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 |
|---|---|---|---|---|
330,488 | <p>On the complex plane <span class="math-container">$\mathbb C$</span> consider the half-open square <span class="math-container">$$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$</span> </p>
<p>Observe that for every <span class="math-container">$z\in \mathbb C$</span> and <span class="math-container"... | Nate Eldredge | 4,832 | <p>(This addresses a misinterpretation of the question, where <span class="math-container">$p$</span> can be chosen. I'll try to fix it.)</p>
<p>This seems too easy, so maybe I've misunderstood the question, but: let <span class="math-container">$L$</span> be the diagonal line <span class="math-container">$\{z : \Re(... |
2,149,006 | <p>While learning the power rule, one thing popped up in my mind which is confusing me. We know what the power rule states :</p>
<p>$$\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}$$ where $n$ is a real number.</p>
<blockquote>
<p>But instead of $n$, if we have a trig function like $\sin(x)$, <strong>will the powe... | Dr. Sonnhard Graubner | 175,066 | <p>one possibility is taking the logarithm on both sides
$$\ln(y)=\sin(x)\ln(x)$$ and by the chain rule we get
$$\frac{y'}{y}=\cos(x)\ln(x)+\sin(x)\cdot \frac{1}{x}$$
you must multiply this equation by $$y(x)$$</p>
|
213,513 | <p>I need help to extrapolate these data:</p>
<pre><code>θ = {20.7, 28.62, 32.04};
ω = {5, 6, 7};
</code></pre>
<p>using this equation:</p>
<p>θ(ω)=θ(ω⟶∞)+ c /ω^n to know what the values of c and n</p>
<p>I found a result using the below plot:</p>
<pre><code>ListLinePlot[
Transpose[{ω^-4,#}]&/@{θ},
Fr... | Henrik Schumacher | 38,178 | <p>Assuming that you meant the third key of your input to be <code>{{a1, a2}, {a1, a1}}</code>, this might do want you ask for.</p>
<pre><code>a = <|{{a1, a1}, {a2, a2}} -> 1, {{a1, a1}, {a1, a2}} -> 10, {{a1, a2}, {a1, a1}} -> 3|>;
f = Sort;
Merge[Thread[Rule[f /@ Keys[a], Values[a]]], Total]
</code></... |
213,513 | <p>I need help to extrapolate these data:</p>
<pre><code>θ = {20.7, 28.62, 32.04};
ω = {5, 6, 7};
</code></pre>
<p>using this equation:</p>
<p>θ(ω)=θ(ω⟶∞)+ c /ω^n to know what the values of c and n</p>
<p>I found a result using the below plot:</p>
<pre><code>ListLinePlot[
Transpose[{ω^-4,#}]&/@{θ},
Fr... | Carl Woll | 45,431 | <p>You can use the <a href="http://reference.wolfram.com/language/ref/ResourceFunction" rel="nofollow noreferrer"><code>ResourceFunction</code></a> <a href="https://resources.wolframcloud.com/FunctionRepository/resources/GroupByList" rel="nofollow noreferrer">"GroupByList"</a> to do this:</p>
<pre><code>ResourceFuncti... |
2,063,038 | <p>Let <span class="math-container">$S$</span> be the region in the plane that is inside the circle <span class="math-container">$(x-1)^2 + y^2 = 1$</span> and outside the circle <span class="math-container">$x^2 + y^2 = 1 $</span>. I want to calculate the area of <span class="math-container">$S$</span>.</p>
<h3>Try:</... | Omar Eafoe | 227,292 | <p>It will be complicated if you tried to solve such problems with cartesian coordinates it's much easier to solve by polar coordinates , and in that type of problems try to draw them if you draw you will find that the area of the upper half is same of the area of lower half so you can calculate 1 of them and mul... |
1,734,819 | <p>I think I'm on the right track with constructing this proof. Please let me know.</p>
<p>Claim: Prove that there exists a unique real number $x$ between $0$ and $1$ such that
$x^{3}+x^{2} -1=0$</p>
<p>Using the intermediate value theorem we get
$$r^{3}+r^{2}-1=c^{3}+c^{2}-1$$
......
$$r^{3}+r^{2}-c^{3}-c^{2}=0$$</... | Hagen von Eitzen | 39,174 | <p>You might rewrite the factor $(r^2+rc+c^2+r+c)$ as
$$r^2+rc+c^2+r+c = \underbrace{(r-c)^2}_{\ge0}+\underbrace{3rc}_{\ge0}+\underbrace r_{\ge0}+\underbrace c_{\ge0}\ge0 $$
with equality only if $r=c=0$.</p>
|
174,075 | <p>What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?</p>
| Prajjawal | 59,632 | <p>Normal is 90 degree to a surface but perpendicular is 90 degree to a line</p>
|
1,966,128 | <p>I know that by De Morgan's law that it is false. But how to disprove it?</p>
| RJM | 376,273 | <p>Suppose x $\in$ $A^c \cup B^c$ assume wlog that x $\in$ $A^c$. Then x $\in$ $A^c \cup B$, which implies x $\notin$ $(A \cup B)^c$</p>
|
4,490,778 | <p>Let <span class="math-container">$\mu$</span> be a sigma finite positive measure on <span class="math-container">$(X,\mathcal{A})$</span>. then exists <span class="math-container">$w\in L^1(\mu)$</span> such that <span class="math-container">$0< w(x) < 1$</span> for all <span class="math-container">$x\in X$</s... | Joe | 524,659 | <p>Since our measure is <span class="math-container">$\sigma$</span>-finite, we may write <span class="math-container">$X = \bigcup_{n=1}^\infty E_n$</span>, where <span class="math-container">$\mu(E_n) < \infty$</span> And <span class="math-container">$E_n \subseteq E_{n+1}$</span> for all <span class="math-contain... |
4,490,778 | <p>Let <span class="math-container">$\mu$</span> be a sigma finite positive measure on <span class="math-container">$(X,\mathcal{A})$</span>. then exists <span class="math-container">$w\in L^1(\mu)$</span> such that <span class="math-container">$0< w(x) < 1$</span> for all <span class="math-container">$x\in X$</s... | Davide Giraudo | 9,849 | <p>Actually, you can modify your construction by choosing the sequence <span class="math-container">$(E_n)$</span> to be pairwise disjoint instead of increasing. In this way,
<span class="math-container">$$
\int_X w(x)d\mu(x)=\sum_{n\geqslant 1}\int_X w_n(x)d\mu(x)\leqslant \sum_{n\geqslant 1}2^{-n}
$$</span>
and since... |
1,650,277 | <p>Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.)</p>
<p>If not, what is a category which is "close" to the category of partial orders (e.g. it should consists of special order-theoretic constructs) and has a subobject classifier? Bonus question: Is there also such an e... | drhab | 75,923 | <p>Normally the disjoint union of topological spaces <span class="math-container">$(X_\alpha)_{\alpha\in A}$</span> is defined as: <span class="math-container">$$X:=\bigcup_{\alpha\in A}\left(X_{\alpha}\times\{\alpha\}\right)$$</span>and is accompanied by injections <span class="math-container">$\iota_{\alpha}:X_{\alp... |
1,307,280 | <p>I m in the point X. I m 2 blocks up from a point A and 3 blocks down from my home H. Every time I walk one block i drop a coin.</p>
<p>H
.
.
.
X
.
.
A</p>
<p>If the coin is face I go one block up and if it is not face I go one block down.</p>
<p>Which is the probability of arriving home before the point A?</p>
... | Ittay Weiss | 30,953 | <p>You are correct. You present a nuance of the 'working definition', a special case where "getting closer" is misleading. You should interpret "getting closer" as "getting as close as you like". This is a more accurate 'working definition' in any case. Then the constant function scenario works just fine. You can get a... |
165,487 | <p>After I use Simplify on an expression I get$\dfrac{1}{2}\sqrt{-\dfrac{\sqrt{(-b^2+16|c|^2)(4|c|^2+b\Im(c))^2}}{4a(4|c|^2+b\Im(c)])}}$. This expression can clearly be simplified further by noticing that the square bracket term in the numerator cancels the other bracket term in the denominator so $\dfrac{1}{2}\sqrt{-\... | Ulrich Neumann | 53,677 | <p>The "bracket" you want to simplify is complex! </p>
<pre><code>bracket = r Exp[I φ];(* stands for (4 c Conjugate[c] + b Im[c])*)
expr = Sqrt[bracket^2]/bracket ;
FullSimplify[ expr, {Element[{r, φ}, Reals], r > 0 }]
(* E^(-I φ) Sqrt[E^(2 I φ)] *)
</code></pre>
<p>Further simplification needs information ab... |
1,906,013 | <blockquote>
<p>Let $f$ be a smooth function such that $f'(0) = f''(0) = 1$. Let $g(x) = f(x^{10})$. Find $g^{(10)}(x)$ and $g^{(11)}(x)$ when $x=0$.</p>
</blockquote>
<p>I tried applying chain rule multiple times:</p>
<p>$$g'(x) = f'(x^{10})(10x^9)$$</p>
<p>$$g''(x) = \color{red}{f'(x^{10})(90x^8)}+\color{blue}{(... | Robert Z | 299,698 | <p>We have that $f(x)=f(0)+x+x^2/2+o(x^2)$. Therefore the expansion of $g$ at $0$ is
$$g(x)=f(x^{10})=f(0)+x^{10}+x^{20}/2+o(x^{20}).$$
Hence $g^{(10)}(0)/10!$, which is the coefficient of $x^{10}$, is equal to $1$, and we conclude that $g^{(10)}(0)=10!$. Are you able now to find $g^{(11)}(0)$?</p>
|
27,951 | <p>Something I notice is when there's an advanced/specialized question, it often receives very few upvotes. Even if it is seemingly well written. I try to upvote advanced questions <strong>that I might not even understand</strong>, if they appear well written. </p>
<p>Is this good behaviour? Should we encourage upvoti... | Aloizio Macedo | 59,234 | <p>First off, you explicitly <a href="https://math.meta.stackexchange.com/a/25118/59234">mentioned this suggestion a while ago</a>, and the community seemed to think it is valuable.</p>
<p>That said, as the comments imply, it can be a little risky... but I don't think it is a significant risk. From my personal experie... |
2,891,444 | <p>For the intersection of two line segments, how was it know to use the determinants shown <a href="http://mathworld.wolfram.com/Line-LineIntersection.html" rel="nofollow noreferrer">here</a>? </p>
<p>I'm trying to determine how it was shown that they could be used to compute the intersection point.</p>
| V. Vancak | 230,329 | <p>The probability that the first proofreader will miss the error is $0.02$. Same probability has the second proofreader. The complementary event of the event of interest is "both will miss the error". Due to independence, the probability of such event is multiplication of the two probabilities of missing the error, i.... |
2,678,406 | <p>I'm trying to compute the distance between a point and a plane of the form
<span class="math-container">$$
ax+bx+cz = d
$$</span>
not using the standard formula for analytical geometry.</p>
<ul>
<li>I am trying to compute it by coming up with the projection matrix onto the normal of the plane passing through the ori... | knight5478 | 1,129,053 | <p>you can calculate the distance between those points is by that formula:</p>
<p>1)find two linearly independent vectors from the plane (<span class="math-container">$\vec u$</span>, <span class="math-container">$\vec v$</span>)</p>
<p>2)find a vector from any point on the plane to the desired point on the line (<span... |
392,020 | <p>It is well known that the minimum number of monochromatic triangles in a red/blue coloring of the edges of the complete graph <span class="math-container">$K_n$</span> is given by Goodman's formula
<span class="math-container">$$M(n)=\binom n3-\left\lfloor\frac n2\left\lfloor\left(\frac{n-1}2\right)^2\right\rfloor\r... | Thomas Bloom | 385 | <p>Suppose that the red degree of each vertex is denoted by <span class="math-container">$r_x$</span>, and the total number of red edges is <span class="math-container">$R=\frac{1}{2}\sum_x r_x$</span>. Then the number of monochromatic triangles is exactly
<span class="math-container">$$ \binom{n}{3}-(n-1)R+\frac{1}{2}... |
70,500 | <p>I am trying to find $\cosh^{-1}1$ I end up with something that looks like $e^y+e^{-y}=2x$. I followed the formula correctly so I believe that is correct up to this point. I then plug in $1$ for $x$ and I get $e^y+e^{-y}=2$ which, according to my mathematical knowledge, is still correct. From here I have absolutely n... | APW | 490,629 | <p>To find the inverse for any $x$, we are looking for
$$ y = \cosh^{-1} x, $$
i.e. $$x = \cosh y = \frac{1}{2} (e^y + e^{-y}). $$
Multiplying through by $2e^y$ gives
$$ (e^y)^2 -2x\,e^y + 1 = 0, $$
which is a quadratic in $e^y$. You can then use the quadratic formula, or here completing the square
$$ (e^y-x)^2 - x^2... |
3,857,698 | <p>Let <span class="math-container">$D_1, ..., D_n$</span> be arbitrary <span class="math-container">$n$</span> sets where <span class="math-container">$D_i \cap D_j \neq \emptyset$</span>. In the simplified case where <span class="math-container">$n = 2$</span>, we have that
<span class="math-container">$$
\begin{spli... | dafinguzman | 123,170 | <p>You can arrive at complex arithmetic from geometric intuition if you start with transformations of the plane.</p>
<p>It is well known that matrices which preserve angles (i.e. map shapes to similar shapes) and orientation are of the form <span class="math-container">$cR(\theta)$</span>, where <span class="math-conta... |
3,645,263 | <p>I have recently started studying Set Theory in a self-thaught way, for that purpose I have been following Kunen's book: Set Theory: An Introduction to Independence Proofs. I'm in Chapter I section 7 and it has been defined the ordinals addition but I don't quite understand that definition. I have seen that in other ... | Community | -1 | <p>As mentioned by Brian, it is essentially the lexicographic ordering. </p>
<p>For example, say <span class="math-container">$2=\{0_2,1_2\}$</span> and <span class="math-container">$3=\{0_3,1_3,2_3\}$</span>. </p>
<p>According to the definition, we first extend <span class="math-container">$2$</span> and <span class... |
245,756 | <p>I'm supposed to show that the jordan content is $0$. The definition for a set $S$ having jordan content zero I have to work with is :$\forall\epsilon>0$ there is a finite collection of generalized rectangles in $\mathbb{R}^n$ that covers $S$ the sum of these rectangles volumes being less that $\epsilon$.</p>
<p>... | André Nicolas | 6,312 | <p><strong>Hint:</strong> In the context of a calculus course, I think you are first expected to argue informally that such a maximal cylinder must have axis that goes through the center of the circle, and that without loss of generality that axis is the $z$-axis.</p>
<p>So now suppose that the cylinder meets the $x$-... |
245,756 | <p>I'm supposed to show that the jordan content is $0$. The definition for a set $S$ having jordan content zero I have to work with is :$\forall\epsilon>0$ there is a finite collection of generalized rectangles in $\mathbb{R}^n$ that covers $S$ the sum of these rectangles volumes being less that $\epsilon$.</p>
<p>... | Till Hoffmann | 23,894 | <p>Let $R$ be the radius of the sphere and let $h$ be the height of the cylinder centered on the center of the sphere. By the Pythagorean theorem, the radius of the cylinder is given by
$$
r^2 = R^2 - \left(\frac{h}{2}\right)^2.
$$</p>
<p>The volume of the cylinder is hence
$$
\begin{align}
V &= \pi r^2 h\\
&... |
1,090,658 | <p>I'm doing some previous exams sets whilst preparing for an exam in Algebra.</p>
<p>I'm stuck with doing the below question in a trial-and-error manner:</p>
<p>Find all $ x \in \mathbb{Z}$ where $ 0 \le x \lt 11$ that satisfy $2x^2 \equiv 7 \pmod{11}$</p>
<p>Since 11 is prime (and therefore not composite), the Ch... | HSN | 58,629 | <p>A first step in this could be to find the inverse of $2$, which turns out to be $6$, yielding $6\cdot2x^2\equiv x^2\equiv 6\cdot 7\mod 11\equiv 9\mod 11$. The obvious solutions to this are $3$ and $-3\equiv 8$. An important remark is that since $11$ is prime, there are at most two solutions to any given quadratic eq... |
3,021,631 | <p>I've been strongly drawn recently to the matter of the fundamental definition of the exponential function, & how it connects with its properties such as the exponential of a sum being the product of the exponentials, and it's being the eigenfunction of simple differentiation, etc. I've seen various posts inwhich... | Ira Gessel | 437,380 | <p>Here's another proof that <span class="math-container">$\log\bigl((1+x)(1+y)\bigr) = \log(1+x) + \log(1+y)$</span> as formal power series. This assumes some facts about derivatives of formal power series that are not difficult to verify (e.g., a very simple special case of the chain rule). It follows directly from t... |
197,393 | <p>Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?</p>
| lab bhattacharjee | 33,337 | <p>$$\tan^{-1}(2)+\tan^{-1}(3)=\tan^{-1}{\left(\frac{2+3}{1-2\cdot 3}\right)}=\tan^{-1}(-1)=n\pi-\frac \pi 4,$$ where $n$ is any integer.</p>
<p>Now the principal value of $\tan^{-1}(x)$ lies in $[-\frac \pi 2, \frac \pi 2]$ precisely in $(0, \frac \pi 2)$ if finite $x>0$.
So, the principal value of $\tan^{-1}(2)+\... |
197,393 | <p>Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?</p>
| kennytm | 171 | <p><img src="https://i.stack.imgur.com/zyXUh.png" alt="enter image description here"> </p>
<p>Consider $O=(0,0)$, $A=(1,1)$, $B=(-1,3)$, $D=(1,-3)$, $E=(1,0)$.</p>
<p>\begin{align}
2 &= \frac{AB}{AO} = \tan \angle AOB \\
1 &= \frac{AE}{EO} = \tan \angle AOE \\
3 &= \frac{DE}{DO} = \tan \angle DOE
\end{ali... |
119,561 | <p>I am interested in determining the <strong>minimum</strong> and <strong>maximum</strong> values of the real roots of polynomials of form $P(x)=\sum_{k=0}^n a_{k} x^k$ where $n$ will have a defined value (say 3,4,5...) and $a_k$ are chosen from the set $\{-1,1\}$ with equal probability.</p>
<p>I have tried creating ... | Bob Hanlon | 9,362 | <p>To find the exact values for min and max roots</p>
<pre><code>roots = DeleteDuplicates@
Flatten[x /.
Solve[# == 0, x, Reals] & /@
(Tuples[{-1, 1}, {4}].{1, x, x^2, x^3})] //
SortBy[#, N] &;
{min, max} = roots[[{1, -1}]] // ToRadicals
</code></pre>
<p><a href="https://i.stack.imgur.co... |
1,041,134 | <p>I need to show if $a$ is in $\mathbb{R}$ but not equal to $0$, and $a+\dfrac{1}{a}$ is integer, $a^t+\dfrac{1}{a^t}$ is also an integer for all $t\in\mathbb N$.
Can you provide me some hints please?</p>
| Henry | 6,460 | <p>Hint: If $\displaystyle a+\frac1a$ is an integer then $\displaystyle \left(a+\frac1a\right)^2,\left(a+\frac1a\right)^3, \ldots $ are integers.</p>
<p>Multiply the powers out and you should be able to see why $a^t+\dfrac1{a^t}$ is going to be an integer for positive integer $t$, using a combination of symmetry and ... |
3,105,482 | <p>For the formula:</p>
<p><span class="math-container">$$1 = \sqrt{x^2 + y^2 + z^2 + w^2}$$</span></p>
<p>How to rewrite it to find <span class="math-container">$w$</span>?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>We have <span class="math-container">$$w^2=1-(x^2+y^2+z^2)=w^2$$</span> so we obtain <span class="math-container">$$w=\pm\sqrt{1-(x^2+y^2+z^2)}$$</span> for <span class="math-container">$$1\geq x^2+y^2+z^2$$</span></p>
|
2,761,658 | <blockquote>
<p>Two numbers $x$ and $y$ are chosen at random from the numbers $1,2,3,4,\ldots,2004$. The probability that $x^3+y^3$ is divisible by $3$ is?</p>
</blockquote>
<p>The correct answer is $\dfrac13$ while mine is $\dfrac{445}{2003}$</p>
<p>My attempt:</p>
<p>For $x^3+y^3$ to be divisible by $3$, EITHER ... | Piquito | 219,998 | <p>It is immediate (by using arithmetic progression) to get that there are equally $668$ numbers of residues $0,1,2$ modulo $3$. We have
$$(3n)^3+(3n)^3\equiv 0\pmod3\\(3n)^3+(3n+1)^3\equiv 1\pmod3\\(3n)^3+(3n+2)^3\equiv 2\pmod3\\(3n+1)^3+(3n+1)^3\equiv 2\pmod3\\(3n+1)^3+(3n+2)^3\equiv 0\pmod3$$
It follows$$\binom{668... |
108,110 | <p>How can I use <em>Mathematica</em> to expand such a product (only need a finite number of terms):</p>
<p>$$\prod^{\infty}_{n=1}\frac{({1-yq^{n+1}})({1-y^{-1}q^n})}{(1-q^n)^2}$$</p>
| Lukas | 21,606 | <p>Given that you only need a finite number of terms, I would do something like this:</p>
<pre><code>ClearAll[prod, expand];
prod[num_?NumericQ] :=Product[(1 - y*q^(n + 1))*(1 - q^n/y)/(1 -q^n)^2, {n, 1, num}];
expand[num_?NumericQ] := Total[Total[#] & /@ Numerator[#]/Flatten[DeleteDuplicates[#] & /@ Denominat... |
3,754,225 | <p>Let S be the circle with centre <span class="math-container">$(0,0)$</span>, radius <span class="math-container">$r$</span> units. The chord <span class="math-container">$C$</span> of the circle S subtends an angle of 2π/3 at its center. If R represents the region consisting of all points inside S which are closer t... | Math Lover | 801,574 | <p><a href="https://i.stack.imgur.com/J1wrS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/J1wrS.jpg" alt="enter image description here" /></a></p>
<p>When you draw the circle and the chord, it should become simpler to understand. This is a bit difficult to visualize otherwise.</p>
<p>You could pick... |
3,271,891 | <p>In <a href="https://en.wikipedia.org/wiki/Boolean_algebra_(structure)" rel="nofollow noreferrer">Wikipedia, the Boolean algebra</a> is defined as a 6-tuple
<span class="math-container">$(A,\wedge,\vee,\neg,0,1)$</span>. In Kuratowski1976, on the other side in the definition on page 34, there is no <span class="math... | Con | 682,304 | <p>I have no access to Kuratowski's definition currently, but I assume that he might want to define something less restrictive (depending on whether all the other axioms are the usual ones or equivalent) that we would usually not call boolean algebra and afterwards introduce the notion of <span class="math-container">$... |
4,629,824 | <p>Let <span class="math-container">$k$</span> be a given positive integer. I want to solve the following system of Diophantine equations: <span class="math-container">$$\begin{cases} a^2 + b^2 + c^2 = k^2 \\ b^2 = ac \end{cases}$$</span>
where <span class="math-container">$a, b, c \in \mathbb{N}$</span> are non-zero.<... | Will Jagy | 10,400 | <p>we may demand <span class="math-container">$ \gcd(a,b,c) = 1$</span> by dividing through by any common factor. This works because of homogeneity.</p>
<p>Next, <span class="math-container">$ac=b^2$</span> implies that
<span class="math-container">$$ a = x^2, \; \; b = xy, \; \; c = y^2 $$</span></p>
<p>At this ... |
785,844 | <p>I've got the following limit to solve:</p>
<p>$$\lim_{s\to 1} \frac{\sqrt{s}-s^2}{1-\sqrt{s}}$$</p>
<p>I was taught to multiply by the conjugate to get rid of roots, but that doesn't help, or at least I don't know what to do once I do it. I can't find a way to make the denominator not be zero when replacing $s$ fo... | corporal | 148,632 | <p>You were on the right track using the <em>conjugate</em>.</p>
<p>$$\begin{align*}
\frac{\sqrt{s}-s^2}{1-\sqrt{s}}&=\frac{\sqrt{s}-s^2}{1-\sqrt{s}}\times\frac{1+\sqrt{s}}{1+\sqrt{s}}\\
&=\frac{\sqrt{s}-s^2+s-s^{5/2}}{1-s}\\
&=\frac{s-s^2+\sqrt{s}-s^{5/2}}{1-s}\\
&=\frac{s(1-s)+\sqrt{s}(1-s^{2})}{1-s}... |
2,704,290 | <p>I'm prepared for the competitive exam like <a href="http://csirhrdg.res.in/mathCEN_June2015.pdf" rel="nofollow noreferrer">this</a> (a sample question).</p>
<p>In order to solve the problems, first to familiarize with the prerequisite for the each concept. It's ok! </p>
<p>My problem is: If I'm working certain pro... | Joonas Ilmavirta | 166,535 | <p>Forgetting something is normal.
What you have to do then is to remind yourself of what you once remembered actively.
It is much easier the second time.
It is easy to forget things after focusing on them for a week.
A more permanent understanding of the topics needs a longer time frame than a couple of weeks.</p>
<p... |
981,748 | <p>We know a closed-form of the first two powers of the integral of <a href="http://mathworld.wolfram.com/Trilogarithm.html" rel="nofollow noreferrer">trilogarithm function</a> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. From the result <a href="https://math.stackexchang... | Alexander Vlasev | 11,998 | <p>Here's my approach. To reduce clutter let</p>
<p>$$f_m(x) = Li_m(x)$$</p>
<p>and let</p>
<p>$$f_M(x) = f_{m_1}(x)\dots f_{m_n}(x)$$</p>
<p>where $M$ has $n$ entries. Then let</p>
<p>$$I_k(M) = \int f_M(x) dx$$</p>
<p>where we can interpret this as an indefinite or definite integral. Consider the following deri... |
3,956,112 | <p>I tried to evaluate
<span class="math-container">$$
\int_{-1}^1(x^2-\frac{1}{x^2}+3) \, dx
$$</span>
in the following way:
<span class="math-container">$$
\left[\frac{x^3}{3}+\frac{1}{x}+3x\right]_{-1}^1=\frac{26}{3} \, .
$$</span>
But when I typed the integral into a calculator I got a math error. Why did this happ... | Jan Eerland | 226,665 | <p>Well, what happends to:</p>
<p><span class="math-container">$$f(x)=\frac{1}{x^2}$$</span></p>
<p>When <span class="math-container">$x\in\left[-1,1\right]$</span>?</p>
|
3,956,112 | <p>I tried to evaluate
<span class="math-container">$$
\int_{-1}^1(x^2-\frac{1}{x^2}+3) \, dx
$$</span>
in the following way:
<span class="math-container">$$
\left[\frac{x^3}{3}+\frac{1}{x}+3x\right]_{-1}^1=\frac{26}{3} \, .
$$</span>
But when I typed the integral into a calculator I got a math error. Why did this happ... | Joe | 623,665 | <p>Say <span class="math-container">$F$</span> is an antiderivative of <span class="math-container">$f$</span>—that is, if you differentiate <span class="math-container">$F$</span>, then you get <span class="math-container">$f$</span>. What you're doing to compute the integral is the following:
<span class="math-contai... |
3,956,112 | <p>I tried to evaluate
<span class="math-container">$$
\int_{-1}^1(x^2-\frac{1}{x^2}+3) \, dx
$$</span>
in the following way:
<span class="math-container">$$
\left[\frac{x^3}{3}+\frac{1}{x}+3x\right]_{-1}^1=\frac{26}{3} \, .
$$</span>
But when I typed the integral into a calculator I got a math error. Why did this happ... | FFjet | 597,771 | <p>Recall when a function is Riemann integrable:</p>
<blockquote>
<p>A <strong>bounded function</strong> on a compact interval <span class="math-container">$[a, b]$</span> is Riemann integrable if and only if it is continuous almost everywhere.</p>
</blockquote>
<p>So, is <span class="math-container">$x^2-\dfrac{1}{x^2... |
1,270,584 | <p>I tried googling for simple proofs that some number is transcendental, sadly I couldn't find any I could understand.</p>
<p>Do any of you guys know a simple transcendentality (if that's a word) proof?</p>
<p>E: What I meant is that I wanted a rather simple proof that some particular number is transcendental ($e$ o... | Michael Hardy | 11,667 | <p>Cantor showed a simple way to list the real algebraic numbers in a sequence $a_1,a_2,a_3,\ldots$. Every term has only finitely many terms before it, and every algebraic number will be reached after only finitely many steps. Now let us seek a transcendental number in the interval $(0,1)$. Find the first number in... |
3,657,106 | <blockquote>
<p>Sam was adding the integers from <span class="math-container">$1$</span> to <span class="math-container">$20$</span>. In his rush, he skipped one of the numbers and forgot to add it. His final sum was a multiple of <span class="math-container">$20$</span>. What number did he forget to add?</p>
</block... | Toby Mak | 285,313 | <p>You don't actually need to calculate the sum to find the missing number.</p>
<p>Using Gauss's trick again, <span class="math-container">$1 + 19 = 20$</span>, <span class="math-container">$2 + 18 = 20$</span>, <span class="math-container">$3 + 17 = 20$</span> and so on, all the way up to <span class="math-container"... |
3,163,355 | <p>I found that the Wieferich prime <span class="math-container">$1093$</span> divides <span class="math-container">$3^{1036}-1$</span>.
Does <span class="math-container">$1093$</span> divides infinitely many <span class="math-container">$3^{k}-1$</span>, with k a positive integer? And what features must <span class="m... | Carl Schildkraut | 253,966 | <p>We see that <span class="math-container">$1093|3^7-1$</span>. Now, </p>
<p><span class="math-container">$$1093|3^k-1\Leftrightarrow 1093|(3^k-1)-(3^7-1) \Leftrightarrow 1093|3^k-3^7 \Leftrightarrow 1093|3^{k-7}-1.$$</span></p>
<p>We can see, by repeatedly subtracting <span class="math-container">$7$</span>, that <... |
3,276,124 | <p>Consider the equation</p>
<p><span class="math-container">$$-242.0404+0.26639x-0.043941y+(5.9313\times10^{-5})\times xy-(3.9303\times{10^{-6}})\times y^2-7000=0$$</span></p>
<p>with <span class="math-container">$x,y>0$</span>. If you plot it, it'll look like below:</p>
<p><a href="https://i.stack.imgur.com/Oqf... | Adrian Keister | 30,813 | <p>Following Calvin Khor's line of reasoning, we will use the following algorithm:</p>
<ol>
<li>Find the center of the hyperbola, and translate the hyperbola so that the center coincides with the origin.</li>
<li>Find the angle of rotation necessary to put the hyperbola into the canonical form <span class="math-contain... |
3,276,124 | <p>Consider the equation</p>
<p><span class="math-container">$$-242.0404+0.26639x-0.043941y+(5.9313\times10^{-5})\times xy-(3.9303\times{10^{-6}})\times y^2-7000=0$$</span></p>
<p>with <span class="math-container">$x,y>0$</span>. If you plot it, it'll look like below:</p>
<p><a href="https://i.stack.imgur.com/Oqf... | quarague | 169,704 | <p>You could also use more high powered maths as follows. </p>
<p>a) Find a parametrization <span class="math-container">$t \mapsto \gamma(t)=(x(t), y(t))$</span> of your curve.</p>
<p>b) Reparametrize to get a cuve parametrized by arc length, that is <span class="math-container">$x'(t)^2+y'(t)^2=1$</span> for all <s... |
2,885,453 | <p>Evaluate $$\lim _{x \to 0} \left[{\frac{x^2}{\sin x \tan x}} \right]$$ where $[\cdot]$ denotes the greatest integer function.</p>
<p>Can anyone give me a hint to proceed?</p>
<p>I know that $$\frac {\sin x}{x} < 1$$ for all $x \in (-\pi/2 ,\pi/2) \setminus \{0\}$ and $$\frac {\tan x}{x} > 1$$ for all $x \in ... | Kavi Rama Murthy | 142,385 | <p>Using the inequalities $\cos x < 1-\frac {x^{2}} {2} + \frac {x^{4}} {24}$ and $\sin x > x -\frac {x^{3}} {6}$ you can check that $0\leq \frac {x^{2}} {\sin x \tan x} <1$ for all $x>0$ sufficiently small. Hence the integer part of this fraction is $0$ for such $x$. The function is even so the limit from ... |
3,107,858 | <p>I have to find the parametric equation for the line <span class="math-container">$M_1$</span> with the following info: </p>
<ul>
<li><span class="math-container">$M_1$</span> goes through the point <span class="math-container">$P(1,2,2)$</span></li>
<li>Is parallel with the plane <span class="math-container">$x + 3... | lulu | 252,071 | <p>I expect there is a typo in the expression for <span class="math-container">$M_2$</span>. You've written <span class="math-container">$-1+2$</span> for the third coordinate which seems incorrect. I'll guess you meant <span class="math-container">$-1+2t$</span> and compute from there. If you meant something else, y... |
181,839 | <p>Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF.</p>
<p>Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$.</p>
<p>Suppose $\forall n\in \omega, {F_n}^o=\emptyset$ ($o$ denotes interior)</p>
<p>Fix $x_0 \in \mathbb{R}^k$
... | Asaf Karagila | 622 | <p>This is essentially the Baire category theorem. Indeed in ZF it holds for separable complete metric spaces.</p>
<p>The argument is as follows:</p>
<p>Suppose that $(X,d)$ is a separable complete metric space, and $\{a_k\mid k\in\omega\}=D\subseteq X$ is a countable dense subset.</p>
<p>By contradiction assume tha... |
3,714,418 | <p>I need to calculate the angle between two 3D vectors. There are plenty of examples available of how to do that but the result is always in the range <span class="math-container">$0-\pi$</span>. I need a result in the range <span class="math-container">$\pi-2\pi$</span>.</p>
<p>Let's say that <span class="math-conta... | fleablood | 280,126 | <p>That's straightforward element chasing and in my opinion the way you did it <em>IS</em> the best proof.</p>
<p>We can go to "concepts" but this assume the reader has a strong intuition which... I'm learning not all do ... and what good is a proof if it's hard to follow:</p>
<p>For any sets <span class="math-conta... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Henry Segerman | 6,015 | <p>(I learned this puzzle from Ravi Vakil.) Suppose you have an infinite grid of squares, and in each square there is an arrow, pointing in one of the 8 cardinal directions, with the condition that any two orthogonally adjacent arrows can differ by at most 45 degrees.</p>
<p>Can there be a closed cycle? (i.e. start at... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p><strong>Fork in the road 1</strong></p>
<p>You're on a path on an island, come to a fork in the road. Both paths lead to villages of natives; the entire village either always tells the truth or always lies <em>(both villages could be truth-telling or lying villages, or one of each)</em>. There are two natives at ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p><strong>Fork in the road 2</strong></p>
<p>You're once again at a fork in the road, and again, one path leads to safety, the other to doom.</p>
<p>There are three natives at the fork. One is from a village of truth-tellers, one from a village of liars, one from a village of random answerers. Of course you don't ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | reb | 50 | <p>What is the resistance between 2 adjacent vertices of an infinite checkerboard if every edge is a 1 ohm resistor? </p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Tom Boardman | 5,869 | <p>Okay, so it's somewhat more numeric than the others, but I quite enjoy the simplicity of:</p>
<blockquote>
<h3>Simplify:</h3>
<p>$$\sqrt{2+\sqrt3}$$</p>
</blockquote>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Ken Fan | 7,434 | <p>Here's a balance scale problem that I decided to post because a little bit of googling around for it came up negative. It differs from most balance scale puzzles I've seen because it doesn't involve "bad weights". I learned of it from a friend of mine who is an engineer.</p>
<p>There are 10 balls which come in two... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | schnitzi | 7,740 | <p>Okay, I've got one, and as far as I know it hasn't been analyzed before.</p>
<p>I was watching a travel show the other night -- they were in Korea, and a group of people were playing a drinking game. It works like this:</p>
<p><em>One person is "it". This person says something like, "ready, set..." then points a... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Tracy Hall | 7,936 | <p>You are the captain of a team of <i>N</i> players, in charge of choosing a strategy that your adversary will overhear (and therefore rig the game for you to lose unless the strategy is perfect). To play the game, the adversary writes a distinct name on each player's forehead and you are brought into a situation whe... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | klaraspina | 6,882 | <p>There are some dwarves approaching to a bridge. They have to cross it to come back home from the cave where they work. Unfortunately, a dragon has just decided to reside under that bridge, and it's hungry. But it's also bored, so it doesn't want just to eat the dwarves, but proposes them a game: it will put on each ... |
557,963 | <p>Suppose that $V = X ⊕Y$, and let $P$ be the projector onto $X$ along
$Y$. Prove that
$R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$.</p>
<p>I know that from $V = X ⊕Y$ I got $v=x+y$ for $v,x,y$ are element of $V,X,Y$ and the intersect of $X$ and $Y$ is zero. But I don't know what to do next.</p>
| Cameron Buie | 28,900 | <p><strong>Hint</strong>: For any $v\in V$, we have $v=P(v)+y,$ where $y=v-P(v)=(I-P)(v).$</p>
<p>Recall that for any linear transformation $T:V\to W,$ we have $$N(T)=\{v\in V:T(v)=0_W\}$$ and $$R(T)=\{w\in W:T(v)=w\text{ for some }v\in V\}.$$</p>
<p>In particular, then, recalling that all $v\in V$ can be uniquely wr... |
557,963 | <p>Suppose that $V = X ⊕Y$, and let $P$ be the projector onto $X$ along
$Y$. Prove that
$R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$.</p>
<p>I know that from $V = X ⊕Y$ I got $v=x+y$ for $v,x,y$ are element of $V,X,Y$ and the intersect of $X$ and $Y$ is zero. But I don't know what to do next.</p>
| Tom | 103,715 | <p>Hint: For a projection $P$, $P^2 = P$. So, for example, $P(I-P)x = Px - P^2x = Px - Px = 0$.</p>
|
4,344,139 | <p>I have broken down this equation into factorials, but I'm unsure of where to go from here. This may not even be the right approach to solve this binomial transform. Any help would be appreciated.</p>
<p><a href="https://i.stack.imgur.com/oqNqm.png" rel="noreferrer">Binomial transform identity</a>:</p>
<p><span class... | Carl Schildkraut | 253,966 | <p>Recall that, if <span class="math-container">$p(x)$</span> is a polynomial of degree <span class="math-container">$n$</span>,
<span class="math-container">$$\sum_{k=0}^n(-1)^{n-k}\binom nkp(x+k)$$</span>
is <span class="math-container">$n!$</span> times the leading coefficient of <span class="math-container">$p(x)$<... |
4,344,139 | <p>I have broken down this equation into factorials, but I'm unsure of where to go from here. This may not even be the right approach to solve this binomial transform. Any help would be appreciated.</p>
<p><a href="https://i.stack.imgur.com/oqNqm.png" rel="noreferrer">Binomial transform identity</a>:</p>
<p><span class... | Marko Riedel | 44,883 | <p>In trying to prove</p>
<p><span class="math-container">$$\sum_{k=0}^n (-1)^{n-k} {n\choose k} {n+jk\choose jk} = j^n$$</span></p>
<p>we start with</p>
<p><span class="math-container">$$\sum_{k=0}^n (-1)^{n-k} {n\choose k} {n+jk\choose n}
\\ = [z^n] (1+z)^n
\sum_{k=0}^n (-1)^{n-k} {n\choose k} (1+z)^{jk}
\\ = [z^n] (... |
4,245,205 | <p>Suppose <span class="math-container">$X$</span> is a CW-complex of dimension <span class="math-container">$n$</span>. If <span class="math-container">$e_i$</span> is an <span class="math-container">$n$</span>-cell then is <span class="math-container">$H_n(X) \to H_n(X, X \setminus e_i)$</span> the zero map? If not, ... | user965894 | 965,894 | <p>No, take <span class="math-container">$X = S^n = e_0 \cup e_n$</span> as a counterexample. The map <span class="math-container">$H_n(X) \to H_n(X, X \setminus \{x\})$</span> is an isomorphism.</p>
<p>More generally your map <span class="math-container">$X \to X/(X \setminus e_n) \cong S^n$</span> is a cellular map. ... |
3,530,285 | <blockquote>
<p>Let <span class="math-container">$\text{char}(\mathbb{K}) = 0$</span>. It then follows that <span class="math-container">$AB-BA \ne 1 \, (A, B \in \Bbb K^{n \times n})$</span>.</p>
</blockquote>
<p>I first showed that <span class="math-container">$\text{trace}(AB) = \text{trace}(BA)$</span> for every... | G. Chiusole | 436,096 | <p><span class="math-container">$S$</span> is not closed in <span class="math-container">$\mathbb{Q}$</span>: </p>
<p>Assume it were, then for any sequence <span class="math-container">$(x_n)_{n \in \mathbb{N}} \subseteq S$</span> which converges to <span class="math-container">$a \in \mathbb{Q}$</span> we would have ... |
4,242,093 | <p><em><strong>Question:</strong></em></p>
<blockquote>
<p>Let <span class="math-container">$G=(V_n,E_n)$</span> such that:</p>
<ul>
<li>G's vertices are words over <span class="math-container">$\sigma=\{a,b,c,d\}$</span> with length of <span class="math-container">$n$</span>, such that there aren't two adjacent equal ... | ploosu2 | 111,594 | <p>For part A use the words</p>
<p><span class="math-container">$$w = a(bc)^k$$</span></p>
<p>and</p>
<p><span class="math-container">$$w = a(bc)^kb$$</span></p>
<p>for <span class="math-container">$n>2$</span> odd and even respectively.</p>
<p>These have odd degrees since the first <span class="math-container">$b$<... |
78,946 | <p>I want to find min of the function
$$\frac{1}{\sqrt{2
x^2+\left(3+\sqrt{3}\right)
x+3}}+\frac{1}{\sqrt{2
x^2+\left(3-\sqrt{3}\right)
x+3}}+\sqrt{\frac{1}{3}
\left(2 x^2+2 x+1\right)}.$$
I know, the exact value minimum is $\sqrt{3}$ at $x = 0$. With <em>Mathematica</em>, I tried </p>
<pre><code>A = 1... | k_v | 24,727 | <pre><code>a[x_] := Sqrt[1 + 2 x + 2 x^2]/Sqrt[3] + 1/Sqrt[
3 + (3 - Sqrt[3]) x + 2 x^2] + 1/Sqrt[3 + (3 + Sqrt[3]) x + 2 x^2]
Plot[a[x], {x, -5, 5}]
</code></pre>
<p><img src="https://i.stack.imgur.com/LBzs5.jpg" alt="enter image description here"></p>
<pre><code>extr = NSolve[a'[x] == 0, x, Reals]
</code></pre>
... |
3,298,282 | <p>Find the number of terms in the expansion <span class="math-container">$(1+a^3+a^{-3})^{100}$</span></p>
<p>I used the concept <span class="math-container">$a^3+a^{-3}=T$</span>, while using this I have 101 terms, from <span class="math-container">$T^2$</span> to <span class="math-container">$T^{100}$</span> how do... | Mike | 544,150 | <p>Well, note that </p>
<p><span class="math-container">$$(a^{-3} + 1+a^3)^{100} = (a^{-3}+a^0+a^{3})^{100} = \sum_{i=-100}^{100} c_ia^{3i},$$</span></p>
<p>for some <span class="math-container">$c_i$</span> and each <span class="math-container">$c_i$</span> is strictly positive. In fact, each <span class="math-conta... |
1,930,933 | <blockquote>
<p>Does there exist an $n \in \mathbb{N}$ greater than $1$ such that $\sqrt[n]{n!}$ is an integer?</p>
</blockquote>
<p>The expression seems to be increasing, so I was wondering if it is ever an integer. How could we prove that or what is the smallest value where it is an integer?</p>
| robjohn | 13,854 | <p>In <a href="https://math.stackexchange.com/a/590901">this answer</a>, it is shown that the number of factors of $p$ that divide $n!$ is
$$
\frac{n-\sigma_p(n)}{p-1}\tag{1}
$$
where $\sigma_p(n)$ is the sum of the base-$p$ digits of $n$.</p>
<p>For $n!$ be an $n^{\text{th}}$ power, $(1)$ must be a multiple of $n$ fo... |
121,784 | <p>I hope this is not too trivial for this forum. I was wondering if someone has come across this polytope.</p>
<p>You start with the <a href="http://mathworld.wolfram.com/RhombicDodecahedron.html" rel="nofollow noreferrer">rhombic dodecahedron</a>, subdivide it into four parallellepipeds, <a href="https://i.stack.img... | Joseph O'Rourke | 6,094 | <p>Is this it?
<br />
<img src="https://i.stack.imgur.com/lTqnQ.jpg" alt="Rhombic Dodecahedron + Tetrahedron" />
<br /></p>
|
121,784 | <p>I hope this is not too trivial for this forum. I was wondering if someone has come across this polytope.</p>
<p>You start with the <a href="http://mathworld.wolfram.com/RhombicDodecahedron.html" rel="nofollow noreferrer">rhombic dodecahedron</a>, subdivide it into four parallellepipeds, <a href="https://i.stack.img... | F. C. | 10,881 | <p>It seems to ressemble the "Self-Dual Icosioctahedron #4" :</p>
<p><a href="http://dmccooey.com/polyhedra/SelfDualIcosioctahedron4.html" rel="nofollow noreferrer">http://dmccooey.com/polyhedra/SelfDualIcosioctahedron4.html</a></p>
<p>Some code:</p>
<pre><code>sage: P = polytopes.rhombic_dodecahedron()
sage: Q = po... |
72,651 | <p>$G$ is a group and $H$ is a subgroup of $G$ such that $\forall a, b$ in $G, ab\in H\implies ba\in H$. Show that $H$ is normal in $G$</p>
| Arturo Magidin | 742 | <p>Let $G$ be a group, and let $H$ be a subgroup. In analogy to the definition of congruence for integers, $a\equiv b\pmod{m}$ if and only if $m|a-b$, if and only if $a-b\in\langle m\rangle$, define:</p>
<ul>
<li>$a$ is congruent on the left modulo $H$ to $b$, $a\mathrel{{}_H{\equiv}} b$, if and only if $a^{-1}b\in H$... |
3,831,702 | <p>One can prove that for <span class="math-container">$x\in \mathbb{R}$</span>, the sequence
<span class="math-container">$$
u_0=x\text{ and } \forall n\in \mathbb{N},\qquad u_{n+1}=\frac{e^{u_n}}{n+1}
$$</span>
converges to <span class="math-container">$0$</span> if <span class="math-container">$x \in ]-\infty,\delta... | dan_fulea | 550,003 | <p>I am proving something about <span class="math-container">$\delta$</span>, namely the <em>divergence</em> of the sequence starting from <span class="math-container">$\delta$</span>, see the <em>Result</em> below. This was the question in the OP. Some numerical aid is added to show how the "sequence works",... |
220,618 | <p>The cyclic group of $\mathbb{C}- \{ 0\}$ of complex numbers under multiplication generated by $(1+i)/\sqrt{2}$</p>
<p>I just wrote that this is $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$ making a polar angle of $\pi/4$. I am not sure what to do next. My book say there are 8 elements. </p>
<p>Working backwards, maybe... | Community | -1 | <p><strong>HINT</strong> Find the minimum $k$ such that $\left(\dfrac1{\sqrt{2}} + \dfrac{i}{\sqrt{2}} \right)^k = 1$. It will be of help to write $\dfrac1{\sqrt{2}} + \dfrac{i}{\sqrt{2}}$ as $\exp(i \pi/4)$.</p>
<p>The element in the group generated by $r = \dfrac1{\sqrt{2}} + \dfrac{i}{\sqrt{2}}$ will then be $$\{r^... |
3,353,483 | <p>I have just begun studying finite fields today, and it is clear in GF(2) why 1+1=0. (I just show that 1+1 can't equal 1, or 1=0, which contradicts an axiom that states that 1 is not 0).</p>
<p>If we interpreted these symbols "1", "+", "1", "0" as we would in primary school, clearly this breaks arithmetic rules in R... | Robert Shore | 640,080 | <p>This actually brings up a subtle point. What do we mean by <span class="math-container">$5$</span> in a finite field? Or if you choose to define <span class="math-container">$5$</span> in terms of <span class="math-container">$1 ~(5=1+1+1+1+1)$</span>, then what do we mean by <span class="math-container">$1$</span... |
2,046,957 | <p>The random variable $X$ is $N(5,2)$ and $y=2X+4$.
Find:</p>
<p>a) $\eta_y$</p>
<p>b) $\sigma_y$</p>
<p>c) $f_Y(y)$</p>
<p>My attempt:</p>
<p>I have solved a and b as follow:</p>
<p>a) $\eta_y = 2\eta_X+4 = 14$</p>
<p>b) $\sigma_y^2 = 4\sigma_{x}^2 = 16, \sigma_y = 4$</p>
<p>c) how can I solve $f_Y(y)$?</p>
| caverac | 384,830 | <p>You need to guarantee that </p>
<p>$$
f_X(x)dx = f_Y(y)dy ~~~\Rightarrow~~~ f_Y(y) = f_X(x)\left| \frac{dx}{dy} \right|
$$</p>
<p>where the absolute value is used here to ensure that $f_Y(y) \ge 0$. In this case $dx/dy = 1/2$ therefore</p>
<p>$$
f_Y(y) = \frac{1}{2}\frac{1}{\sqrt{2\pi \sigma_x^2}} e^{-\left(\frac... |
3,361,489 | <p><strong>Question:</strong></p>
<p>Do there exist functions <span class="math-container">$f$</span> and <span class="math-container">$g$</span> such that
<span class="math-container">$$\lim_{x \to c} f(x) = 1 \text{ and } \lim_{x \to c} f(x) g(x) - g(x) \neq 0 \, ?$$</span>
(Allowing, of course, for <span class="ma... | David C. Ullrich | 248,223 | <p><span class="math-container">$c=0$</span>, <span class="math-container">$f(x)=1+x$</span>, <span class="math-container">$g(x)=1/x$</span>.</p>
|
434,614 | <p>Is there a name for a type of grid you might find in Battleship? Where coordinates don't relate to points on a grid but rather the squares themselves?</p>
| Will Orrick | 3,736 | <p>How about <a href="http://en.wikipedia.org/wiki/Alpha-numeric_grid" rel="nofollow">alpha-numeric grid</a>? That the coordinates refer to squares rather than vertices is no big deal. Squares can be identified with vertices in the <a href="http://en.wikipedia.org/wiki/Dual_graph" rel="nofollow">dual</a> of the squar... |
3,532,173 | <p>I have seen this problem somewhere on the internet but I could not prove it.</p>
<p>Let <span class="math-container">$$I_{0}=\int^{\infty}_{0}\frac{\sin x}{x}dx$$</span> and then define
<span class="math-container">$$I_{n+1}=\int^{I_{n}}_{0}\frac{\sin x}{x}dx.$$</span></p>
<p>Show that
<span class="math-container... | xpaul | 66,420 | <p>It is easy to see that <span class="math-container">$I_n\to0$</span> as <span class="math-container">$n\to\infty$</span>. So by Stolze's Theorem, one has
<span class="math-container">\begin{eqnarray}
\lim_{n\rightarrow\infty}n I^2_{n}&=&\lim_{n\rightarrow\infty}\frac{n}{I^{-2}_{n}}\\
&=&\lim_{n\right... |
2,767,392 | <p>I have the following curve:</p>
<p>$x^4=a^2(x^2-y^2)$</p>
<p>Prove that the area of its loop is $\frac{2a^2}{3}$.</p>
<p><strong>My approach</strong></p>
<p>This curve has four loops. So the required area should be:</p>
<p>$4\int_{0}^{a}\frac{x}{a}\sqrt{a^2-x^2} dx$</p>
<p>But, After solving, the area turned o... | Sri-Amirthan Theivendran | 302,692 | <p>We use the generalized hockey-stick identity
$$
\sum_{i=0}^k\binom{m+i}{i}=\binom{m+k+1}{k}\quad(m\in\mathbb{C})\tag{H.S.}
$$
which follows from the regular hockey-stick identity via the polynomial method to prove that
$$
\sum_{i=0}^k(-1)^{i}\binom{n}{i}
=(-1)^k\binom{n-1}{k}.$$
Indeed, since $(-1)^{i}\binom{n}{i}=(... |
2,438,795 | <blockquote>
<p>Why are any two initial objects of a category equivalent?</p>
</blockquote>
<p>By definition:</p>
<p>If $A,B$ are initial objects of a category $C$, then for each $X \in \text{obj } C$, there exists a unique morphisms $f : A \rightarrow X, g : B \rightarrow X$.</p>
<p>How can these be equivalent? ... | Community | -1 | <ul>
<li>Since $A$ is initial, there is a unique map $A \to B$. </li>
<li>Since $B$ is initial, there is a unique map $B \to A$. </li>
</ul>
<p>Furthermore, they are inverses; the products have to be identities because</p>
<ul>
<li>Since $A$ is initial, there is a unique map $A \to A$. </li>
<li>Since $B$ is initial,... |
158,483 | <p>When I started mathematica, this message popped up.</p>
<pre><code>Part::partw: Part 5 of PacletManager`Utils`Private`$taskData[2] does not exist.
</code></pre>
<p>Does anyone know what this is? My version is mma 11.2.</p>
| Stefan R | 5,678 | <p>We believe this should be fixed in <em>Mathematica</em> 12.0. We were never able to reproduce this internally so we're not completely sure about this, though. Please let us know if this continues to happen in 12.0 for you.</p>
|
651,731 | <blockquote>
<p>Let $\{f_n\}$be sequence of bounded real valued functions on $[0,1]$ converging at all points of this interval. Then
If $\int^1_0 |f_n(t)-f(t)|dt\, \to 0$ as $n \to \infty$,does $\lim_{n \to \infty} \int^1_0 f_n(t) dt\,=\int^1_0 f(t)dt\,$</p>
</blockquote>
<p>I just know that if somehow we can sho... | Did | 6,179 | <p>$$\left|\int^1_0 f_n(t)\,\mathrm dt-\int^1_0 f(t)\,\mathrm dt\right|\leqslant\int^1_0 \left|f_n(t) - f(t)\right|\,\mathrm dt$$</p>
|
2,555,200 | <p>Let $l$ be the smallest positive linear combination of $a,b\in \mathbb{Z}^+$ i.e.,$$l := \min\{ax+by >0 : x,y\in\mathbb{Z}\}.$$
Now, according to @Brahadeesh's answer here, <a href="https://math.stackexchange.com/questions/2553324/proof-for-gcd-being-the-smallest-linear-combination-of-a-b-in-mathbb-z">Proof for $... | user | 505,767 | <p>Since the g.c.d. is in the form:</p>
<p>$$l=ax+by$$</p>
<p>Every other $k>l$ in the form:</p>
<p>$$k=ax'+by'$$</p>
<p>is a multiple of $l$,</p>
<p>thus it can't divide a and b simultaneously.</p>
|
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Dick Palais | 7,311 | <p>Here is my rather personal answer. I like Wikipedia---both in general and the mathematically oriented part, and I would like to collaborate in its improvement and contribute in any way I can. But I don't know where to begin. With MO, when I have free time I can start "at the top" so to speak, and continue on down un... |
361,060 | <blockquote>
<p>Consider the ring of Gaussian integers $D=\lbrace a+bi\mid a,b \in \mathbb{Z \rbrace}$, where $i \in \mathbb{C}$ such that $i^2=-1$. Consider the map $f$ from $D$ to $\mathbb{Z}[x]/(x^2+1)$ sending $i$ to the class of $x$ modulo $x^2+1$. Show that $f$ is a ring isomorphism.</p>
</blockquote>
<p>I got... | Andreas Caranti | 58,401 | <p>As the others have said, you have also to specify what happens to the integers.</p>
<p>And then, it's the <em>inverse</em> map that occurs more naturally in the context of the structure of simple extensions. </p>
<p>Consider the ring homomorphism
$$
\begin{align}
\varphi : & \Bbb{Z}[x] \to \Bbb{Z}[i]\\
&a ... |
597,949 | <p>$\begin{cases}x\equiv 1 \pmod{3}\\
x\equiv 2 \pmod{5}\\
x\equiv 3 \pmod{7}\\
x\equiv 4 \pmod{9}\\
x\equiv 5 \pmod{11}\end{cases}$ </p>
<p>I am supposed to solve the system using the Chinese remainder theorem but $(3,5,7,9,11)\neq 1$
How can I transform the system so that I will be able to use the theorem?</p>
| John Butnor | 185,327 | <p>Solving the first two equations simultaneously you get X = 7(mod 15).</p>
<p>Solving the third and fourth simultaneously you get X = 31(mod 63).</p>
<p>Solving these two results simultaneously you get X = 157(mod 315).</p>
<p>Solving this result with the fifth equation simultaneously,</p>
<p>you get the final an... |
538,870 | <p>Evaluate</p>
<p>$$\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right).$$</p>
<p>This sequence looks extremely horrible and it makes me crazy. How can we evaluate this?</p>
| Start wearing purple | 73,025 | <p>Use <a href="http://en.wikipedia.org/wiki/Stirling_formula">Stirling's approximation</a>:
$$n!\sim\left(\frac{n}{e}\right)^{n}\sqrt{2\pi n}=\left(\frac{n}{e}\right)^{n}e^{\frac12 \ln 2\pi n}$$
It transforms your limit into
$$\lim_{n\rightarrow\infty}\frac{n}{\ln n}\frac{e^{\frac{\ln n}{2n}}-1}{e}=\lim_{x\rightarrow ... |
538,870 | <p>Evaluate</p>
<p>$$\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right).$$</p>
<p>This sequence looks extremely horrible and it makes me crazy. How can we evaluate this?</p>
| Mikhail Katz | 72,694 | <p>There is a gap in @O.L.'s solution as I explained in a comment below that solution. I don't have time now to provide a complete solution but the OP should be aware of this. I see that the wiki page provides a formula with an error term, but the $O(1/n)$ occurs inside an argument and is not stated as a theorem, nor i... |
354,885 | <p>Let <span class="math-container">$X_1,...,X_n$</span> be iid normal random variables. </p>
<p>I am looking for a strategy to establish the following limit for fraction of expectation values</p>
<p><span class="math-container">$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i < j\le n} \vert X_i-X_j \vert^{1/n... | Matt F. | 44,143 | <p>The numerator and denominator can also be written as
<span class="math-container">$$E\left[\exp\left(\frac1n\sum\ln|X_i-X_j|\right)\right]$$</span>
where the numerator has <span class="math-container">$n(n-1)/2$</span> summands and the denominator has <span class="math-container">$(n-1)(n-2)/2$</span> summands.</p>... |
1,299,552 | <p>Can you tell me why my answer is wrong?</p>
<p>$$\frac {x+y} {x-y} + \frac 1 {x+y} - \frac {x^2+y^2} {y^2-x^2} = \frac {x^2 + y^2} {x^2-y^2} + \frac {x-y} {x^2-y^2} + \frac {x^2+y^2} {x^2-y^2} = 2x^2 + 2y^2 + x-y$$</p>
| abel | 9,252 | <p>on the second line, the first expression you have $$\frac{x+y}{x-y} = \frac{x^2+y^2}{x^2-y^2} $$ it should be $$\frac{x+y}{x-y} = \frac{(x+y)^2}{x^2-y^2} = \frac{x^2+2xy + y^2}{x^2-y^2}$$ instead. the rest of them looks good. </p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | ThudanBlunder | 1,770 | <p>A regular tetrahedron and a regular square pyramid both have unit length. If a triangular face of the tetrahedron is glued to a triangular face of the square pyramid, the resulting shape has how many edges? </p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Oscar Lanzi | 248,217 | <p>A time bomb has 25 switches engaged, all in a row and numbered from 1 to 25. A spy has told you that you could defuse the bomb by flipping every switch, then flipping every multpile of 2, then flipping every multiple of 3, etc through multiples of 25 (then you are done, there are no numbers greater than equal to 26... |
3,255,174 | <p>How can I find the volume bounded between <span class="math-container">$z=(x^2+y^2)^2$</span> and <span class="math-container">$z=x$</span>? </p>
<p>My idea so far is to use cylindrical polar coordinates and <span class="math-container">$z$</span> limit is from <span class="math-container">$(x^2+y^2)^2$</span> to <... | user10354138 | 592,552 | <p>Clearly <span class="math-container">$x>0$</span>, or equivalently <span class="math-container">$\theta\in(-\frac\pi2,\frac\pi2)$</span>. Then
<span class="math-container">$$
(x^2+y^2)^2<x\iff r^4<r\cos\theta\iff 0<r^3<\cos\theta.
$$</span></p>
|
387,519 | <p>The domain of the following function $$y=2$$ is just 2? And the image of it?</p>
<p>I don't think I quiet understand what the image of a function means. The domain is all values that it can assume, correct?</p>
<p>Could you please try to define the image of this equation too: $$y = 2x - 6$$ so I can try to underst... | rurouniwallace | 35,878 | <p>An image is a subset of the co-domain with respect to a certain pre-image, which is a subset of the domain. For the function $y=2x-6$, for example, given a pre-image of $[2,10]$, the image is $[-2,14]$. For the function $y=2$, since any input value in the domain will result in $y=2$, besides the null set, the only ... |
387,519 | <p>The domain of the following function $$y=2$$ is just 2? And the image of it?</p>
<p>I don't think I quiet understand what the image of a function means. The domain is all values that it can assume, correct?</p>
<p>Could you please try to define the image of this equation too: $$y = 2x - 6$$ so I can try to underst... | Wishwas | 19,992 | <p>The value of f when applied to x is the image of x under f. y is alternatively known as the output of f for argument x.
Now Y=2 is a constant function. The image of Y is always 2 for any value of x.
So the Image of function Y=2 is the set containing element 2.
Image of 2, under Y=2x−6, is the set containing the elem... |
387,519 | <p>The domain of the following function $$y=2$$ is just 2? And the image of it?</p>
<p>I don't think I quiet understand what the image of a function means. The domain is all values that it can assume, correct?</p>
<p>Could you please try to define the image of this equation too: $$y = 2x - 6$$ so I can try to underst... | Alistair Savage | 74,366 | <p>This really depends on the context of the question. I'm going to make the assumption here that you're discussing functions from the real numbers to the real numbers. Strictly speaking, the domain of the function should be given explicitly when you define the function. However, often one defines a function and the... |
506,767 | <p>What is the largest number on multiplying with itself gives the same number as last digits of the product?</p>
<p>i.e., $(376 \times 376) = 141376$</p>
<p>i.e., $(25\times 25) = 625$</p>
<p>If the largest number cant be found out can you prove that there is always a number greater than any given number? (only in ... | Leen Droogendijk | 95,972 | <p>The numbers can be arbitrarily large.
Take an arbitrary number $k$, we will produce a number with the desired property that is
at least as big as $5^k$.</p>
<p>$2^k$ and $5^k$ are coprime, so we can find integers $p,q$ such that $p2^k+q5^k=1$, where we
may choose $q$ to be a positive number less than $2^k$.
Now $q5... |
4,196,125 | <p>Can someone tell me where this calculation goes wrong?
I get (2 3 4)(1 2 3 4 5 6)^-1 = (1 6 5 2).
My book and Mathematica get (1 6 5 4).
I have read several explanations of how to multiply permutations in cycle notation and have
worked dozens of examples successfully, but I always get this one wrong.</p>
<p>(2 3 4)(... | Henry Lee | 541,220 | <p>yep so what you have is the circle:
<span class="math-container">$$x^2+(y-2)^2=1$$</span>
rotated about the <span class="math-container">$y$</span>-axis. Rearranging you have:
<span class="math-container">$$y=\pm\sqrt{1-x^2}+2$$</span>
just remember to account for above and below the axis. I am not sure where the <s... |
222,105 | <p>I want to use geometric shapes in Mathematica to build complex shapes and use my raytracing algorithm on it. I have a working example where we can get the intersections from a combination of a <code>Cone[]</code> and <code>Cuboid[]</code>, e.g </p>
<pre><code>shape1 = Cone[];
shape2 = Cuboid[];
(* add shapes in thi... | Tim Laska | 61,809 | <p>This is not a direct answer to your question, but an alternate approach. You could create a list of primitives and a build function that contains the Computational Solid Geometry (CSG).</p>
<pre><code>square = Cuboid[];
ball = Ball[{0, 0, 1}, 1];
buildList = {square, ball};
(* Constraints *)
buildFn = ¬ #2 ∧ #1 &a... |
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