qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
4,788 | <p>Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial of degree n has <strong>at most</strong> n roots) is not considered fundamental by algebraists as it's not needed fo... | T.. | 467 | <p>It's easy to be jaded about FTA after mathematical history has run for another two centuries. Sure, it is not the number one most important result any more, or the center of any research program (though understanding the algebraic closure of Q could be considered as half of number theory). But consider the situati... |
438,440 | <p>The determinant map $\det$ sending an $n\times n$ real matrix to its determiant is continuous since it's a polynomial in the coefficients. Is it also uniformly continuous?</p>
| Julien | 38,053 | <p>For $n=1$, clearly yes as $\det (a)=a$ is linear.</p>
<p>For $n\geq 2$, no. Consider the sequence of diagonal matrices $D_n=\mbox{diag}(\sqrt{n},\sqrt{n},1\ldots,1)$ and use, say, the operator norm inherited from the $\ell^2$ norm. Then $\|D_{n+1}-D_n\|=\frac{1}{\sqrt{n}+\sqrt{n+1}}\longrightarrow 0$ while $\det D_... |
3,465,963 | <p>A question from the CodeChef December Long Challenge:</p>
<blockquote>
<p>Given two binary numbers <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, each with <span class="math-container">$N$</span> bits. We may reorder the bits of <span class="math-container">$A$</span> in an arbi... | wendy.krieger | 78,024 | <p>You are free to select any four numbers. It is not clear if the same number can be selected twice. But we get</p>
<p><span class="math-container">$499 \cdot 498 \cdot 497 \cdot 496$</span> different sets of four numbers. It then suffices to add a number that makes this a multiple of five.</p>
<p>The fifth numbe... |
167,846 | <p>The question is the following:</p>
<p>Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: </p>
<p>When do we have $\widehat{HF}(Y,\mathfrak{s})\cong \mathbb{Z}$? If yes, what do we know about the d-... | christian | 17,492 | <p>Let assume $Y$ is a closed oriented rational homology sphere.</p>
<p>The set of spin$^c$ structure of $Y$ is in one to one correspondence with $H_1(Y,\mathbb{Z})$. So if $H_1(Y,\mathbb{Z}) \neq 0$ then there are more than one spin$^c$ structure. </p>
<p>The question then becomes: When does $\widehat{HF}(Y,\mathfr... |
4,236,968 | <p>I want to define a function implicitly by <span class="math-container">$$x^y = y^x$$</span> with <span class="math-container">$$x \ne y$$</span>
such that <span class="math-container">$f: X \rightarrow Y$</span> is constrained by <span class="math-container">$X = \{ x \in \mathbb{R} | x > 1 \}$</span> and <span c... | Prasun Biswas | 215,900 | <p>Since <span class="math-container">$x,y\gt 0$</span> under consideration, we can use logarithms here.</p>
<p><span class="math-container">$$x^y=y^x\implies y\ln x=x\ln y\implies \frac{\ln x}x=\frac{\ln y}y$$</span></p>
<p>Now, define the map <span class="math-container">$\varphi\colon t\mapsto\dfrac{\ln t}t$</span> ... |
4,472,705 | <p>Say that <span class="math-container">$x \notin A \lor x \in B$</span>. Is <span class="math-container">$x\in(A\setminus B)^c$</span> ?</p>
<p>The answer is apparently yes, but I can't think of what logical intermediary steps I must take in order to reach this conclusion. Would kindly appreciate your assistance.</p>... | 温泽海 | 948,121 | <p><span class="math-container">$(A\setminus B)^c=(A \cap B^c)^c=A^c \cup B$</span>. The first equation comes from the definition of <span class="math-container">$\setminus$</span>. The second equation is de Morgan's law.</p>
|
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | TCL | 3,249 | <p>Consider $x-f(x)$ and use Intermediate Value Theorem.</p>
|
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | Alex Basson | 506 | <p>Let $g(x) = f(x) - x$. Then $g$ is continuous and $g(a) \geq 0$ while $g(b) \leq 0$. By the Intermediate Value Theorem, $g$ has at least one zero on $[a, b]$.</p>
|
9,801 | <p>If I write:</p>
<pre><code>Histogram[{1, 1, 1, 2, 2, 2}]
</code></pre>
<p>I get a nice histogram chart</p>
<p>but if I write</p>
<pre><code>Histogram[{"A", "A", "A", "B", "B", "B"}]
</code></pre>
<p>I get an <strong>empty</strong> chart!!</p>
<p>How can I tell mathematica to generate an histogram from <strong>... | Ajasja | 745 | <p>One posibility is to use <code>Tally</code> with <code>BarChart</code>.</p>
<p>For example:</p>
<pre><code>hist = Tally[{"A", "A", "B", "B", "B"}];
BarChart[hist[[All, 2]], ChartLabels -> hist[[All, 1]]]
</code></pre>
<p><img src="https://i.stack.imgur.com/hjLjf.png" alt="bar chart"></p>
|
9,801 | <p>If I write:</p>
<pre><code>Histogram[{1, 1, 1, 2, 2, 2}]
</code></pre>
<p>I get a nice histogram chart</p>
<p>but if I write</p>
<pre><code>Histogram[{"A", "A", "A", "B", "B", "B"}]
</code></pre>
<p>I get an <strong>empty</strong> chart!!</p>
<p>How can I tell mathematica to generate an histogram from <strong>... | kglr | 125 | <p>Here is a variation that works with multiple data sets:</p>
<p><strong>BarChart: pure categorical variables</strong></p>
<pre><code> ClearAll[mapCount]
mapCount[dataset_] :=
Map[Function[{elem}, {elem, Count[#, elem]}], Union @@ dataset] & /@ dataset
data2 = {{"A", "A", "B", "B", "B"}, {"A", "A", "A", "A... |
255,416 | <p><em>(<strong>Update</strong>)</em>:
Courtesy of Myerson's and Elkies' answers, we find a second <em>simple</em> cyclic quintic for $\cos\frac{\pi}{p}$ with $p=10m+1$ as,
$$F(z)=z^5 - 10 p z^3 + 20 n^2 p z^2 - 5 p (3 n^4 - 25 n^2 - 625) z + 4 n^2 p(n^4 - 25 n^2 - 125)=0$$
where $p=n^4 + 25 n^2 + 125$. Its discriminan... | Noam D. Elkies | 14,830 | <p>Question 1: Yes; in fact
$$
\sum_{n\bmod 5} y_n y_{n+1} = \! \sum_{n\bmod 5} y_n y_{n+2} = -p/5
$$
for the "depressed" $y_n$, using an order consistent with the
action of the cyclic Galois group. Likewise "for other values of 5".</p>
<p>This follows from the formula $|\tau|^2 = p$ for the absolute value of
a nontr... |
259,150 | <p>An airline is given permission to fly $4$ new routes of its choice. The airline is
considering $12$ new routes: $4$ routes in Florida, $5$ routes in California, and $3$ routes
in Texas. If the airline selects the $4$ new routes at random from the $12$
possibilities, determine the probability that:</p>
<p>a) $2$ are... | ashley | 50,188 | <p>a.) $(\dbinom{4}{2}+\dbinom{3}{1})/\dbinom{12}{4}$,</p>
<p>b.) $(\dbinom{5}{3}+\dbinom{4}{1})/\dbinom{12}{4}$,</p>
<p>c.) $(\dbinom{4}{1}+\dbinom{5}{1}+\dbinom{3}{2})/\dbinom{12}{4}$,</p>
<p>b.) $1-(\dbinom{9}{4}/\dbinom{12}{4}$.</p>
<p>EDIT: added the TeX tags. </p>
|
205,871 | <p><strong>Question.</strong> Do you know a specific example which demonstrates that the tensor product of monoids (as defined below) is not associative?</p>
<hr>
<p>Let $C$ be the category of algebraic structures of a fixed type, and let us denote by $|~|$ the underlying functor $C \to \mathsf{Set}$. For $M,N \in C$... | HeinrichD | 369,530 | <p><i>Claim.</i> The tensor product of monoids is not associative: We have <span class="math-container">$\mathbb{N} \otimes (\mathbb{N}^2 \otimes \mathbb{N}^2) \not\cong (\mathbb{N} \otimes \mathbb{N}^2) \otimes \mathbb{N}^2$</span>.</p>
<p>First, let us look in general what <span class="math-container">$\mathbb{N} \o... |
98,565 | <pre><code>as = <|1 -> 2.2, 2 -> 3.4, 3 -> 8.1|>;
</code></pre>
<p>I don't understand the following:</p>
<pre><code>as[[1]]
</code></pre>
<blockquote>
<p>2.2</p>
</blockquote>
<p>But I expected to see</p>
<blockquote>
<p><|1 -> 2.2|></p>
</blockquote>
<p>On the other hand:</p>
<pre><code>as... | Graumagier | 25,421 | <p>The difference could be related to the fact that <code>Part</code> (<code>as[[1]]</code>) treats associations transparently while <code>as[[1;;1]]</code> (i.e. <code>as[[1~Span~1]]</code>) is equivalent to <code>Take[as[1,1]]</code>, and <code>Take</code> does not treat associations transparently.</p>
|
306,754 | <p>Looked around a bit and all I see are proofs using the limit definition of a derivative. This is not for an assignment, I could just use the limit definition if I wanted to, but I was wondering how you could go about proving this using the epsilon-delta definition of a derivative ($\forall \epsilon >0$, $\exists ... | thelionkingrafiki | 116,382 | <p>I know this post is somewhat old, but I'll post a thorough answer anyway so it may benefit future readers.</p>
<p>To prove this, we will want to use the limits $\frac {\sin x}x \to 1$ and $\frac {1 - \cos x}x \to 0$ when $x \to 0$. By the $\epsilon-\delta$ definition, we can write (quantifiers left out, assuming re... |
3,213,362 | <p><span class="math-container">$$f(x)=\frac{x+2}{1-2x}$$</span>
<span class="math-container">$$g(x)=\frac{2x+1}{2-x}$$</span></p>
<p>Find
<span class="math-container">$$(fofofo...ofOgogo...og)=\frac{1}{x}$$</span>
{fofo... are 101 times and gogo.. are 100 times}</p>
<p>Then Find <span class="math-container">$x$</spa... | Allawonder | 145,126 | <p>Composition of functions is not -- in general -- a commutative process, so what you've computed is apparently different from what the problem specifies. You are not to rearrange the order of composition for different functions unless you've first shown that it is allowable.</p>
<p>What you need may be a little tena... |
853,143 | <p>Let $J$ be a canonical Jordan form (real or complex). Is it true that the $2$-norm of $J$ is equal to its spectral radius?</p>
| cirpis | 152,276 | <p>If $n\geq k+1$ then one of the factors of $n!$ is obviously going to be $k+1$ since the factorial of $n$ is the product of all natural numbers less than or equal to $n$, thus you can always factor out $k+1$ in $n!+k+1$ if $n\geq k+1$.</p>
|
2,929,182 | <p>Let <span class="math-container">$p,q>1$</span>, <span class="math-container">$\frac{1}{p}+\frac{1}{q}=1$</span> and <span class="math-container">$x\in[0,1]$</span>. I would like to show that <span class="math-container">$x^{\tfrac{1}{p}}\leq \frac{1}{p}x+\frac{1}{q}$</span>.</p>
<p>I tried to rewrite both sides... | user | 505,767 | <p>We have that </p>
<p><span class="math-container">$$ x^{\tfrac{1}{p}}\le \frac{1}{p}x+\frac{1}{q}\iff \frac1p \log x +\frac1q \log 1 \le \log \left(\frac1p x+\frac1q \right)$$</span></p>
<p>then recall that <span class="math-container">$\log$</span> function is concave and refer to <a href="https://artofproblemsol... |
2,828,060 | <blockquote>
<p>Let $f(x)=x^3 +4x^2 +6x$ and $g(x)$ be its inverse. Then the value of $g'(-4)$ is?</p>
</blockquote>
<p>My attempt at the solution:
$$
g(x) = f^{-1}(x) \implies f(g(x))=x,
$$
<em>differentiating both sides,</em>
$$
f'(g(x)) \cdot g'(x)=1 \implies g'(x)=\frac1{f'(g(x))}.
$$
I am stuck here as we do no... | joriki | 6,622 | <p>In the image, the curve seems to be the envelope of the lines whose $x$ and $y$ intercepts add to $1$. These lines are given by</p>
<p>$$
y=y_0\left(1-\frac x{1-y_0}\right)\;,
$$</p>
<p>where $y_0$ is the $y$ intercept.</p>
<p>The envelope maximizes $y$ with respect to $y_0$ for given $x$. Setting the derivative ... |
463,353 | <p>How do I need to modify this in order for it to be correct?</p>
<p>The center of $S_n$ (for $n\geq$ 3) is the trivial identity. Proof: Assume the center of $S_n$ is $C = \{ id , \tau \}$ where $ \tau \in S_n$ and $\tau \neq \ id$. Then for some $n$ the factor group $S_n\backslash C$ is abelian and solvable, a contr... | Souvik Dey | 42,041 | <p>"Let $S$ be a set having more than two elements , then the center of $A(S)$ , the permutation group of $S$ , is trivial" . </p>
<p>Let $ id \ne \sigma \in A(S) $ , then $\exists a \in S $ such that $b:=\sigma(a)\ne a$ . As $S$ has more
than two elements , </p>
<p>$\exists c \in S $ such that $c \ne a , c \ne b$ .... |
50,864 | <p>I have a column vector of the form $\{1, 1, 2, 3, 2, 3\}^t$ and an associated matrix that has the same number of rows as the column vector, $6$, and a number of columns that is equal to the maximum of the values in the vector, $3$.</p>
<p>I would like to construct a matrix that has a specific value at the positions... | ciao | 11,467 | <pre><code>(* make some col vec *)
cv = Transpose[{{1, 1, 2, 3, 2, 3}}]
(*Use it to fill positions in associated array with e.g. twos *)
result=Normal@SparseArray[
Flatten /@ Transpose[{Range@Length@#, #}] -> #2, {Length@#, Max@#}] &[cv, 2]
(* {{2, 0, 0}, {2, 0, 0}, {0, 2, 0}, {0, 0, 2}, {0, 2, 0}, {0,... |
581,605 | <p>Source: Miklos Bona, A Walk Through Combinatorics.</p>
<p>$$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$</p>
<p>The RHS is the upper bound of the Ramsey number $R(k,k)$.</p>
<p>How can I prove the inequality without using mathematical induction? I've merely expanded the LHS to obtain $\frac{(2k-2)!}{(k-1)!(k-... | lab bhattacharjee | 33,337 | <p>$$\text{From }\frac{n+1}{(n+2)!},$$
$$=\frac{n+2-1}{(n+2)!}=\frac1{(n+1)!}-\frac1{(n+2)!}$$</p>
<p>Can you identify the <a href="http://en.wikipedia.org/wiki/Telescoping_series">Telescoping series</a>?</p>
<p>The survivor will be $$\frac1{2!}$$</p>
|
185,527 | <p>I would love to understand the famous formula <span class="math-container">$g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(\|x\|^3)$</span>, which is valid in Riemannian normal coordinates and possibly more general situations.</p>
<p>I'm aware of 2 proofs: One using Jacobi fields [cf. e.g. S.Sternberg's &quo... | dennis | 485,792 | <p>There is the following way of doing it. Let <span class="math-container">$\Gamma_{ijk}\equiv g_{is}\Gamma^{s}_{jk}$</span>. Then (see <a href="https://math.stackexchange.com/questions/4631639/metric-in-terms-of-the-connection-on-mathbb-r3">here</a>)
<span class="math-container">$$
g_{jk}(x)=\delta_{jk}+\int_0^1 \alp... |
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | Rocket Man | 85,096 | <p>It is known that
$$\sum_{k=1}^nk=\frac{n(n+1)}{2}.$$
Thus the value of your sum would be
$$\sum_{k=1}^nk+\sum_{k=1}^{n-1}k=\frac{n(n+1)}{2}+\frac{(n-1)(n)}{2}=\frac{n^2+n+n^2-n}{2}=\frac{2n^2}{2}=n^2.$$</p>
|
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | Argon | 27,624 | <p>In Zeilberger fashion: Plug in $n=2, 3$ into the LHS to get $4, 9$. Fit a quadratic to that and get $n^2$. Then to complete the proof, simply note</p>
<p>$$
\left(\sum^{n+1}_{i = 1}i + \sum^{n}_{i=1}i\right) - \left(\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i\right) = n+n+1 = (n+1)^2-n^2
$$</p>
|
2,421,896 | <p>I have the following integral to find:</p>
<p>$$\int 12x^2(3+2x)^5 dx$$</p>
<p>Now, I am aware of the integration by parts property - </p>
<p>$$\int \ u \frac{dv}{dx} = uv - \int v\frac{du}{dx}$$</p>
<p>Now, my question is the following - </p>
<p>When I make $u = 12x^2$, I find a different answer to when I make... | Vasili | 469,083 | <p>It does not matter to the final result what function you pick as $u$. You just need to determine what substitution will let you to get the result faster.</p>
|
379,140 | <p>I have this irregular line and I want to split it in, for example, ten equal parts. How can I do that?</p>
<p><img src="https://i.stack.imgur.com/4AcBP.png" alt="Example"></p>
<p>Thank you!</p>
| Abel | 71,157 | <p>Since the $D_i$ are assumed to be open, we have the following:</p>
<p>Let $f\colon S^1\to \bigcup_i D_i$, then $\mathrm{im}(f)$ is compact and covered by $\bigcup_iD_i$. Hence there is a finite set $I\subset \mathbb{N}$ such that $\mathrm{im}(f)\subseteq\bigcup_{i\in I} D_i = D_{\max(I)}$.</p>
<p>Since $D_{\max(I... |
339,806 | <p>The definition of the integral I was given (which after searching around seems like the common definition) is the value of the inf{upper sums across all dissections} (integral exists when this coincides with the sup{lower sums across all dissections}). </p>
<p>Now, when I searched online of how to do the integral i... | 2'5 9'2 | 11,123 | <p>There is a <a href="http://www.proofwiki.org/wiki/Continuous_Function_is_Riemann_Integrable" rel="nofollow">theorem</a> that proves that continuous functions over closed intervals really are <em>Riemann-integrable</em>. The definition of that term, is that the infimum and supremum you speak of really will equal each... |
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | UltiKhopdi 86 | 365,645 | <p>Your scenario : -
1 - 1 * 5 votes
2 - 2 * 3 votes
3 - 3 * 1 vote
4 - 4 * 17 votes
5 - 5 * 2 votes</p>
<p>Your to total number of votes casted are :-
5+3+1+17+2 = 28 </p>
<p>so my friend to calculate the correct average divisor has to be 28
Now just add all up - 1*5 + 2*3 + 3*1 + 4*17 + 5*2 = 92</p>
<p>So you av... |
588,057 | <p>I just attempted to solve a problem that involved the title. </p>
<p>So, it turns out I was getting an incorrect answer b/c I computed the probability of getting a 3 as 2/6 as opposed to 2/7. I don't fully understand why the probability of getting a 3 would be 2/7 though. </p>
<p>If I think about it like if you ha... | Alex Botev | 39,340 | <p>Okay so you understood probability good, but to some extend you are asking a wrong question. The problem is, that when we say a dice which comes up 3 twice more often than any other side, you imagine a standard 6 sided die. So the question you must be asking is in fact, how can at all there exists a dice that lands ... |
588,057 | <p>I just attempted to solve a problem that involved the title. </p>
<p>So, it turns out I was getting an incorrect answer b/c I computed the probability of getting a 3 as 2/6 as opposed to 2/7. I don't fully understand why the probability of getting a 3 would be 2/7 though. </p>
<p>If I think about it like if you ha... | Community | -1 | <p>Hint:</p>
<pre><code>Dice number | Relative Probability
------------+---------------------
1 | 1
2 | 1
3 | 2
4 | 1
5 | 1
6 | 1
------------+---------------------
Total | 7
</code></pre>
|
3,688,208 | <p>Let <span class="math-container">$f: ]0,1[ \to \mathbb{R}$</span> be a function. Suppose that for every sequence <span class="math-container">$(\epsilon_n)_n$</span> in <span class="math-container">$]0,1[$</span> with <span class="math-container">$\epsilon_n \searrow 0$</span> we have that <span class="math-containe... | Will Jagy | 10,400 | <p>I find it clarifies things to use Lagrange Multipliers on the ratio. In this case, because of the absolute sign, we stick to the first quadrant. Multipliers says take</p>
<p><span class="math-container">$$ x = A t^3 \; , \; \; y = B t^2 $$</span></p>
<p>so the fraction gives
<span class="math-container">$$ \fra... |
1,393,423 | <p>Given the function</p>
<p>$$y=Ax + B\sqrt x$$</p>
<p>where $A$ and $B$ are real constants, $x$ is real and $x > 0$</p>
<p>I want to find the inverse where $x$ is a function of $y$. ButI don't believe that's possible without approximating the square root term, right?</p>
<p>Either
$$y=B\sqrt x$$
or
$$y=Ax $$</... | Terra Hyde | 253,768 | <p>It may not work in a very pretty way, but you could write
$$0=Ax+B\sqrt{x}-y$$
and solve this for $\sqrt{x}$ using the quadratic formula to find
$$\sqrt{x}=\frac{-B\pm\sqrt{B^2+4Ay}}{2A}$$</p>
<hr>
<p>EDIT: from there, simply square and take the domain of definition based on $A$, $B$, and $y$. You get
$$x=\frac{2B... |
1,188,196 | <p>This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at it.</p>
<p>Considering the image below, I am trying to work out a set of formulas that will specify either the radi... | GEdgar | 442 | <p>I got this... I used Maple. </p>
<p><img src="https://i.stack.imgur.com/jADXo.gif" alt="circs"></p>
<p>let $f$ be the solution of $Z^8+191 Z^7-840 Z^6+1366 Z^5-1051 Z^4+369 Z^3-22 Z^2-20 Z+4=0$<br>
with $f \approx 0.490892451047172$. </p>
<p>The red circle has center $(0,0)$ and radius $1$. </p>
<p>The orange... |
3,812,841 | <p>I would like to prove the following: <span class="math-container">$$\sum_{m=i}^{n}2^{n-m}\binom{m}{i} = \binom{n+1}{i+1}+\binom{n+1}{i+2}+\ldots+\binom{n+1}{n+1} = \sum_{m=i}^{n}\binom{n+1}{m+1}$$</span>
which is quite easy to prove by induction. I'm looking for an algebraic or combinatoric approach. This shouldn't ... | Arthur | 15,500 | <p><strong>Hint:</strong> Consider organizing the subsets by what the <span class="math-container">$(i+1)$</span>th element is.</p>
|
3,812,841 | <p>I would like to prove the following: <span class="math-container">$$\sum_{m=i}^{n}2^{n-m}\binom{m}{i} = \binom{n+1}{i+1}+\binom{n+1}{i+2}+\ldots+\binom{n+1}{n+1} = \sum_{m=i}^{n}\binom{n+1}{m+1}$$</span>
which is quite easy to prove by induction. I'm looking for an algebraic or combinatoric approach. This shouldn't ... | Marko Riedel | 44,883 | <p>Here is an algebraic proof. We get for the RHS</p>
<p><span class="math-container">$$\sum_{m=q}^n {n+1\choose m+1} =
\sum_{m=q}^n {n+1\choose n-m}
= [z^n] \sum_{m=q}^n z^m (1+z)^{n+1}.$$</span></p>
<p>Here the coefficient extractor enforces the range and we get</p>
<p><span class="math-container">$$[z^n] (1+z)^{n+1}... |
739,019 | <p>Can we evaluate $c$ such that the following series converges using the definition of limit of a sequence?
$$
s{_n}=\sum_{t=1}^{n}tc^{t}, c\in \mathbb{R}
$$</p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>You could write $$s{_n}=\sum_{t=1}^{n}tc^{t}=c\sum_{t=1}^{n}tc^{t-1}$$ and recognize that the right hand side summation is just the derivative of $ z=\sum_{t=1}^{n}c^{t}$ with respect ot $c$ and that this is a geometric progression. So, compute $z$, diffrentiate it with respect to $c$, ... |
494,239 | <p>How would you find the 4th term in the expansion $(1+2x)^2 (1-6x)^{15}$?</p>
<p>Is there a simple way to do so?</p>
<p>Any help would be appreciated</p>
| Adi Dani | 12,848 | <p>$$(1+2x)^2 (1-6x)^{15}=\sum_{i=0}^{2}\binom{2}{i}(2x)^i\sum_{j=0}^{15}\binom{15}{j}(-6x)^j=$$
using $i+j=3$ for fourth term we get
$$\sum_{i+j=3}\binom{2}{i}2^i\binom{15}{j}(-6)^jx^3=$$
$$=\left(\binom{15}{3}(-6)^3+4\binom{15}{2}(-6)^2+4\binom{15}{1}(-6)^1\right)x^3=$$
$$=(-98280+15120-360)x^3=-83520x^3$$</p>
|
3,807,296 | <p>Suppose you have a triangulated region in the plane, the triangulation consisting of <span class="math-container">$n$</span> triangles. Take an arbitrary triangle of this triangulation and call it <span class="math-container">$\Delta_i$</span> with <span class="math-container">$1\leq i\leq n$</span>.</p>
<p>The neig... | Trebor | 584,396 | <p>Note that the property you stated only refers to "local" properties of the triangulation. However, the 4-color theorem is a global property (i.e. involving all parts of the triangulation at the same time). So in general it is not a valid logical inference.</p>
<p>Indeed, the 4-color theorem depends in an e... |
3,691,993 | <p>According to the figure, CA is tangent to the circle, centre O, at A. ABT and POT are straight lines. </p>
<p><strong>Question:</strong></p>
<p>Given that BT is equal to the radius of the circle, prove that:</p>
<ol>
<li><span class="math-container">$\angle ABP = 3 \angle OBP$</span> </li>
<li><span class="math-c... | sirous | 346,566 | <p>a):</p>
<p><span class="math-container">$BOT=2PBO$</span></p>
<p><span class="math-container">$ABO=2BOT=4PBO$</span></p>
<p><span class="math-container">$ABP=ABO-PBO=4PBO-PBI=3PBO$</span></p>
<p>b):</p>
<p><span class="math-container">$ABP+\frac{1}{3}ABP=2BOT$</span></p>
<p><span class="math-container">$\frac{... |
3,198,705 | <p>I have this problem:</p>
<blockquote>
<p>In a game, the probability of win is <span class="math-container">$1/3$</span> and of lose is <span class="math-container">$2/3$</span>, ¿What
is the probabilty of win at least 1 prize playing 3 times?</p>
</blockquote>
<p>The probability of win at least 1 prize, playin... | Tom Chen | 117,529 | <p><a href="http://stats.stackexchange.com/questions/7977/how-to-generate-uniformly-distributed-points-on-the-surface-of-the-3-d-unit-sphe#comment13055_7984">It is known that</a> the uniform distribution on the unit <span class="math-container">$(n-1)$</span>-sphere can be represented as a standard multivariate Gaussia... |
3,103,639 | <p>Let's say I want to evaluate the following integral using complex methods - </p>
<p><span class="math-container">$$\displaystyle\int_0^{2\pi} \frac {1}{1+\cos\theta}d\theta$$</span></p>
<p>So I assume this is not very hard to be solved using real analysis methods, but let's transform the problem for the real plane... | Paras Khosla | 478,779 | <p>Maybe you're confusing what <span class="math-container">$\mid z\mid =1$</span> means. It simply is the equation of a circle of radius <span class="math-container">$1$</span> and it is slightly ill-defined in the context of the integral. When you make that transform, you must specify the bounds on <span class="math-... |
2,002,014 | <p>Is the following set of vectors independent or dependent:</p>
<p>$$(1,-2,5,-3)^T, \;\;(2,3,1,-4)^T \;\; and \;\; (3,8,-3,5)^T \;\;?$$</p>
<p>How do I determine whether this set is dependent or independent ?</p>
| Parcly Taxel | 357,390 | <p>$$|AP|^2=(x+1)^2+(y-3)^2$$
$$|BP|^2=(x-2)^2+(y-7)^2$$
The equation of $C$ may be worked out as follows:
$$|AP|=\alpha|BP|$$
$$|AP|^2=\alpha^2|BP|^2$$
$$(x+1)^2+(y-3)^2=\alpha^2((x-2)^2+(y-7)^2)$$
$$x^2+2x+1+y^2-6y+9=\alpha^2(x^2-4x+4+y^2-14y+49)$$
$$(\alpha^2-1)x^2+(\alpha^2-1)y^2-(\alpha^2+\tfrac12)(4x)-(\alpha^2-\... |
2,696,400 | <p>The Peano axioms are intended to be able to prove very general statements about arithmetic, such as "all natural numbers can be written as the sum of two primes".</p>
<p>However, how can we use the peano axioms to mathematically derive all the rules that are being taught to primary school children, about how to add... | Bram28 | 256,001 | <p>I would say there is a big difference between arithmetical <em>theorems</em> that you can derive from the Peano axioms, and the <em>algorithms</em> (i.e. <em>computations</em>) that we are taught in elementary school to do things like multiplication or long division. </p>
<p>I suppose you could still show that thos... |
123,844 | <p>Show that the solution to
$$T(n) = 2T\left(\biggl\lfloor \frac n 2 \biggr\rfloor+17\right)+n$$
is $\Theta(n \log n)$?</p>
<p>So the induction hypothesis is
$$ T \left( \frac n 2 \right) = c\cdot \frac n2 \cdot \log \frac n2.$$
Hence,
$$ T(n) = 2c \cdot \frac n2 \cdot \log \frac n2 + 17 + n $$</p>
<p>but how do ... | dato datuashvili | 3,196 | <p>Write it as: $$T(n)=2\cdot T\bigg(\frac{n+34}{2}\bigg)+n$$
Alternative form: if we continue, we get:
$$\frac1{2} \bigg(-34+n\bigg) c+\frac1{2} \bigg(-34+n\bigg) \bigg(68 \bigg(1-\frac1{-34+n}\bigg)+\frac{(2 \log(-34+n))}{\log2}\bigg)$$ </p>
|
954,377 | <p>Let $M\in GL_n(\mathbb{R})$ such that all its coefficients are non zero.</p>
<p>How can one show that $M^{-1}$ has at most $n^2-2n$ coefficients equal to zero ?</p>
<p>I have no idea how to tackle that problem, I've tried drawing some contradiction if $M^{-1}$ had $n^2+1-2n$ zero coefficients but couldn't find any... | Najib Idrissi | 10,014 | <p>Equivalently you need to prove that $M^{-1}$ has at least $2n$ nonzero entries. Let's in fact prove that every row of $M^{-1}$ had at least two nonzero entries. Suppose that the $i$th row of $M^{-1}$ has less than two nonzero entries; of course it cannot have a zero row (otherwise it would be singular), so it has ex... |
2,414,640 | <p>I'm trying to implement Bowyer-Watson algorithm. Currently performance is stuck behind calculating the circumsphere of a tetrahedron.</p>
<p>I tried using this <a href="http://mathworld.wolfram.com/Circumsphere.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/Circumsphere.html</a>. It works but it is sl... | achille hui | 59,379 | <p>Following are some alternatives.</p>
<p><strong>Alternative I</strong> - compute barycentric coordinates of circumcenter $O$ directly. </p>
<p>Given a tetrahedron $T$ with vertices $v_0, v_1, v_2, v_3$. Let</p>
<ul>
<li>$V$ and $R$ be the volume and circumradius of $T$.</li>
<li>$S_i$ and $R_i$ be the area and ci... |
2,414,640 | <p>I'm trying to implement Bowyer-Watson algorithm. Currently performance is stuck behind calculating the circumsphere of a tetrahedron.</p>
<p>I tried using this <a href="http://mathworld.wolfram.com/Circumsphere.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/Circumsphere.html</a>. It works but it is sl... | J. M. ain't a mathematician | 498 | <p>I'm fond of <a href="http://dx.doi.org/10.1017/CBO9780511973611" rel="nofollow noreferrer">Miroslav Fiedler</a>'s method for determining the circumsphere. The starting point is the usual <a href="http://mathworld.wolfram.com/Cayley-MengerDeterminant.html" rel="nofollow noreferrer">Cayley-Menger matrix</a>:</p>
<p>$... |
459,677 | <p>How do I determine a Standard deviation with the mean and range known?</p>
| Clement C. | 75,808 | <p>You cannot; below are a few examples demonstrating the variance can be pretty much arbitrary.</p>
<p>Mean: $\frac{1}{2}$. Range: $[0,1]$.</p>
<ul>
<li><p>Example #1: $\operatorname{Bernoulli}(\frac12)$. Variance: $\frac14$.</p></li>
<li><p>Example #2: $\operatorname{Uniform}([0,1])$. Variance: $\frac{1}{12}$.</p><... |
1,703,491 | <p>I have the following problem:
$$\int\left(\frac{x+2}{x^2+x+1}\right)dx$$
I received this by simplification of another integral. But my question is how to procede from here. Is there a way to simplify this?</p>
| Ángel Mario Gallegos | 67,622 | <p><strong>Hint:</strong></p>
<p>You can re-write the integral as
\begin{align*}
\int\frac{x+\frac{1}{2}}{x^2+x+1}dx+\int\frac{\frac{3}{2}}{x^2+x+1}dx&=\frac{1}{2}\int\frac{d(x^2+x+1)}{x^2+x+1}+\frac{3}{2}\int\frac{1}{(x+\frac12)^2+\frac34}dx
\end{align*}</p>
|
1,599,840 | <p>Please help me write down a step by step solution to this question:</p>
<p>Let $T$ be a linear transformation from $V$ to $V$. Prove that there exists a non-zero linear transformation from $S$ from $V$ to $V$ such that $TS=0$ if and only if there exists a non-zero vector $v\in V$ such that $T(v)=0$. </p>
<p>Assume... | JMoravitz | 179,297 | <p><em>Assuming $V$ is finite-dimensional</em></p>
<p>$\Rightarrow)$ Via contrapositive:</p>
<p>Suppose that there does <em>not</em> exist a nonzero vector $v$ such that $T(v)=0$.</p>
<p>This directly implies that $T$ is invertible.</p>
<p>Suppose also that $TS=0$. Since $T$ is invertible we can left-multiply by ... |
2,842,177 | <blockquote>
<p>$n| 3^n +1$</p>
</blockquote>
<p>My progress so far:</p>
<ol>
<li><p>$3^n + 1$ is even , thus $n$ is also even</p></li>
<li><p>$3^n + 1 \equiv n \equiv 2 \mod 10$ or $3^n + 1 \equiv n \equiv 0 \mod 10$</p></li>
<li><p>$3^n + 1 \equiv n \equiv 1 \mod 3$ / <em>I'm not quite sure about this</em></p></l... | Joffan | 206,402 | <p>First we can say that clearly $n$ is not divisible by $3$.</p>
<p>And clearly $n=1$ is a solution because $1$ divides every integer.</p>
<p>So $n\mid 3^n+1 \implies 3^n\equiv -1 \bmod n$. We know from <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow noreferrer">Fermat's Little Theor... |
2,842,177 | <blockquote>
<p>$n| 3^n +1$</p>
</blockquote>
<p>My progress so far:</p>
<ol>
<li><p>$3^n + 1$ is even , thus $n$ is also even</p></li>
<li><p>$3^n + 1 \equiv n \equiv 2 \mod 10$ or $3^n + 1 \equiv n \equiv 0 \mod 10$</p></li>
<li><p>$3^n + 1 \equiv n \equiv 1 \mod 3$ / <em>I'm not quite sure about this</em></p></l... | sirous | 346,566 | <p>In addition to your solutions,Suppose n is odd, then :</p>
<p>$3^n+1=(4-1)^n+1= 4 k$. </p>
<p>So we must have $n|k$, that is n can be any factor of a composite odd number like k. </p>
|
1,710,762 | <p>I understand a counterexample is when $a=4, b=-3$ and $x=y=2$.</p>
<p>Yet, I get confused because according to this Bezout's identity it states that $ax+by=(a,b)$ Therefore shouldn't the above statement be true or am I misunderstanding the identity. </p>
| carmichael561 | 314,708 | <p>$d=(a,b)$ is the smallest natural number for which there exist $x,y\in\mathbb{Z}$ such that $ax+by=d$, but it is not the only natural number that has this property. In fact, all multiples of $d$ will. So if $d=1$, as in your example, then all natural numbers have this property.</p>
|
2,546,222 | <blockquote>
<p>If $p(x)$ is an irreducible polynomial of degree n in $F[x]$ then $F[x]/\langle p(x)\rangle \cong F(c)$ where c is a root of $p(x)$. Prove every element of $F(c)$ can be written uniquely as $a_0+a_1c+...+a_{n-}c^{n-1}$ for some $a_0,...,a_{n-1}\in F$.</p>
</blockquote>
<p>So to be unique there is onl... | Nightgap | 506,645 | <p>Choose a root $c$ of $p(x)$ and show that the homomorphism $F[x]\rightarrow F, \ q\mapsto q(c)$ has kernel $<p(x)>$. Then use the fundamental homomorphism theorem to obtain the desired isomorphism.</p>
<p>If you have an element $y\in F(c)$ the isomorphism above gives a corresponding polynomial $q$ with $q(c)... |
1,029,489 | <p>I am studying the book "introduction to set theory", by Donald Monk, and I am having difficulties to solve some exercises about proper classes, could anybody help me?</p>
<p>here they are:</p>
<p>Prove that:
there are proper classes A, B such that $A \cap B= 0$<br>
there are proper classes A, B such that $A \subs... | DeepSea | 101,504 | <p>Apply Cauchy-Schwarz inequality twice: $x^4 + y^4 \geq \dfrac{1}{2}\left(x^2+y^2\right)^2 \geq \dfrac{1}{2}\left(\dfrac{1}{2}\left(x+y\right)^2\right)^2 = \dfrac{1}{8}\left(x+y\right)^4$.</p>
|
500,678 | <blockquote>
<p>Let $a$ and $b$ belong to a group $G$. Find an $x$ in $G$ such that $xabx^{-1}= ba$.</p>
</blockquote>
<p>This is what I have done so far, but I am stuck and not sure if I am in the right direction:</p>
<p>$xabx^{-1} = ba$</p>
<p>Multiply both sides on the right by $x$.</p>
<p>$xabx^{-1}x = bax$.<... | mr nobody | 96,079 | <p>$x=a^{-1}$ or $x=b.$ </p>
<p>Proof: $xabx^{-1}=ba.$ By a little manipulation we get $xa=b(ax)b^{-1.}$ Now as $x,a,b$ are in $G$ and the structure of the equation is similar to the one in question, we get by comparing the two as $x=b.$ Similarly, $bx^{-1}=a^{-1}(x^{-1}b)a.$ So by comparison again we get $x^{-1}=a$ o... |
4,569,871 | <p>I have a question on the equality of the ring of regular functions <span class="math-container">$\mathcal{O}_X(X)$</span> on <span class="math-container">$X\subset\mathbb{A}^n$</span> and the affine coordinate ring <span class="math-container">$A(X)$</span>. Some sources state it is as an isomorphism (Harthshorne, c... | lhf | 589 | <p>The claim is false.</p>
<p>Take <span class="math-container">$g$</span> of even order and <span class="math-container">$h=g$</span>. Then <span class="math-container">$\operatorname{ord}(gh)=\operatorname{ord}(g^2)=\operatorname{ord}(g)/2\ne \operatorname{ord}(g)=\operatorname{lcm}(\operatorname{ord}(g), \operatorna... |
4,569,871 | <p>I have a question on the equality of the ring of regular functions <span class="math-container">$\mathcal{O}_X(X)$</span> on <span class="math-container">$X\subset\mathbb{A}^n$</span> and the affine coordinate ring <span class="math-container">$A(X)$</span>. Some sources state it is as an isomorphism (Harthshorne, c... | Mike | 544,150 | <p>Another way yet to see this claim is false:</p>
<ol>
<li><p>Let <span class="math-container">$\pi$</span> be any element of a group with <span class="math-container">$\pi \not = e$</span>. Then <span class="math-container">$\operatorname{ord}(\pi) = \operatorname{ord}(\pi^{-1}) \not = 1$</span>. What is <span class=... |
4,569,871 | <p>I have a question on the equality of the ring of regular functions <span class="math-container">$\mathcal{O}_X(X)$</span> on <span class="math-container">$X\subset\mathbb{A}^n$</span> and the affine coordinate ring <span class="math-container">$A(X)$</span>. Some sources state it is as an isomorphism (Harthshorne, c... | bof | 111,012 | <p>Let <span class="math-container">$X$</span> be a set with at least four elements, and let <span class="math-container">$\operatorname{Sym}(X)$</span> be the group of all bijections <span class="math-container">$f:X\to X$</span>. Every element of <span class="math-container">$\operatorname{Sym}(X)$</span> is the prod... |
3,059,444 | <p>Exercise asks to verify that the sum of three quantities x, y, z, whose product is a constant k, is minimum when these three quantities are equal.</p>
<p>This is my amateurish attempt:</p>
<ol>
<li><span class="math-container">$x + y + z = S$</span>;</li>
<li><span class="math-container">$x*y*z=k$</span>;</li>
<li... | Thomas Davis | 458,276 | <p>Note: if you have complex coefficients instead, the statement isn’t true.</p>
<p>As an example, consider <span class="math-container">$f(z) = (z-1)(z+1)(z-i)$</span>. <span class="math-container">$f$</span> has two real roots, but one complex root.</p>
|
177,102 | <p>I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real solutions. The equations were derived as the gradients of a sum-of-squares cost function, which I am attempting to find a... | Dima Pasechnik | 11,100 | <p>As far as I can see, you'd like to find a global minimum of $F_0(x)=\sum_{k=1}^{M_0} f_k(x)^2$, where $x=(x_1,\dots,x_{12})$. Equivalently, the problem is to find maximal $\mu$ so that $F(\mu,x):=F_0(x)-\mu\geq 0$ for all $x_1,\dots,x_{12}$.</p>
<p>Now, a sufficient condition for $F(\mu,x)\geq 0$ is that
$F(\mu,x)=... |
1,858,002 | <p>Is there any standard solution or way to solve the following integration</p>
<p>$$I=\int_0^{\pi} \left(\cos(\theta)\right)^n \cos(p\theta) d\theta$$</p>
<p>where, $n=0, 1, 2,\dots$ and $p=0, 1, 2,\dots$ and $p> n$</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
2,841,198 | <p>I have to find an upper bound as described in the title. First I'll give some background of the question.</p>
<p>For $k \in \mathbb{N}$, let $\alpha_k$ be the boundary of the square with vertices $k\pi(1+i)$, $k\pi(-1+i)$, $k\pi(-1-i)$ and $k\pi(1-i)$. Furthermore let $D = \{ z \in \mathbb{C}: z \neq 0 \quad \text{... | user | 505,767 | <p>We have that</p>
<p>$$AX=B$$</p>
<p>and by left multiplication by $A^{-1}$ and associativity we obtain</p>
<p>$$A^{-1}(AX)=A^{-1}B\implies (A^{-1}A)X=A^{-1}B\implies X=A^{-1}B$$</p>
<p>Recall also that matrix multiplication is not commutative and moreover in that case the multiplication $BA^{-1}$ is not well def... |
2,841,198 | <p>I have to find an upper bound as described in the title. First I'll give some background of the question.</p>
<p>For $k \in \mathbb{N}$, let $\alpha_k$ be the boundary of the square with vertices $k\pi(1+i)$, $k\pi(-1+i)$, $k\pi(-1-i)$ and $k\pi(1-i)$. Furthermore let $D = \{ z \in \mathbb{C}: z \neq 0 \quad \text{... | Cameron Buie | 28,900 | <p>There's no need to multiply by anything at all. An $m\times n$ matrix multiplied by an $n\times k$ matrix will be an $m\times k$ matrix. We cannot multiply an $m\times n$ matrix by a $p\times k$ matrix unless $n=p.$ Use this to find the size of $X,$ instead.</p>
|
1,563,820 | <p>In a question of Probability, I got the answer
$$\frac{\sum_{j=0}^N(j/N)^{n+1}}{\sum_{j=0}^N(j/N)^{n}}$$</p>
<p>Now I have to prove when $N\to\infty$ above expression approximately is $\frac {n+1}{n+2}.$ No idea how to proceed, kindly help in this regards.</p>
| Thomas Andrews | 7,933 | <p>Hint:</p>
<p>$$\frac{1}{N}\sum_{j=0}^N(j/N)^{n+1}\to \int_{0}^1 x^{n+1}\,dx$$</p>
<p>and:</p>
<p>$$\frac{1}{N}\sum_{j=0}^N(j/N)^{n}\to \int_{0}^1 x^{n}\,dx$$</p>
|
382,170 | <blockquote>
<p>Show that every $2\times 2$ matrix $A$, for which $A^2=-I$, is similar to $\begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}$.</p>
</blockquote>
<p>I need help proving this. Don't have any idea how to proceed. Obviously look at the matrix as a linear operator...</p>
| Ma Ming | 16,340 | <p>Your statement is wrong. Let $A=\pmatrix{\pm i&0\\0&\pm i}$, there are at least two different forms. It would be true if you assume $A$ is a real matrix, in that case the eigenvalues of $A$ should be different, obtaining $i$ and $-i$, the statement follows.</p>
|
1,877,254 | <p>For the integration of $\frac{x^3}{\sqrt{1+x^2}}$, can we use $tan(\theta)$ to substitute $x$, and then use $u= sec(\theta)$ later this proof? I got a solution which is $\frac{((sec^3 (tan^{-1} x))}{3}$$ - (sec (tan^{-1} (x)) + C)$. Is this method correct? </p>
| egreg | 62,967 | <p>Let's do the substitution $x=\tan\theta$, for $\theta\in(-\pi/2,\pi/2)$, so
$$
\sqrt{1+\tan^2\theta}=\frac{1}{\cos\theta}
$$
and
$$
dx=\frac{1}{\cos^2\theta}\,d\theta
$$
Then the integral is
$$
\int\tan^3\theta\cos\theta\frac{1}{\cos^2\theta}\,d\theta=
\int\frac{\sin^3\theta}{\cos^4\theta}\,d\theta
$$
Now use $\sin^... |
2,632,527 | <p>Suppose the product $z_1z_2$ of two complex numbers is a nonzero real constant. Show that $z_2$ = $k\overline{z}_1$, where $k$ is a real number.</p>
<p>Hint: $z_2$=$k$ conjugate of $z_1$</p>
| Deepak | 151,732 | <p>Let $z_1 = a + bi, z_2 = c + di$ where $a,b,c,d \in \mathbb{R}$.</p>
<p>So $z_1z_2 = (ac-bd) + i(bc+ad)$.</p>
<p>If the result is purely real, then $bc = -ad$. Either $b = d = 0$ in which case $z_1$ and $z_2$ are both real and the result immediately follows or $\frac db = - \frac ca = -k$ (say) where $k$ is a real... |
711,122 | <p>I'm trying to derive the equation </p>
<p>$$y = (2x-6)^4$$</p>
<p>I thought that it would be</p>
<p>$$\frac{dy}{dx} = 8(2x-6)^3$$</p>
<p>Wolframalpha says $dy/dx = 64(x-3)^3$</p>
<p>Who's correct? I thought it would be a simple calc-1 chain rule. </p>
| rlartiga | 93,314 | <p>$$[f(g(x))]'=g'(x)f'(g(x))$$
Taking $f(x)=x^4$ and $g(x)=2x-6$:
$$[(2x-6)^4]'=(2x-6)'4(2x-6)^3=8(2x-6)^3=8(2(x-3))^3=64(x-3)^3$$</p>
<p>As you can see they are equivalent</p>
|
26,019 | <p>This is a little bit of a niche topic.</p>
<p>I've dealt with a pretty bad dose of long COVID that has caused some serious gaps in my mathematics (basically causing terrible arithmetic skills and a really shaky foundation). Here's essentially what I'm dealing with. I've seen the mathematics in old courses (Ross' Ana... | Mark Fantini | 47 | <p>I would like to complement existing answers by suggesting <a href="https://en.wikipedia.org/wiki/Spaced_repetition" rel="nofollow noreferrer">spaced repetition</a>. I have been applying it myself and have found myself feeling like I had newfound powers.</p>
<p>I use a free app called <a href="https://mochi.cards/" r... |
3,670,178 | <p>Simplification: <span class="math-container">$x(t)=\exp{(-m\cdot t)}\cdot (a\cdot cos(k\cdot t) + b\cdot sin(k\cdot t)) = A\cdot \exp{(-m\cdot t)}\cdot cos(k\cdot t + c)$</span>.</p>
<p>There is no further explanation in the script, but I found this theorem: <a href="https://mathworld.wolfram.com/HarmonicAdditionTh... | Doug M | 317,162 | <p>There are two ways you can go by.</p>
<p>We can say <span class="math-container">$A = \text {sgn(a)}\sqrt {a^2+b^2}$</span> in wich case, <span class="math-container">$c\in (-\frac {\pi}{2}, \frac {\pi}{2}]$</span></p>
<p>Or we can say <span class="math-container">$A = \sqrt {a^2+b^2}$</span> in wich case the sign... |
2,298,032 | <p>A factory manufactures products in two grades of quality. 90% of its revenues comes from manufacturing the first-rate product. How will the revenues change if production of the first-rate product is increased by 10% while the production of second-rate product is decreased by 10%?</p>
| Community | -1 | <p>$$90\%\cdot r=r_1\implies10\%\cdot r=r_2$$</p>
<p>$$r'=110\%\cdot r_1+90\%\cdot r_2=(110\%\cdot90\%+90\%\cdot10\%)\cdot r$$</p>
|
1,231,156 | <p>I am supposed to give a 30-minute class presentation on any PDE subjects as end-of-semester project. Do you have any pet subject you would love to suggest? </p>
<p>I have very little applied maths in my background, therefore I am not very advanced in PDE; I took this class as elective only. Hence I prefer a subject... | user225318 | 225,318 | <p>From what you described, you have not covered Method of Characteristics in any detail at all? Then for a 30 minute presentation a reasonable thing to do would be to introduce the method of characteristics and discuss the shock formation for the <a href="http://en.wikipedia.org/wiki/Burgers%27_equation" rel="nofollow... |
2,860,013 | <p>I'm trying to prove that the following system of congruence equations has a solution:</p>
<p>$X \equiv 2 $ (mod $5^N$)</p>
<p>$X \equiv 1 $ (mod $7^N$)</p>
<p>$X \equiv 4 $ (mod $6^N-4$)</p>
<p>being $N$ an integer number, $N\geq 2$</p>
<p>I guess that this may be answered using the <a href="https://en.wikipedi... | Community | -1 | <p>Let $z:=y-x$ so that $z'=y'-1$. This turns the equation to</p>
<p>$$z(z'+1)=1$$ or $$\frac z{1-z}z'=1,$$</p>
<p>$$-z-\log(1-z)=x+C,$$</p>
<p>$$-y-\log(1-y+x)=C$$ and from the initial condition, $C=0$.</p>
<p>Now we have to solve</p>
<p>$$-1-\log(1-1+a)=0$$
and $a=e^{-1}$.</p>
<hr>
<p>Alternatively:</p>
<p>$... |
2,720,922 | <blockquote>
<p>With $0<a,b,c<1$,
\begin{align}
\begin{bmatrix}
X_n \\
Y_n
\end{bmatrix}=\begin{bmatrix}
a^2 & (1-b)^2 \\
(1-a)^2 & b^2
\end{bmatrix}^n \begin{bmatrix}
c^2 \\
(1-c)^2
\end{bmatrix}
\end{align}
How fast does $X_n+Y_n$ go to zero?</p>
</blockquote>
<p>This is coming from a Marko... | hamam_Abdallah | 369,188 | <p>If $x=y =0$, it gives</p>
<p>$$f (0)=f (0)^2$$
thus $f (0)=0$ or $f (0)=1$.</p>
<p>Assume $f (0)=0$.</p>
<p>with $x=-y $, we get
$$f (0)+x=f (x)f (-x)=-x $$
which is not possible.</p>
<p>so, $f (0)=1$.</p>
<p>with $x=-y $,
$$1+x=f (x)f (-x)=1-x $$
There is no function satsfying that condition.</p>
|
1,466,637 | <p>Solve $3sec^2(x)=4$.</p>
<p>$sec^2(x)=\frac{4}{3}$</p>
<p>$sec(x)=4=\sqrt{\frac{4}{3}}=\frac{4\sqrt{3}}{3}$</p>
<p>How to continue, i.e. how to calculate the value of x for which $sec(x)=\frac{4\sqrt{3}}{3}$</p>
<p>I can rearrange the expression above into $cos(x)=\frac{\sqrt{3}}{4}$, but how to continue to find... | lab bhattacharjee | 33,337 | <p>For squares of any trigonometric function, find double angle of cosine.</p>
<p>$$\cos2x=2\cos^2x-1=2\cdot\dfrac34-1=\dfrac12=\cos\dfrac\pi3$$</p>
<p>$$\implies2x=2m\pi\pm\dfrac\pi3$$ where $m$ is any integer</p>
|
317,294 | <p>Suppose we have a vector space $V$, and $U$, $W$ subspaces of $V$.</p>
<p>Dimension theorem states:
$$ \dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$</p>
<p>My question is:</p>
<p>Why is $U \cap W$ necessary in this theorem?</p>
| Wheepy | 63,287 | <p>$U \cap W$ is the intersection of the vector spaces $U$ and $W$, that is, the set of all vectors of the space $V$ which are in both subspaces $U$ and $W$.</p>
<p>As $U$ and $W$ are both subspaces of $V$, their intersection $U \cap W$ is also a subspace of $V$ (this assertion can be easily proved). Because $U \cap W... |
540,649 | <p>Show that the Dirichlet problem</p>
<p>$$ \left\{ \begin{array}{l} u_{xx}+ u_{yy}=u^3 \ \text{in} \ x^2+y^2 \lt 1 \\ u=0 \ \text{on} \ x^2+y^2 = 1 \end{array} \right.$$</p>
<p>where $u=u(x,y)$, has only the trivial solution $u \equiv 0$</p>
<p>Thanks</p>
| Did | 6,179 | <p>For an unbounded smooth positive function whose improper Riemann integral exists, try the function $f$ defined by
$$
f(x)=\sum_{n\geqslant1}n\mathrm e^{-n^5(x-n)^2}.
$$
Note that
$$
\int_\mathbb Rf(x)\mathrm dx=\sum_{n\geqslant1}n\int_\mathbb R\mathrm e^{-n^5(x-n)^2}\mathrm dx=\sum_{n\geqslant1}\frac1{n^{3/2}}\int_\... |
912,030 | <p><img src="https://i.stack.imgur.com/FTQSg.png" alt=""></p>
<p>Hallo everybody,</p>
<p>I have the following problem regarding shortest paths in $R^2$.</p>
<p>Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture.
I am looking for a path from $p$ to $q$ through a point $c_1$ in the firs... | Dmoreno | 121,008 | <p>After integrating you have:</p>
<blockquote>
<p>$$ - \frac{1}{2} \cos{2x} + \frac{1}{3} \sin{3y} = K, \quad K \in \mathbb{R}, $$</p>
</blockquote>
<p>which is as nice as an explicit solution. Substitute the initial data to find the value of the constant of integration:</p>
<blockquote>
<p>$$ - \frac{1}{2} \co... |
3,767,656 | <p>Using L'hopital rule: <span class="math-container">$\lim_{x \to x_i, y \to y_i} \frac{x-x_i}{\sqrt{(x-x_i)^2 + (y-y_i)^2}} = \lim_{x \to x_i, y \to y_i} \frac{\frac{d(x-x_i)}{dx}}{\frac{d\sqrt{(x-x_i)^2 + (y-y_i)^2}}{dx}} = \lim_{x \to x_i, y \to y_i} \frac{1}{\frac{x-x_i}{\sqrt{(x-x_i)^2 + (y-y_i)^2}}} = \lim_{x \t... | zkutch | 775,801 | <p><strong>Hint:</strong> when we have <span class="math-container">$a = \frac{1}{a}$</span>, then it gives <span class="math-container">$a^2 = 1$</span>.</p>
|
29,945 | <p>It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)_{\aleph_0}^{n+1}$ (or in particular that $(2^{\aleph_0})^+ \to (\aleph_1)_{\aleph_0}^2$) by using forcing? Is it sim... | Vladimir Kanovei | 32,916 | <p>There is also a theorem saying that a sequence of canonical Borel approximations of an analytic non-Borel set from inside can never be of a bounded Borel rank</p>
|
441,980 | <p>Is there a notation to reference a single element within a set? Let's say I have a set <code>n = {1, 2, 4, 8, 16}</code>. If I wanted to use a single element from this set, is there a certain notation to do so? In computer programming, if I have an array <code>int x = {1, 2, 4, 8, 16}</code> I could reference the th... | Vivek Kaushik | 169,367 | <p>Here is a way to understand complex Fourier series representation. Consider $L^2 [-\pi,\pi]$, the set of square integrable complex valued functions on the interval $[-\pi,\pi]$. Define the inner product on this space: $$\langle u(x),v(x)\rangle=\int_{-\pi}^{\pi} u(x)\overline{v(x)}dx.$$
In addition, we can define t... |
335,577 | <p>could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area of unit square on a sphere. </p>
| Roger | 179,034 | <p>Consider a cylinder radius R intersected by a triangular wedge or prism with length 2R width R thickness t: <img src="https://i.stack.imgur.com/YDplX.png" alt="triangular prism intersecting cylinder"></p>
<p>Calculate the area $S$ of the shaded portion by sweeping the arrow from the position shown through $\pi/2$, ... |
2,343,026 | <p>Let $a$ and $b$ be greater than $0$ and let</p>
<p>$$I=\int_0^{\infty}\frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$$</p>
<p>I know I can first manipulate by adding and subtracting $1$ in the numerator. Thus</p>
<p>$$\int_0^{\infty}\frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx=\int_0^{\infty}\frac{1+e^{ax}-1-e^{bx... | Archis Welankar | 275,884 | <p>I dont know if it helps but what you can do is write $\frac {1}{1+e^{bx}}=\frac {e^{-bx}}{1+e^{-bx}} $ and make the substitution $e^{-bx}=u $ that directly results into a basic integrand. Same for the other integral with $e^{ax} $</p>
|
3,044,271 | <p>If <span class="math-container">$x^2+y^2=1$</span>. then the range of expression <span class="math-container">$3x^2-2xy$</span> without trigonometric substitution method</p>
<p>what i have done try here is use arithmetic geometric inequality</p>
<p><span class="math-container">$\displaystyle x^2+y^2\geq 2xy$</sp... | José Carlos Santos | 446,262 | <p>The range that you're after is <span class="math-container">$\left[\frac{3-\sqrt{13}}2,\frac{3+\sqrt{13}}2\right]$</span>. In fact, if <span class="math-container">$x^2+y^2=1$</span>, then<span class="math-container">$$3x^2-2xy=3x^2\pm x\sqrt{1-x^2}.$$</span>So, for each <span class="math-container">$x\in[-1,1]$</sp... |
3,044,271 | <p>If <span class="math-container">$x^2+y^2=1$</span>. then the range of expression <span class="math-container">$3x^2-2xy$</span> without trigonometric substitution method</p>
<p>what i have done try here is use arithmetic geometric inequality</p>
<p><span class="math-container">$\displaystyle x^2+y^2\geq 2xy$</sp... | user10354138 | 592,552 | <p>For a calculs-free approach, note that
<span class="math-container">$$
\frac{3+\sqrt{13}}{2}\left[\frac{2y-(3+\sqrt{13})x}{\sqrt{26+6\sqrt{13}}}\right]^2+\frac{3-\sqrt{13}}{2}\left[\frac{2y-(3-\sqrt{13})x}{\sqrt{26-6\sqrt{13}}}\right]^2=3x^2-2xy
$$</span>
and
<span class="math-container">$$
\left[\frac{2y-(3+\sqrt{1... |
1,771,075 | <p>I have a large number of data sets that have either a unimodal normal distribution or a bimodal normal distribution. I'm not a statistician by any means, so I'm quite limited in my experience.</p>
<p>For the bimodal data sets, I have implemented (through a library) the Expectation-Maximization method for identifyin... | Batman | 127,428 | <p>Normal distributions are always unimodal. It looks like by "bimodal normal data" you mean a Gaussian Mixture Model (GMM) with 2 components (i.e. the PDF of the data is a convex combination of Gaussian PDF's). </p>
<p>There are several techniques to estimate the number of components in a Gaussian Mixture Model -- Ba... |
2,639,570 | <p>I have a very short proof and am questioning one of the logical steps I made.</p>
<p>Let $S$ be a subset of a metric space $E$. </p>
<p>I want to prove that if $p$ is not an interior point of $S^c$, then any ball in $E$ contains points of $S$.</p>
<p>Proof:</p>
<p>Suppose that $\forall\epsilon >0, B_{\epsilon... | Mathemagical | 446,771 | <p>Observe that the $\operatorname{int}(S^c)$ is the complement of $\bar{S}$. </p>
<p>So your point P lies in $\bar{S}$. By definition of the closure, that means all balls centered on $P$ intersect $S$.</p>
|
2,639,570 | <p>I have a very short proof and am questioning one of the logical steps I made.</p>
<p>Let $S$ be a subset of a metric space $E$. </p>
<p>I want to prove that if $p$ is not an interior point of $S^c$, then any ball in $E$ contains points of $S$.</p>
<p>Proof:</p>
<p>Suppose that $\forall\epsilon >0, B_{\epsilon... | polynomial_donut | 109,466 | <p>Yes. This is, because the compliment is taken with respect to $E$, (the 'relative complement' as described on Wikipedia: <a href="https://en.m.wikipedia.org/wiki/Complement_(set_theory)" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Complement_(set_theory)</a>.<br>
So, $S^C$ here is the set of all eleme... |
553,571 | <p>We <a href="http://www.maa.org/programs/maa-awards/writing-awards/the-circle-square-problem-decomposed" rel="noreferrer">know</a> that there is no paper-and-scissors solution to <a href="http://en.wikipedia.org/wiki/Tarski%27s_circle-squaring_problem" rel="noreferrer">Tarski's circle-squaring problem</a> (my six-yea... | Jim Belk | 1,726 | <p>Well, here's a specific infinite family of scissors congruences between large portions of the circle and large portions of the square. I make no claim that these are close to optimal.</p>
<p>We begin by inscribing a regular $n$-gon in the circle. We then cut the $n$-gon into $2n$ triangles, and rearrange these as... |
553,571 | <p>We <a href="http://www.maa.org/programs/maa-awards/writing-awards/the-circle-square-problem-decomposed" rel="noreferrer">know</a> that there is no paper-and-scissors solution to <a href="http://en.wikipedia.org/wiki/Tarski%27s_circle-squaring_problem" rel="noreferrer">Tarski's circle-squaring problem</a> (my six-yea... | Robert Frost | 201,797 | <p>Ok not my greatest piece of work ever, but here's a suggestion in 2 parts:
<img src="https://i.stack.imgur.com/vMIzt.png" alt="enter image description here"></p>
|
817,708 | <p>I had this as part of a question in an exam. And, I reasoned, even when it's arctan(1/0) (undefined), it is pi/2. And, so I said, domain belongs to all Real Numbers. Why isn't it this </p>
| Hubble | 49,109 | <p>The function $f(x) = \arctan(1/x)$ is undefined at $x=0$, you cannot assume it is $\pi/2$ at such value of $x$. The domain of $f$ is $\mathbb{R} \setminus \{0\}$.</p>
|
1,723,331 | <blockquote>
<p>$$\frac{1}{1-x^2}$$</p>
</blockquote>
<p>$$\frac{1}{1-x^2}=\frac{a}{1-x}+\frac{b}{1+x}$$</p>
<p>$$1=a+ax+b-bx$$</p>
<p>$$1=a+b+x(a-b)$$</p>
<p>$a+b=1$ and $x(a-b)=0\Rightarrow a-b=0\Rightarrow a=b$</p>
<p>$$2a=1\Rightarrow a=\frac{1}{2}$$</p>
<p>$b=\frac{1}{2}$</p>
<p>$$\frac{1}{1-x^2}=\frac{1}... | Archis Welankar | 275,884 | <p>Whats the problem take $1/2$ common so you get $1/2(\frac{1+x+1-x}{1-x^2})=\frac{2}{2(1-x^2)}=\frac{1}{1-x^2}$</p>
|
2,484,034 | <p>I want to prove that the intersection between the cone $z^2=x^2+y^2$ and the plane $x+y+2z=2$ is an elispe in this plane. </p>
<p>My work:</p>
<p>I suppose to prove it I have to see that the equation $x+y+2\sqrt{x^2+y^2}=2$ can be rewritten as $\frac{x^2}{a}+\frac{y^2}{b}=c$. To do so I have tried to completing t... | Harry Alli | 378,603 | <p>You cannot put it in this form, because the ellipse is not always centered at the origin, or having its major axis on either the $y$ or $x$ axes.
$$\frac{x+y-2}{2}=z$$
$$\left(\frac{x+y-2}{2}\right)^2=x^2+y^2$$
$$x^2+xy-2x+xy+y^2-2y-2x-2y+4=4x^2+4y^2$$
$$3x^2+3y^2-2xy+4x+4y=4$$</p>
<p>This conic is an ellipse. You... |
1,604,958 | <p>I have the following <a href="https://i.stack.imgur.com/Segc0.png" rel="nofollow noreferrer">Bipartite graph</a>, how can I prove that it is connected?</p>
<p><a href="https://i.stack.imgur.com/R71r0.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/R71r0.jpg" alt="enter image description here"></a... | Frentos | 298,557 | <p>Since you've tagged this under "matrices", let's use the adjacency matrix $A$.</p>
<p>$A_{ij}^n$ (i.e. entry $ij$ in $A^n$) represents the number of walks of length $n$ between vertices $i$ and $j$. In a connected graph, the maximum distance require to get from any node to any other is $V-1$ where $V$ is the numbe... |
1,604,958 | <p>I have the following <a href="https://i.stack.imgur.com/Segc0.png" rel="nofollow noreferrer">Bipartite graph</a>, how can I prove that it is connected?</p>
<p><a href="https://i.stack.imgur.com/R71r0.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/R71r0.jpg" alt="enter image description here"></a... | Rebecca J. Stones | 91,818 | <p>These two graphs have the same degree sequence:</p>
<p><a href="https://i.stack.imgur.com/Yk5Az.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Yk5Az.png" alt="enter image description here"></a></p>
<p>The one on the left is your (connected) graph, whereas the one on the right is disconnected. ... |
1,629,394 | <p>Hi I have recently started studying propositional logic and am finally understanding the truth tables and how to use them. I came across this formula which is confusing me.</p>
<blockquote>
<p>Use truth tables to establish the following:</p>
<p>(a) <span class="math-container">$p ∧ q,\, p ⇒ r\models r$</span>.</p>
<... | Patrick Stevens | 259,262 | <p>The turnstile means "show that the thing on the right is true in all valuations for which all things on the left are true".</p>
<p>For example, $\vDash (p=p)$ is a true statement; $\vDash (p=q)$ is not; $(p=q) \vDash (p=q)$ is true.</p>
|
1,191,237 | <p>I have the list $l = (6, 6, 5, 4)$ and want to how to calculate the possible number of permutations.</p>
<p>By using <em>brute force</em> I know that there are <strong>12</strong> possible permutations:</p>
<p>$$\{(6, 5, 6, 4),
(6, 6, 5, 4),
(5, 6, 6, 4),
(6, 4, 5, 6),
(6, 5, 4, 6),
(4, 6, 6, 5),
(4, 5, 6, 6... | N. F. Taussig | 173,070 | <p>Here is another way of thinking about the problem: There are four positions to fill. We can fill two of them with a $6$ in $\binom{4}{2}$ ways. Once those two positions have been filled, we fill one of the remaining two positions with a $5$ in $\binom{2}{1}$ ways. Once those three positions have been filled, we ... |
4,310,324 | <p>This is theorem 11.5 from CLRS book.
Suppose <span class="math-container">$a\in \mathbb{Z}^*_p, b\in \mathbb{Z}_p$</span>.</p>
<p>Consider two distinct keys <span class="math-container">$k$</span> and <span class="math-container">$l$</span> from <span class="math-container">$\mathbb{Z}_p$</span>, so that <span class... | Alan Abraham | 823,763 | <p>You are pretty much on the right track. The last part is pretty tedious, so it might be better that you just use Wolfram Alpha or possibly some other program. I think you defined your variables a little wrong. For example, if <span class="math-container">$a<b$</span>, then <span class="math-container">$a-b<0$<... |
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