arch diagram</em> of size $n$, I mean a diagram consisting of $n$ arches matching $2n$ points, where the points are ordered on a line running from left to right. An arch diagram is basically just a way of representing a fixed-point free involution in $S_{2n}$, and arch diagrams of size $n$ are counted by t... | Ilya Bogdanov | 17,581 | <p>Let me address <b>Q2</b>.</p>
<p>Paint all right endpoints in white, and all left ones in black. Number white points $1,2,\dots,n$ from left to right, and number the black points according to the numbers of white points joined to them.</p>
<p>All black points lying between two consecutive white ones can be permute... |
4,343,873 | <blockquote>
<p>If <span class="math-container">$A$</span> is a subspace of <span class="math-container">$X$</span> and <span class="math-container">$B$</span> is a subspace of <span class="math-container">$Y$</span>, then the product topology on <span class="math-container">$A\times B$</span> is the same as the topolo... | Nicolas Bourbaki | 103,943 | <p><strong>Abstract Mindset</strong></p>
<p>It is usually better to try to prove things (especially in more modern subjects of mathematics) in a more abstract way. At some point the sets become too complicated to deal with. Especially if it is sets, within sets, within sets, ect. So one should try to a more categorica... |
1,907,275 | <p>Me and a friend could need some assistance:</p>
<p><a href="https://i.stack.imgur.com/UBkDm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UBkDm.png" alt="Given function + Plot"></a></p>
<blockquote>
<p>$$x(t)=t\cos t$$
$$y(t)=t\sin t$$</p>
</blockquote>
<p>We've been given the assignment ... | Parcly Taxel | 357,390 | <p><em>HINT</em>:</p>
<ul>
<li>$\sin t=0$ if $t=0,\pm\pi,\pm2\pi,\dots$</li>
<li>$\cos t=0$ if $t=\pm\frac12\pi,\pm\frac32\pi,\pm\frac52\pi,\dots$</li>
</ul>
<p>Can you take it from here?</p>
|
1,463,457 | <p>Given a function </p>
<p>$$F(x)= \begin{cases} x^2 & \text{when }x \in \mathbb Q \\3x & \text{when }x \in\mathbb Q^c \end{cases}$$</p>
<p>Show that $F$ is continuous or not on $x=3$ with $\epsilon-\delta$.</p>
<p>I tried to deal with problems just like doing on Dirichlet functions. Mistakenly or not, I co... | Babai | 36,789 | <p>Let $\epsilon > 0$ , be a positive number </p>
<p>To find a $\delta$ such that $|x-3|<\delta$ implies $|f(x)-f(3)|<\epsilon$.</p>
<p>Case-I if $x\in \mathbb Q$</p>
<p>Then $|f(x)-f(3)|=|x^2-9|=|(x-3)(x+3)|$</p>
<p>Therefore we choose our $\delta$ to be $<1$, then $|x-3|<1$ implies $-1<x-3<1$... |
147,932 | <p>I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x) = ax^2 + bx + c$ provide the solutions when $f(x) = 0$. What does that $ y = 0$ mean for the solutions, the intercept at the $x$ axis? Why aren't the solutions at $f(x) = 403045$ or some o... | Isaac | 72 | <p>There are a few critical terms floating around in your question: roots, solutions, $x$-intercepts; I'll add <em>zeros</em> to this list. Let's start by defining these terms.</p>
<p>The <strong>roots</strong> of a <em>polynomial</em> are values of the variable that make the polynomial equal to zero.</p>
<p>The <st... |
355,095 | <p>For arbitrary formulas $A,B,C$ it holds that:</p>
<ol>
<li>$\{A,B\} \vDash C $ if $A \vDash (B \Rightarrow C)$</li>
<li>$(A \Rightarrow B) \vDash C$ if $A \vDash (B \Rightarrow C)$</li>
<li>$A \vDash C$ if $A \vDash (B \Rightarrow C)$</li>
</ol>
<p>I know that only first one holds, can someone explain me why?</p>
| Jonathan Rich | 66,596 | <p>Number 3 Should be fairly obvious:</p>
<pre><code>A entails (B implies C)
</code></pre>
<p>Which, in English, becomes:</p>
<pre><code>A entails C or A entails not B
</code></pre>
<p>Number 2 is a bit more complex. The values it fails for are $A, ¬B, ¬C$.</p>
|
355,095 | <p>For arbitrary formulas $A,B,C$ it holds that:</p>
<ol>
<li>$\{A,B\} \vDash C $ if $A \vDash (B \Rightarrow C)$</li>
<li>$(A \Rightarrow B) \vDash C$ if $A \vDash (B \Rightarrow C)$</li>
<li>$A \vDash C$ if $A \vDash (B \Rightarrow C)$</li>
</ol>
<p>I know that only first one holds, can someone explain me why?</p>
| Peter Smith | 35,151 | <p>Here's one approach:</p>
<ol>
<li><p>Note that trivially $\vDash p \to p$, so a fortiori $p \vDash p \to p$. But $p \to p \nvDash p$ (suppose $p$ is false). So we can have an instance of $A \vDash B \to C$ without the corresponding $A \to B \vDash C$.</p></li>
<li><p>Note that trivially $\vDash q \to q$, so a for... |
26,723 | <p>I have a large table (7000 rows × 17 columns) of terse textual data.
In many of the columns, empty entries have been replaced with "." as a marker.
Working from the top of the columns downward, I want to replace each successive "." with the data value (non-".") from above until the next data value is found; at which... | Michael E2 | 4,999 | <p>Row by row:</p>
<pre><code>Clear[datum];
SeedRandom[1];
datum[] := RandomInteger[{0, 9}];
tab = {RandomInteger[{0, 9}, 5]} ~Join~ RandomChoice[{8, 1} -> {".", x}, {9, 5}] /. x :> datum[]
filldown@tab
</code></pre>
<blockquote>
<pre><code>{{ 1, 4, 0, 7, 0},
{".", ".", ".", 8, "."},
{".", ".", "."... |
26,723 | <p>I have a large table (7000 rows × 17 columns) of terse textual data.
In many of the columns, empty entries have been replaced with "." as a marker.
Working from the top of the columns downward, I want to replace each successive "." with the data value (non-".") from above until the next data value is found; at which... | Kuba | 5,478 | <p><code>Map</code> approach:</p>
<pre><code>g["."] := last; g[x_] := x;
Map[(last = g[#]) &, Transpose[data], {2}]\[Transpose]
</code></pre>
<h2>Latest ideas, quite important remarks (in my opinion)</h2>
<p>I have asked Keith about gaps in first row. ...and no one have paid atenttion (me too). It is crucial,... |
26,723 | <p>I have a large table (7000 rows × 17 columns) of terse textual data.
In many of the columns, empty entries have been replaced with "." as a marker.
Working from the top of the columns downward, I want to replace each successive "." with the data value (non-".") from above until the next data value is found; at which... | Mr.Wizard | 121 | <p>This question is arguably a duplicate of: <a href="https://mathematica.stackexchange.com/q/2952/121">Fill out blanks with a upcoming number in a list?</a><br>
There the filling is requested in reverse and for a single list, but the principle is the same.</p>
<p>Simon Woods already posted what is IMHO the most natu... |
56,322 | <p><strong>Bug introduced in 9.0 or earlier and persisting through 12.0</strong></p>
<hr>
<p>While answering <a href="https://mathematica.stackexchange.com/q/56155/121">Change the inner color of markers</a> I came across a behavior that may be a bug.<br>
I would like to know if there is another explanation.</p>
<p>F... | MinHsuan Peng | 1,376 | <p>A workaround to question 2 is to increase legend marker size:</p>
<p><code>LineLegend[Automatic, LegendMarkerSize -> {40, 20}]</code></p>
|
56,322 | <p><strong>Bug introduced in 9.0 or earlier and persisting through 12.0</strong></p>
<hr>
<p>While answering <a href="https://mathematica.stackexchange.com/q/56155/121">Change the inner color of markers</a> I came across a behavior that may be a bug.<br>
I would like to know if there is another explanation.</p>
<p>F... | Alexey Popkov | 280 | <p>One workaround is to use <code>GeometricTransformation</code> instead of <code>Rotate</code>. By specifying coordinates in the <code>Offset</code> form we can achieve consistent size of markers on the plot and in the legend. The cropping of the markers in the legend can be avoided by increasing their sizes as <a hre... |
3,151,429 | <p><span class="math-container">$$
\sqrt{-21+20i}.\sqrt{-21-20i}=\pm(2+5i).\pm(2-5i)=\pm29
$$</span>
But why it is not
<span class="math-container">$$
\sqrt{-21+20i}.\sqrt{-21-20i}=\sqrt{(-21+20i)(-21-20i)}=\sqrt{|z|^2}\\
=\sqrt{441+400}=\sqrt{841}=29
$$</span></p>
<p>Does this has something to do with <span class="ma... | Vlad Z | 287,055 | <p>By definition, <span class="math-container">$\sqrt(x) = y$</span> means that <span class="math-container">$y^2 = x$</span>, so both <span class="math-container">$y$</span> and <span class="math-container">$-y$</span> would work. However, in case of real numbers, square root is conventionally defined as a non-negativ... |
528,068 | <p>This problem has two parts:
$a)$ Let $k>0$, find the minimum of the function $f(x,y)=x+y$ over the set S=$\{(x,y) \in \mathbb R^2_{> 0}:xy=k\}$. $b)$ Prove that for every $(x,y) \in \mathbb R^2_{> 0}$ the inequality $\frac{x+y} {2}\geq \sqrt{xy}$ holds.$$$$
I want to find the minimum of $f$ restricted to t... | Brian M. Scott | 12,042 | <p>For (b), suppose that $x,y\in\Bbb R_{>0}$, and let $k=xy$; from (a) you know that $x+y\ge 2\sqrt{k}$ and hence that $\frac{x+y}2\ge\sqrt{xy}$.</p>
<p>For (a) you don’t really need anything as fancy as Lagrange multipliers. You have $y=\frac{k}x$, so $x+y=x+\frac{k}x$; its derivative with respect to $x$ is $1-\fr... |
3,572,777 | <p>Are there any 2 x 2 matrices A and B that contain real numbers such that <span class="math-container">$(AB+I_2)^2 = I_2 $</span> <span class="math-container">$ $</span> and that <span class="math-container">$ AB\neq O_2$</span> ?</p>
<p>( There comes another question after it, which is: )</p>
<p>Are there any 3 x ... | Rybin Dmitry | 606,842 | <p>Hint:</p>
<p>Try to find <span class="math-container">$X$</span> of size <span class="math-container">$2\times 2$</span> or <span class="math-container">$3\times 3$</span> not equal to identity matrix such that <span class="math-container">$X^2 = I_2$</span> or <span class="math-container">$X^{3} = I_3$</span>. (si... |
909,690 | <p><img src="https://i.stack.imgur.com/Z8Hq1.png" alt="enter image description here"></p>
<p>Part (a) I am familiar with:</p>
<p>(a) P(batch is rejected) = P(X greater than or equal to 3)</p>
<p>and n = 15 and p(defective) = 0.1 </p>
<p>This gives me the correct answer of 0.1841</p>
<p>I am stuck at part 2! I have... | Mary Star | 80,708 | <p>A point on the $y-$axis is of the form $(0,y)$.</p>
<p>The distance between $(0,y)$ and $(5,-5)$ is:
$$\sqrt{(5-0)^2+(-5-y)^2}=\sqrt{25+(5+y)^2}$$</p>
<p>The distance between $(0,y)$ and $(1,1)$ is:
$$\sqrt{(1-0)^2+(1-y)^2}=\sqrt{1+(1-y)^2}$$</p>
<p>The two distances are equal, so also their squares:</p>
<p>$$25... |
3,108,432 | <p>I was practicing limit and I came across this question: </p>
<blockquote>
<p><span class="math-container">$$
\lim_{x \to 0} \frac{|x|^{1/2} \cos(\pi^{1/x^2})}{2 + (x^2 + 3)^{1/2}}
$$</span></p>
</blockquote>
<p>I reasoned the absolute value of <span class="math-container">$x$</span> goes to zero, and the denomi... | Robert Z | 299,698 | <p>Hint. Note that since <span class="math-container">$-1\leq\cos(t)\leq 1$</span> for all <span class="math-container">$t\in\mathbb{R}$</span>, it follows that
<span class="math-container">$$-\frac{|x|^{1/2} }{2 + (x^2 + 3)^{1/2}}\leq \frac{|x|^{1/2} \cos(\pi^{1/x^2})}{2 + (x^2 + 3)^{1/2}}\leq \frac{|x|^{1/2} }{2 + (... |
1,588,361 | <p>I know if a matrix has a left and right inverse then the inverses are the same and are (is) unique and the original matrix is a square matrix, thus if I have a matrix which has multiple left inverses for example then it has no right inverse and is a non-square matrix. But if a matrix has a unique left inverse then ... | math635 | 161,489 | <p>I think this is true. Let $f:V\rightarrow W$ be a linear transformation with a left inverse, i.e., it's injective. If $f$ is not surjective then there are always multiple ways to define a left inverse.</p>
<p>To be more precise, let $\{a_i: i\in I\}$ be a basis of $V$. Then $\{f(a_i): i\in I\}$ is linearly independ... |
2,947,806 | <p>Clearly it isn't, a quick sketch would show it, but I need an analytical proof. The obvious suggestion would be to view the limit on both sides and try to find a inconsistency but how does that work exactly in this case.</p>
<p>Evaluating from both sides the definition of differentiability yields already an intuiti... | symplectomorphic | 23,611 | <p>If I were grading your solution for absolute "rigor," I'd wonder how you'd prove that the one-sided limits are negative and positive, respectively. As it is, you simply assert the difference in sign, but you do not defend those assertions.</p>
<p>To make that analysis explicit, you could either use L'Hopital's rule... |
415,573 | <p>Show that there holds $$\sqrt{1+x} < 1+(x/2)$$ for all $x > 0$ .</p>
<p>I need guidance in doing this question. Can anyone help please? I'll be thankful!</p>
| Pedro | 23,350 | <p>You should use rather something related to Rolle's theorem, which is Lagrange's theorem. A consequence of this is that if $f$ is differentiable and $f'>0$ over some interval $I$; then $f$ is increasing on $I$.</p>
<p>Let $$g(x)=1+\frac x 2 -\sqrt{1+x}$$</p>
<p>Note $g(0)=0$ and $g'(x)=\dfrac 1 2 -\dfrac 1 2\dfr... |
116,727 | <p>I was thinking a bit about isometric embeddings into Hilbert spaces and got the following idea.</p>
<p>First, as we recall, many vector spaces over the reals are isomorphic to $\mathbb{R}^{\alpha}$ for some cardinal number $\alpha$. [EDIT: As Carl Mummert remarked below, not every vector space is of this form, as I... | Michael Hardy | 11,667 | <p>If the rationals were an open set, then each rational would be in some open interval containing only rationals. Therefore <span class="math-container">$\mathbb{Q}$</span> is not open.</p>
<p>If <span class="math-container">$\mathbb{Q}$</span> were closed, then its complement would be open. Then each irrational nu... |
1,399,402 | <p>Let $f:[0,\infty [\to[0,\infty [$ continuous such that $$\lim_{t\to\infty }\frac{f(t)}{t}=\ell\in[0,1).$$</p>
<p>Prove that $f$ has a fixed point, i.e. there is an $x\geq 0$ such that $f(x)=x$. </p>
<p>I don't really know how to solve this problem. My first intension was to use Brouwer, but it's only useable on a ... | Showhat | 231,518 | <p>Consider the function $g(t)=f(t)-t$. Since $f$ is nonnegative, we must have $g(0) \ge 0$. If $g(0)=0$, we are done. Otherwise, we have $\lim_{t \to \infty} \frac{g(t)}{t}=\ell-1<0,$
so $g(t)<0$ for sufficiently large $t$. It immediately follows by the Intermediate Value Theorem that $g$ has a positive root.</p... |
3,163,005 | <blockquote>
<p>If <span class="math-container">$a,b,c\in\mathbb{R}$</span> and <span class="math-container">$2a+b+3c=20.$</span> Then minimum value of <span class="math-container">$a^2+4b^2+c^2$</span> is </p>
</blockquote>
<p>what i try</p>
<p>Cauchy schwarz inequality</p>
<p><span class="math-container">$$(a^2+... | 5xum | 112,884 | <p>An alternative way of solving the problem is using <a href="https://en.wikipedia.org/wiki/Lagrange_multiplier" rel="nofollow noreferrer">Lagrange multipliers</a>, other than that (i.e. without using the CS inequality), I don't see a simple solution, and neither do I see a reason to look for such a solution.</p>
|
3,163,005 | <blockquote>
<p>If <span class="math-container">$a,b,c\in\mathbb{R}$</span> and <span class="math-container">$2a+b+3c=20.$</span> Then minimum value of <span class="math-container">$a^2+4b^2+c^2$</span> is </p>
</blockquote>
<p>what i try</p>
<p>Cauchy schwarz inequality</p>
<p><span class="math-container">$$(a^2+... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$(a,b,c)=\left(\frac{160}{53},\frac{20}{53},\frac{240}{53}\right).$</span></p>
<p>Thus, we get a value <span class="math-container">$\frac{1600}{53}.$</span></p>
<p>We'll prove that it's a minimal value.</p>
<p>Indeed, we need to prove that
<span class="math-container">$$a^2+4b^2+... |
24,569 | <p>I am teaching a combinatorics class in which I introduced the notion of a "mass formula". My terminology is inspired by the <a href="http://en.wikipedia.org/wiki/Smith%25E2%2580%2593Minkowski%25E2%2580%2593Siegel_mass_formula" rel="nofollow">Smith–Minkowski–Siegel mass formula</a> for the total mass of positive-def... | Pete L. Clark | 1,149 | <p>I do call such things "mass formulas", but then again I am a number theorist, and one of my colleagues is a quadratic form theorist who specializes in such things. So this is mostly an expression of my specific mathematical culture. </p>
<p>I do not think that it is a standard term, at least not the only standard... |
56,373 | <p>ellipse described about the circle in which a regular pentagon is constructed mapped on an ellipse<img src="https://i.stack.imgur.com/aF9gG.jpg" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/7pK7f.jpg" alt="enter image description here"></p>
<p>The surface can be calculated from my... | André Nicolas | 6,312 | <p>The above is exactly the method suggested in one of the answers to your <a href="https://math.stackexchange.com/questions/55986/how-to-partition-area-of-an-ellipse-into-odd-number-of-regions">earlier question</a> about dividing an ellipse into an odd number of parts. But the picture certainly improves the presentat... |
172,871 | <p>I have another problem that I can't figure out.</p>
<p>A triangle har 2 of its summits in points (7.788,0,1.95), (0,7.788,1.95), and the last one on the curve with all the points (7.788,7.788,a^2+1.95), a is a real number. Calculate the area f(a) of the triangle as a function of a and calculate where it takes its m... | Michael E2 | 4,999 | <p>Perhaps the great (!?) thing about recent advances in <em>Mathematica</em> is that you can solve such problems without having hardly to worry about the mathematics at all. Just <a href="http://reference.wolfram.com/language/ref/Join.html" rel="nofollow noreferrer"><code>Join</code></a> the given point to two points ... |
3,074,454 | <p>Find the following limit</p>
<p><span class="math-container">$$\lim_{x \to 0}\frac{[x]}{x}$$</span></p>
<p>Where <span class="math-container">$[x]$</span> is the greatest integer function.</p>
<p>I tried using the squeeze theorem on this but couldn’t come up with the appropriate functions, can someone help me out... | Harsh Wasnik | 455,835 | <p>if the GIF function contains a value that tends to infinity then the gif function can be removed.
hence [1/x] in the second expression will simply become 1/x.
now if the GIF contains some other expression then we must understand that the limit is applied on value given by GIF and not on the expression in the GIF.
Ri... |
3,074,454 | <p>Find the following limit</p>
<p><span class="math-container">$$\lim_{x \to 0}\frac{[x]}{x}$$</span></p>
<p>Where <span class="math-container">$[x]$</span> is the greatest integer function.</p>
<p>I tried using the squeeze theorem on this but couldn’t come up with the appropriate functions, can someone help me out... | Mostafa Ayaz | 518,023 | <p>Defining <span class="math-container">$u={1\over x}$</span> we must obtain two limits<span class="math-container">$$\lim_{u\to \infty}{\lfloor u\rfloor\over u}$$</span>and <span class="math-container">$$\lim_{u\to -\infty}{\lfloor u\rfloor\over u}$$</span>for <span class="math-container">$u\to \infty$</span> we can ... |
3,375,280 | <p>Let A and B two bonded sets from <span class="math-container">$R$</span> i want to prove that the set <span class="math-container">$AB=\{ab,a\in A,b\in B\}$</span> is bounded </p>
<p>Let <span class="math-container">$c\in AB$</span> then there exists <span class="math-container">$a\in A$</span> and <span class="mat... | Calvin Lin | 54,563 | <p>Your claim is not true in general. For example, consider <span class="math-container">$ A = \{ 1, -1 \}, B = \{ 2, 1 \}$</span>.<br>
We have <span class="math-container">$\inf (A) \times \inf (B) = -1 \times 1 = -1 > -2 \in AB $</span>.</p>
<p>If both of the sets are positive, then your claim is true. Can you pr... |
4,331,390 | <p>Let <span class="math-container">$f_a(x) = axa^{-1}$</span> be an inner automorphism of G, where <span class="math-container">$a,x$</span> <span class="math-container">$ \in$</span> G. What is <span class="math-container">$f_a^{-1}$</span>?</p>
<p>Method 1.<br />
<span class="math-container">$f_a^{-1}(x) = (axa^{-1}... | Andrea B. | 686,323 | <p>In the first method, you are doing <span class="math-container">$(f_a(x))^{-1}$</span>; that is, you are inverting the output of a homomorphism. By homomorphism properties, this is also the same as <span class="math-container">$f_a(x^{-1})$</span>.</p>
<p>In the second method, you are actually solving for the invers... |
690,209 | <p>I have a question that says this:</p>
<blockquote>
<p>Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.</p>
</blockquote>
<p>I would like to see how it is correctly answered. This is not homework; I'd... | i. m. soloveichik | 32,940 | <p>The Smith Normal Form of the matrix $\begin{pmatrix} 5&0&0\\ 0&4&0\\0&0&8\\1&1&1\\\end{pmatrix}$ (which gives a presentation for the group) has diagonal entries 1, 1, 4;
thus the group is isomorphic to $Z_4$.</p>
|
3,567,076 | <p>I know that to determine the radius of convergence of the series
<span class="math-container">$$ \sum_{n=0}^\infty a_nx^n $$</span>
I need to find
<span class="math-container">$$ \lim_{k\rightarrow \infty} \left| \frac{a_{k+1}}{a_k} \right| = c$$</span>
Then the radius of convergence <span class="math-container">$R$... | herb steinberg | 501,262 | <p>For your problem, let <span class="math-container">$y=x^B$</span>, so the series becomes <span class="math-container">$a_ny^n$</span> and the ratio test gives the radius of convergence for <span class="math-container">$y$</span> as <span class="math-container">$\frac{1}{c}$</span>, so the radius of convergence for <... |
3,567,076 | <p>I know that to determine the radius of convergence of the series
<span class="math-container">$$ \sum_{n=0}^\infty a_nx^n $$</span>
I need to find
<span class="math-container">$$ \lim_{k\rightarrow \infty} \left| \frac{a_{k+1}}{a_k} \right| = c$$</span>
Then the radius of convergence <span class="math-container">$R$... | vonbrand | 43,946 | <p>Consider the series:</p>
<p><span class="math-container">$\begin{equation*}
\sum_{n \ge 0} a_n x^{B n}
\end{equation*}$</span></p>
<p>From the respective theory, you know that for the series:</p>
<p><span class="math-container">$\begin{equation*}
\sum_{n \ge 0} a_n y^n
\end{equation*}$</span></p>
<p>there is a <... |
3,212,371 | <p>Let <span class="math-container">$(X,d)$</span> be a metric space and let <span class="math-container">$x_n$</span> be a sequence in <span class="math-container">$X$</span>. Let <span class="math-container">$x\in X$</span>. Define a sequence <span class="math-container">$\{y_n\}$</span> by <span class="math-contain... | E.H.E | 187,799 | <p>Hint:
<span class="math-container">$$\frac{n^n}{(1+n)^n}=\frac{1}{(1+\frac{1}{n})^n}$$</span></p>
|
1,708,090 | <p>Suppose we have the original Riemann sum with no removed partitions, where $f(x)$ is continuous and reimmen integratable on the closed interval $[a,b]$.
$$\lim_{n\to\infty}\sum_{i=1}^{n}f\left(a+\left(\frac{b-a}{n}\right)i\right)\left(\frac{b-a}{n}\right)$$</p>
<p>If we remove $s$ partitions for every $d$ partition... | ibnAbu | 334,224 | <p>For Riemann integrable function $f$ ,the proof hinges on three facts:</p>
<ol>
<li><p>The theorem is valid for step function:
Because removing any fractional number of partitions from a number of partitions of equal size results in the value of the integral being reduced by the same fraction as the size of the... |
602,143 | <p>Find the area bounded by the curves $y=e^x$, $y=xe^x$, and $x=0$</p>
<p>I know how to solve the integral for this, but I'm getting hung up trying to find the points of intersection for the two equations.</p>
<p>So I did what I normally do for these problems:</p>
<p>$$e^x = xe^x$$</p>
<p>I then did this (and this... | Stefan4024 | 67,746 | <p>Check two cases.</p>
<p>First let $e^x \neq 0$. This implies $x \neq 1$. Divide both sides by $e^x$ and you'll get $x=1$. A contradiction.</p>
<p>Now check the case when $e^x = 0$. This case provide one solution, that's $x=1$</p>
<p>Also you can continue where you've stopped:</p>
<p>$$\ln e^x = \ln xe^x$$
$$x = ... |
4,377,170 | <p>I have the following system:</p>
<p><span class="math-container">$$
(a + tb)\cos \theta = x \\
(b + ta) \sin \theta = y \\
$$</span></p>
<p>with the constraints</p>
<p><span class="math-container">$$a > b$$</span> <span class="math-container">$$\theta \in (-\pi, \pi]$$</span> <span class="math-container">$$t >... | Vasili | 469,083 | <p>Hint: write <span class="math-container">$$
\cos \theta = \frac{x}{a+tb} \\
\sin \theta = \frac{y}{b + ta} \\
$$</span>
Then solve <span class="math-container">$(\frac{x}{a+tb})^2+(\frac{y}{b + ta})^2=1$</span>. Still pretty ugly but finding roots of quartic polynomial is a well-known problem that has a solution.</p... |
600,411 | <blockquote>
<p>Evaluate $ \iiint_D (x^2+y^2) \, ,\mathrm{d}V $, where $D$ is the
region bounded by the graphs of $y=x^2$, $z=4-y$, and $z=0$.</p>
</blockquote>
<p>How would I do this problem? I can't even visualize the region D. I tried plotting it into Wolfram Alpha but it doesn't understand that I need to 3D pl... | Lame-Ov2.0 | 114,476 | <p>To start, it might make the problem easier to convert to cylindrical coordinates and then attempt to visualize the region. From that, you just switch your limits of integration accordingly and just integrate like normal (but don't forget about the Jacobian!).</p>
|
4,306,842 | <p>I've tried to prove that
<span class="math-container">$$1+e^{-1/x}+2x(e^{-1/x} - 1)>0$$</span>
whenever <span class="math-container">$ x>0$</span> .</p>
<p><strong>My attempt</strong>: By setting <span class="math-container">$f(x) =1+e^{\frac{-1}{x}}+2x(e^{\frac{-1}{x}}-1)$</span> and then I tried to see where... | user2661923 | 464,411 | <p>If I understand correctly, I am allowed to assume that for all <span class="math-container">$y$</span> such that <span class="math-container">$1 \leq y < 2$</span>, you are able to <em>satisfactorily</em> express <span class="math-container">$\ln(y)$</span> as a series.</p>
<p>Given <span class="math-container">$... |
2,942,617 | <p>finding points of discontinuity of the function <span class="math-container">$f(x) = \frac{\frac{1}{x} - \frac{1}{x+1}}{ \frac{1}{x-1} - \frac{1}{x}}$</span>.</p>
<p><strong>My answer is:</strong></p>
<p>After simplifying the function it becomed <span class="math-container">$$f(x) = \frac{x-1}{x+1}$$</span> and I ... | Lee Mosher | 26,501 | <p>Algebra has allowed you to change the formula of the function, simplifying it.</p>
<p>But there was a cost: not only did you change the formula, <em>you changed the function itself</em>. Before applying the algebra and changing the formula, it is clear from the formula that plugging in <span class="math-container">... |
3,431,903 | <p>If there is some continuous function in <span class="math-container">$\mathbb{R}$</span> that satisfies <span class="math-container">$f(x)\notin \mathbb{Q}$</span> for every x. Is f then a constant function? How would I show this?</p>
| Franklin Pezzuti Dyer | 438,055 | <p>Suppose <span class="math-container">$f$</span> is continuous and takes on two distinct values <span class="math-container">$a,b\in\mathbb R$</span> with <span class="math-container">$a<b$</span>. By the intermediate value theorem, it also takes on all values in the interval <span class="math-container">$(a,b)$</... |
2,412,789 | <p>I am given</p>
<p>Prove the statement, for all integers $n$ if $5n$ is odd, then $n$ is odd</p>
<p>We know that this is not true for $n=4$ simply by building a table for values for $n$ and $5n$...</p>
<p>Is that all I need to do? I've shown a table up to $n=4$ and, by contradiction of the original statement, $n$ ... | user577215664 | 475,762 | <p>Well if 5n is odd think about the multiplication of two odds .What is the result ? Odd or even ? And 5 is odd...and 5n by hypothesis is odd too..</p>
<p>let's consider s odd and t odd too. Multiply both numbers st, and see what you get.. </p>
<p>$ s=2k+1, t=2j+1 $</p>
<p>$ st= (2h+1)(2j+1)= 4hj+2h+2j+1 $ therefor... |
3,788,418 | <p>If you take a bus to work in the morning there is a 20% chance you’ll arrive late. When you go
by bicycle there is a 10% chance you’ll be late. 70% of the time you go by bike, and 30% by
bus. Given that you arrive late, what is the probability you took the bus?</p>
<p>P(took the bus | being late) = <strong>P(took th... | Community | -1 | <p>P(took the bus and being late)= P(being late | took the bus) <span class="math-container">$\times $</span> P(took the bus)</p>
|
1,385,026 | <p>I came across something related to the degree of a splitting field for a polynomial over a field $K$. Let's suppose $p \in K[x]$ with degree $n$, and $p$ has irreducible factors $f_{1}, \ldots, f_{c}$ with respective degrees $d_{1}, \ldots, d_{c}$.</p>
<p>Ok, I know we can construct the splitting field as a tower ... | Kaj Hansen | 138,538 | <p>It is certainly achievable. For a concrete example:</p>
<p>Let $f(x) \in \mathbb{Q}[x]$ be an irreducible quintic polynomial, and let $K$ be its splitting field. Furthermore, suppose $f$ has exactly $2$ complex roots. It will be convenient to think about $\operatorname{Gal}(K/\mathbb{Q})$ as a permutation group ... |
1,889,002 | <p>I have been going through and doing some (non-assessed) homework questions, but am getting really stuck on conditional probability. The following problem is one that I simply cannot get my head around.</p>
<p>Question: Die A has four red and two blue faces, and die B has two red and four blue faces. One of the dice... | Parcly Taxel | 357,390 | <p>There is a hidden assumption in the statement "one of the dice is selected randomly before use": <em>the same dice is used in all three throws</em>. If the dice is chosen individually before each throw, as some might interpret it, the throws would be independent and the answer to (ii) $\frac12$, as I had it in an ea... |
3,245,841 | <blockquote>
<p>Let <span class="math-container">$a,b$</span> be two positive numbers. Prove or disprove the statement:</p>
<p>If <span class="math-container">$a+b \leq \frac{1}{2}$</span>, then <span class="math-container">$\dfrac{1-a}{a} \dfrac{1-b}{b} \geq 1$</span>.</p>
</blockquote>
<p>True. Assume <span c... | J. W. Tanner | 615,567 | <p>You have succeeded in proving that <span class="math-container">$\dfrac{1-a}{a} \dfrac{1-b}{b} \geq 1$</span>
if <span class="math-container">$a,b>0$</span> and <span class="math-container">$a+b \leq 1.$</span> </p>
<p>Therefore, since <span class="math-container">$\frac12<1$</span>, certainly <span class="... |
916,688 | <p>Suppose I have two sets A and B:</p>
<p>$$ A = \lbrace 2k-1 : k \in \mathbb{Z}\rbrace$$
$$ B = \lbrace 2l+1 : l \in \mathbb{Z}\rbrace$$ </p>
<p>I need to prove that A = B. </p>
<p>I know that to prove equality between two sets I need to prove both:</p>
<p>$$ A \subseteq B $$</p>
<p>and</p>
<p>$$ A \supseteq B... | Kim Jong Un | 136,641 | <p><strong>Hint</strong>: </p>
<ul>
<li>$2k-1=2l+1$ where $l=(k-1)$;</li>
<li>$2l+1=2k-1$ where $k=l+1$.</li>
</ul>
|
916,688 | <p>Suppose I have two sets A and B:</p>
<p>$$ A = \lbrace 2k-1 : k \in \mathbb{Z}\rbrace$$
$$ B = \lbrace 2l+1 : l \in \mathbb{Z}\rbrace$$ </p>
<p>I need to prove that A = B. </p>
<p>I know that to prove equality between two sets I need to prove both:</p>
<p>$$ A \subseteq B $$</p>
<p>and</p>
<p>$$ A \supseteq B... | creative | 166,713 | <p>Let $x\in A.$ Then $x=2k-1$ for some $k\in \mathbb{Z}.$ Also since $k\in \mathbb{Z}, k=l+1$ for some $l\in \mathbb{Z}.$ </p>
<p>So $x=2(l+1)-1=2l+2-1=2l+1$ so that $A\subseteq B.$ Similarly, </p>
<p>let $x\in B,$ then $x=2l+1$ for some $l\in \mathbb{Z}.$ Again by same arguement as above we can show that $B\subse... |
916,688 | <p>Suppose I have two sets A and B:</p>
<p>$$ A = \lbrace 2k-1 : k \in \mathbb{Z}\rbrace$$
$$ B = \lbrace 2l+1 : l \in \mathbb{Z}\rbrace$$ </p>
<p>I need to prove that A = B. </p>
<p>I know that to prove equality between two sets I need to prove both:</p>
<p>$$ A \subseteq B $$</p>
<p>and</p>
<p>$$ A \supseteq B... | creative | 166,713 | <p>An alternate way of doing the same thing is to use the set definitions : </p>
<p>$A=\{2k-1:k\in \mathbb{Z}\}$ </p>
<p>$ $ $ $ $ $ $ $ $=\{2(l+1)-1:l\in \mathbb{Z}\}$ </p>
<p>$ $ $ $ $ $ $ $ $=\{2l+2-1:l\in \mathbb{Z}\}$ </p>
<p>$ $ $ $ $ $ $ $ $=\{2l+1:l\in \mathbb{Z}\}$ </p>
<p>$ $ $ $ $ $ $ $ $=B$ </... |
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