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 |
|---|---|---|---|---|
1,491,805 | <p>What is the simplified value of $(\tan15)(\tan30)(\tan45)(\tan60)(\tan75)$</p>
<p>I am trying to find this value without the use of a calculator but there are certain values I do not know like $(\tan15)$ and $(\tan75)$</p>
<p>Can someone give me any advice?</p>
| Angelo Mark | 280,637 | <p>By using $\tan(90^0-x)=\cot(x)$</p>
<p>Observe that $\tan75^0=\tan(90^0-15^0)=\cot(15^0)$</p>
<p>And $\tan60^0=\tan(90^0-30^0)=\cot(30^0)$</p>
<p>So we have </p>
<p>$\tan15^0\cdot \tan30^0 \cdot \tan45^0 \cdot \cot30^0 \cdot \cot15^0 =\tan45^0=1$</p>
|
3,593,700 | <p>I have run into a thinking error while trying to solve the following two exercises, here they are:</p>
<p>1) There are 30 fish in the lake. 5 of them have been taken out and marked and then put back into the lake. Later 7 fish have been picked from the lake. What is the probability that 2 out of 7 picked fish were ... | Alain Remillard | 278,299 | <p>When doing probability, the important thing is to consider order (or not) in both numerator (favorable outcomes) and denominator (total outcomes).</p>
<p>Regarding the first problem. If order is important, then total outcomes are
<span class="math-container">$$A_7^{30}=\frac{30!}{23!}$$</span>
and favorable outcom... |
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>
| Mariano Suárez-Álvarez | 274 | <p>The determinant function on the set of $n\times n$ matrices is a non-zero polynomial which is homogeneous of degree $n$. </p>
<p>Since the restriction of a uniformly continuous function to a subspace is still uniformly continuous, it is enough to show that the restriction of $\det$ to the line spanned by any matrix... |
2,895,477 | <blockquote>
<p>What is the number of injective functions from a set of size $n$ into a set of size $m$, with $n\le m$?</p>
</blockquote>
<p>I am thinking along the lines of, let a set $A = \{1,\dotsc,n\}$ and set $B = \{1,\dotsc,m\}$. </p>
<p>Then $f(1)$ can take $m$ values, $f(2)$ can take $m-1$ values, …, $f(n)$... | Arthur | 15,500 | <p>You're absolutely on the right track. Now you just need to combine $m, m-1,\ldots,m-(n-1)$ in the right way, and you're done.</p>
|
2,895,477 | <blockquote>
<p>What is the number of injective functions from a set of size $n$ into a set of size $m$, with $n\le m$?</p>
</blockquote>
<p>I am thinking along the lines of, let a set $A = \{1,\dotsc,n\}$ and set $B = \{1,\dotsc,m\}$. </p>
<p>Then $f(1)$ can take $m$ values, $f(2)$ can take $m-1$ values, …, $f(n)$... | Community | -1 | <p>This is $P^m_n=\frac{m!}{(m-n)!}$, the number of permutations of $n$ objects from $m$ objects. </p>
|
3,798,729 | <p>Consider some nonzero vector <span class="math-container">$x^1=(x_1,x_2) \in \mathbb Z^2$</span> with integer components. Then by defining
<span class="math-container">$$x^2 = (x_2, -x_1)$$</span>
we can find a vector with integer components orthogonal to <span class="math-container">$x^1$</span>. I am wondering whe... | Alex Kruckman | 7,062 | <p>There are two ways to interpret your first question. One is purely algebraic and straightforward: the group <span class="math-container">$2M^{*}$</span> has exponent <span class="math-container">$2$</span> (because for all <span class="math-container">$a\in 2M^*$</span>, <span class="math-container">$a = 2b$</span> ... |
1,540,069 | <p>If $\lim \limits _{x \to x_0} (f(x) + g(x))$ exists, can I write: $\lim \limits _{x \to x_0} (f(x) + g(x)) = \lim \limits _{x \to x_0} f(x) + \lim \limits _{x \to x_0} g(x)$ ? <br>I mean, to write this do I have to know that the other limits exist? Because they tell me that $\lim \limits _{x \to x_0} f(x)$ exists an... | Farewell | 278,893 | <p>I am certainly not an expert and here I will only give some of my thoughts on the questions you raised and I will do it in the form of a story that will, I hope, touch at least partially every of the questions you raised on the general principles of exponentiation when we are exponentiating in the field of real and ... |
2,137,801 | <blockquote>
<p>Evaluate:
$$\int \frac{e^x}{e^{2x} + 3e^x + 2} dx$$</p>
</blockquote>
<p><em>My solution</em>: Let $u = e^x$, then $$\int \frac{u}{u^2+3u+2} du=\dots$$</p>
<p>and $\frac{u}{(u+1)(u+2)} = \frac{A}{u+2} + \frac{B}{u+1}$ with $A = 2, B = -1$. So,
$$\dots=-2 \ln |u + 2| - \ln |u+1| + C$$ resubstitute... | user414998 | 414,998 | <p>Hint. What can you say about two angles $A$ and $B$ if $\tan A = \tan B$? Looking at the graph of the tangent function may help you answer this question. </p>
|
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... | Christian Blatter | 1,303 | <p>A <span class="math-container">$5$</span>-element subset of <span class="math-container">$[500]$</span> is called a <em>hand</em>. A hand is <em>good</em> if the sum of its elements is a multiple of <span class="math-container">$5$</span>. We are told to determine the number <span class="math-container">$N$</span> o... |
76,006 | <p>In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of generalized Springer fibers. </p>
<p>Briefly, let $G$ be a sufficiently nice algebraic group, and choose a semisimple $s \in... | Ben Webster | 66 | <p>Borel-Moore homology isn't a homotopy invariant, but it is an isotopy invariant; if two homeomorphisms are homotopic through a homotopy which is a homeomorphism over any point in [0,1], they induce the same map on Borel-Moore homology.</p>
<p>One can see this instantly by writing Borel-Moore homology as homology of... |
1,744,858 | <blockquote>
<p>Let $$A=53\cdot 83\cdot109+40\cdot66\cdot96$$
Is this number prime or composite?</p>
</blockquote>
<p>I'm sure it's a composite number. But I do not know how to prove it.</p>
| Mithlesh Upadhyay | 234,055 | <p>Given number is $A=53\cdot 83\cdot109+40\cdot66\cdot96= 732931$</p>
<p>No, $732931$ is not a prime number, it is a composite number.</p>
<blockquote>
<p>Why $732931$ is not a prime number?</p>
</blockquote>
<p>Because $732931$ has divisors rather than $1$ and itself. $732931$ can be divided by $149$ and $4919$... |
364,394 | <p>I was asked to find the minimum and maximum values of the functions:</p>
<blockquote>
<ol>
<li>$y=\sin^2x/(1+\cos^2x)$;</li>
<li>$y=\sin^2x-\cos^4x$.</li>
</ol>
</blockquote>
<p>What I did so far:</p>
<ol>
<li><p>$y' = 2\sin(2x)/(1+\cos^2x)^2$<br />
How do I check if they are suspicious extrema points? ... | Brian Rushton | 51,970 | <p>The union of two connected spaces that share at least one point is connected. You can use the fact that the sphere minus a point is homeomorphic to $\mathbb{R}^3$ and go from there.</p>
|
1,413,022 | <p>In Guillemin and Pollack's <em>Differential Topology</em>, they give as an exercise (#1.8.14) to prove the following generalization of the Inverse Function Theorem:</p>
<blockquote>
<p>Use a partition-of-unity technique to prove a noncompact version of
[the Inverse Function Theorem]. Suppose that the derivative... | user149792 | 149,792 | <p>Suppose $f: X \to Y$ maps $Z \subset Y$ diffeomorphically onto $f(Z) \subset Y$.</p>
<p>Since $df_x: T_x(X) \to T_y(y)$ is an isomorphism for each $x \in Z$, there exist open sets $x \in U_x \subset X$ and $V_x \subset Y$ such that $f$ maps $U_x$ diffeomorphically onto $V_x$ by the Inverse Function Theorem presente... |
4,642,102 | <blockquote>
<p><strong>Edits:</strong> Parcly Taxel first discovered that the length of cycle cannot be constrained by the 3-connected condition. However, I think
the construction proposed by kabenyuk later is wonderful. Therefore, I
choose it as the best answer. But this does not mean that Parcly
Taxel's construction... | kamills | 497,007 | <p>There is another way to do this problem, namely by showing that for the space <span class="math-container">$X = S^n \times S^m$</span> (take <span class="math-container">$n,m \neq 0$</span> to avoid a trivial problem), there is no map <span class="math-container">$f:A \vee B \to X$</span> inducing an isomorphism on ... |
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... | André Nicolas | 6,312 | <p>There are $\dbinom{12}{4}$ ways to choose $4$ routes from the $12$. Under our assumptions, they are equally likely.</p>
<p>(a) There are $\dbinom{4}{2}$ ways to choose $2$ Florida routes from the $4$ available. For <strong>each</strong> of these ways, there are $\dbinom{3}{2}$ ways to choose $2$ Texas routes, for a... |
154,215 | <p>Prove that</p>
<ol>
<li>Each field of characteristic zero contains a copy of the rational number field.</li>
<li>For an $n$ by $n$ matrix $A,$ if it is not invertible, then there exists an $n$ by $n$ matrix $B$ such that $AB=0$ but $B\ne0.$</li>
</ol>
<p>For (1), I think I have to use the fact that each subfield o... | Alex J Best | 31,917 | <p>For 2 try showing that as $A$ is non-invertible, so is has a nonzero nullity which means there is at least one nonzero vector which is mapped to 0. Now taking $B$ to be the matrix formed by taking its columns as the vector described above, this will multiply to give the nxn zero matrix.</p>
|
44,746 | <p>Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me.</p>
<p>I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - \frac{1}{\infty} = 1$ as well? Because $\frac{1}{\infty} $ would be infinitely close to $0$, perhaps as $1^{-\in... | Michael Hardy | 11,667 | <p>Whether one defines $1/\infty$ may be a matter of convention, but one can say that $1/x$ approaches $0$ as $x$ approaches $\infty$, and what that means is that $1/x$ can be made as close as desired to $0$ by making $x$ big enough. How big is big enough depends on how close you want to make $x$ to $0$. If that's wh... |
3,214,662 | <p><strong>Q1</strong> Prove that every simple subgroup of <span class="math-container">$S_4$</span> is abelian.</p>
<p><strong>Q2</strong> Using the above result, show that if <span class="math-container">$G$</span> is a nonabelian simple group then every proper subgroup of <span class="math-container">$G$</span> has... | ab123 | 454,871 | <p>By your method: </p>
<p>Shift the origin to <span class="math-container">$B$</span>. Rotate the axes by the original slope of line <span class="math-container">$BC$</span>, and check if <span class="math-container">$y$</span> coordinate of <span class="math-container">$A$</span> is positive and less than <span clas... |
3,043,996 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a normed space. Prove that for all <span class="math-container">$x \in X$</span> there exists <span class="math-container">$f \in X^*$</span>, such that <span class="math-container">$f(x) = \|x\|^2$</span> and <span cla... | Fred | 380,717 | <p>If <span class="math-container">$x=0$</span>, we are done. Now let <span class="math-container">$x \in X$</span> and <span class="math-container">$x \ne 0$</span>. A consequence of the Hahn-Banach theorem is the existence of some <span class="math-container">$g \in X^*$</span> with</p>
<p><span class="math-containe... |
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>
| Debashish | 154,864 | <p>Hint: Try expressing $n!+(k+1)$ as $(k+1)N$ where $N$ is a natural number greater than $1$. That should be pretty simple as $n\geq (k+1)$.</p>
|
2,983,553 | <p>I've been stuck trying to solve this problem for the whole day. Also, I'm trying to translate the problem as good as I can, as my English skills aren't the greatest; sorry for that.</p>
<p>Problem is as follows: <strong>Points OBDE form a quadrilateral. Points B and D are on the line x=1. Find the value of x that m... | David K | 139,123 | <blockquote>
<p>... I'm assuming that the function that I'm supposed to form should be <span class="math-container">$A\left(x\right)=\frac{1}{2}ah+ah$</span></p>
</blockquote>
<p>Notice that you used the same symbols, <span class="math-container">$a$</span> and <span class="math-container">$h$</span>, in both parts ... |
26,636 | <p>I have a function </p>
<pre><code>f[x_, y_, z_] := Exp[-x^2]*Exp[-z^2]
</code></pre>
<p>which traces out a tube along $y$. Is there a way to plot this 4D function, where the plot color is the 4th dimension?</p>
| Kuba | 5,478 | <p>Have I understood correctly? Look at this:</p>
<p>Edit: I forgot <code>ColorFunctionScaling</code>. Now it is ok.</p>
<pre><code>RegionPlot3D[
x^2 + z^2 <= 1, {x, -2, 2}, {y, -12, 12}, {z, -2, 2}, Mesh -> None,
ColorFunctionScaling -> False,
ColorFunction ->
Function[{x, y, z}, ColorData["Rainbo... |
2,855,142 | <p>I'm hosting a bar sports tournament with $10$ teams and $6$ different sports (pool, darts, table tennis, foosball, beer pong and cornhole). Trying to get the fixtures as fair as possible so that each team plays each sport twice and playing against the same team multiple times is minimised. Are there any formulas to ... | joriki | 6,622 | <p>Label the teams $0$ to $9$ and the sports $0$ to $5$. Here $k\,\%\,6$ denotes the remainder of $k$ modulo $6$.</p>
<p>First let each team $i$ play each other team $j$ in the sport $i+j\,\%\,6$:</p>
<p>$$
\matrix{&1&2&3&4&5&0&1&2&3}\\
\matrix{1&&3&4&5&0&1... |
4,497,033 | <p>Let <span class="math-container">$r(t)$</span> be the function:<br />
<span class="math-container">$r(t) = \sqrt{x(t)^2 + y(t)^2}$</span>, where<br />
<span class="math-container">$x(t) = 3b (1 − t)^2 t + 3c (1 − t) t^2 + a t^3$</span>, and<br />
<span class="math-container">$y(t) = a (1 − t)^3 + 3c (1 − t)^2 t + 3b... | njuffa | 114,200 | <p>Modern PCs are so incredibly fast that a simple problem like this readily lends itself to a modified Monte Carlo approach. The idea is to generate many random triples <span class="math-container">$(a, b, c)$</span> in the vicinity of the approximate solution until one finds a parameter triple that reduces the maximu... |
3,513,308 | <p>Let <span class="math-container">$X$</span> be a CW complex, and suppose <span class="math-container">$W$</span> is obtained from <span class="math-container">$X$</span> by attaching an <span class="math-container">$n$</span>-cell to X, where <span class="math-container">$n>1$</span>. Consider the universal cover... | Narasimham | 95,860 | <p>HINT:</p>
<p>First circle has equation</p>
<p><span class="math-container">$$ (x-h1)^2 + (y-k1)^2 = r_1^2 \tag1 $$</span></p>
<p>If you rotate its center through angle <span class="math-container">$\beta$</span></p>
<p><span class="math-container">$$(x-[h1 \cos \beta-k1 \sin \beta])^2+(y-[h1 \sin \beta + k1\cos ... |
2,370,851 | <p>How to solve the Integral $\int{\frac{1}{\sqrt{1-\sqrt{4-x}}}}$dx with steps</p>
<p>I have tried to make the substitution $\frac{du}{dx}=4-x$ but it seems that there is no continuation road.</p>
<p>I have seen that there is a substitutions that gives the following results.</p>
<p>$\int{\frac{1}{2\sqrt{-u}} \fr... | Michael Rozenberg | 190,319 | <p>The domain gives $3<x\leq4$.</p>
<p>Let $x=4\sin^2t$, where $t\in\left(\frac{\pi}{3},\frac{\pi}{2}\right]$.</p>
<p>Hence, $\cos{t}=\sqrt{1-\frac{x}{4}}$, $dx=8\sin{t}\cos{t}dt$ and $$\int\frac{1}{\sqrt{1-\sqrt{1-4x}}}dx=\int\frac{8\sin{t}\cos{t}}{\sqrt{1-2\cos{t}}}dt=-8\int\frac{\cos{t}}{\sqrt{1-2\cos{t}}}d(\co... |
1,057,999 | <p>I'm trying to find the limit: $\displaystyle\lim_{n\to\infty}\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}$</p>
<p>I thought of taking a pretty obvious binding from above expression: $\frac {n^n} {(n+1)^n}$ which is $n$ times the largest numerator and $n$ times the smallest denominator, but this limit isn... | Tintarn | 197,823 | <p>Try to express your term with binomial coefficients!</p>
|
1,057,999 | <p>I'm trying to find the limit: $\displaystyle\lim_{n\to\infty}\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}$</p>
<p>I thought of taking a pretty obvious binding from above expression: $\frac {n^n} {(n+1)^n}$ which is $n$ times the largest numerator and $n$ times the smallest denominator, but this limit isn... | Michael Lugo | 173 | <p>Write your left-hand side as</p>
<p>$${1 \over (n+1)} \times {2 \cdot 3 \cdot \cdots \cdot n \over (n+2)(n+3) \cdot \cdots \cdot 2n}$$</p>
<p>Now the first factor is $1/(n+1)$, and the second factor is obviously less than 1 (since the numerator is smaller than the denominator). So the product is less than $1/(n+1... |
2,738,244 | <p>I'm trying to understand the growth of the term $\binom{n}{k}$ -
I saw <a href="https://math.stackexchange.com/questions/1265519/approximation-of-combination-n-choose-k-theta-left-nk-right">here</a> a proof that $\binom{n}{k} = O(n^k)$. However, if $k$ is quite large (say $k=n$) then this term is not polynomial. I ... | Chappers | 221,811 | <p>For $0<\alpha<1$, we have
$$ \binom{n}{n\alpha}^{-1} = (n+1)\int_0^1 x^{\alpha n} (1-x)^{(1-\alpha)n} \, dx $$
(this is a consequence of the more general relationship $\binom{a+b}{a}^{-1} = (1+a+b)\int_0^1 x^a (1-x)^b \, dx $).</p>
<p>We can use Laplace's Method to find an asymptotic for this integral, whence... |
4,068,830 | <p>My logic is since <span class="math-container">$3$</span> out of <span class="math-container">$4$</span> elements are chosen, each element would appear once.
So a sequence would look like: <span class="math-container">$a\,b\,c\,x\,x\,x\,x\,x\,x\,x$</span></p>
<p>We have <span class="math-container">$7$</span> spots ... | DreiCleaner | 751,382 | <p>Use <span class="math-container">$\binom{4}{3}$</span> to make the selection of <span class="math-container">$3$</span> elements to show up in the sequence.</p>
<p>Now you have to ask, "How many of each should there be?" Since there needs to be at least one of each, we are looking for how many ways there a... |
4,068,830 | <p>My logic is since <span class="math-container">$3$</span> out of <span class="math-container">$4$</span> elements are chosen, each element would appear once.
So a sequence would look like: <span class="math-container">$a\,b\,c\,x\,x\,x\,x\,x\,x\,x$</span></p>
<p>We have <span class="math-container">$7$</span> spots ... | RobPratt | 683,666 | <p>The <a href="https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling number of the second kind</a> <span class="math-container">$\left\{n \atop k\right\}$</span> counts the number of partitions of an <span class="math-container">$n$</span>-set into exactly <span class="m... |
165,492 | <p>Sorry for my English if there is any mistake. The exercice for which I need help is the following:</p>
<p>Compute using complex methods:
$I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}$</p>
<p>i) Choose the complex function to integrate.</p>
<p>I guess it is $f(z)=1/(z^2+1)$</p>
<p>ii) Choose the contour.</p>
<p>I ... | Robert Israel | 8,508 | <p>Note that with $x = 1/u$
$$\int_0^1 \dfrac{dx}{x^2+1} = \int_\infty^1 \dfrac{-du/u^2}{1/u^2+1} = \int_1^\infty \dfrac{du}{1+u^2}$$
so your integral is $\displaystyle \dfrac{1}{2}\int_0^\infty \dfrac{dx}{x^2+1} = \dfrac{1}{4} \int_{-\infty}^\infty \dfrac{dx}{x^2+1} $. Now use the semicircular contour as in DonAnton... |
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | Narasimham | 95,860 | <p>$$ \frac{(n-1) n}{2}+\frac{ n(n+1)}{2} = n^2.$$</p>
|
3,224,342 | <blockquote>
<p>Prove that for any <span class="math-container">$\Delta ABC$</span> we have the following inequality:</p>
<p><span class="math-container">$$ \sin A + \sin B + \sin C \le 3 \sin \left(\frac{A+B+C}{3}\right) $$</span></p>
</blockquote>
<p>Could you use AM-GM to prove that?</p>
| Tojra | 655,621 | <p>We know in a triangle<span class="math-container">$ ABC, 0 \lt A,B,C \lt \pi$</span>. So <span class="math-container">$sin A , sinB, sinC \gt 0$</span>. Moreover in <span class="math-container">$[0,\pi],$</span> so <span class="math-container">$(sinx)''=-sinx \lt 0$</span>, which is concave downwards. Using Jensen's... |
3,224,342 | <blockquote>
<p>Prove that for any <span class="math-container">$\Delta ABC$</span> we have the following inequality:</p>
<p><span class="math-container">$$ \sin A + \sin B + \sin C \le 3 \sin \left(\frac{A+B+C}{3}\right) $$</span></p>
</blockquote>
<p>Could you use AM-GM to prove that?</p>
| QuasarChaser | 557,892 | <p>I'm sure there's a different way to approach this question, but here's one way using the graph of <span class="math-container">$\sin x $</span>:</p>
<p><a href="https://i.stack.imgur.com/6p3qs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6p3qs.png" alt="A graph of y = sin x with a triangle on i... |
339,806 | <p>The definition of the integral I was given (which after searching around seems like the common definition) is the value of the inf{upper sums across all dissections} (integral exists when this coincides with the sup{lower sums across all dissections}). </p>
<p>Now, when I searched online of how to do the integral i... | Dominic Michaelis | 62,278 | <p>As you have that $x^2$ is continuous all partitions will give the same integral.
Hence
$$\int_0^a x^2 =\lim_{n\to \infty} \sum_{i=0}^n \frac{a}{n} \cdot \frac{(ai)^2}{n^2}=\lim_{n\to \infty} \frac{a}{n^3} \sum_{i=0}^n (ai)^2=\lim_{n\to \infty} \frac{a^3}{n^3} \cdot \frac{n \cdot (n+1)\cdot (2n+1)}{6} $$
The limit o... |
339,806 | <p>The definition of the integral I was given (which after searching around seems like the common definition) is the value of the inf{upper sums across all dissections} (integral exists when this coincides with the sup{lower sums across all dissections}). </p>
<p>Now, when I searched online of how to do the integral i... | vonbrand | 43,946 | <p>You are describing the Riemann integral. And you are absolutely right, the definition talks about <em>all</em> divisions, not just division into $N$ equal parts. But you can prove that if the length on the shortest slice is larger than $1 /N$, uniform division into $N$ pieces gives a smaller minimal sum, and a large... |
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | Ethan Bolker | 72,858 | <p>I think what you want is the <em>weighted average</em>, and you've almost correctly invented it.</p>
<p>The formula is (total point sum)/(number of voters). In your example it's
$$
\frac{1 \times 5 + 2 \times 3 + 3 \times 1 + 4 \times 17 + 5 \times 2}
{5 + 3 + 1 + 17 + 2} .
$$</p>
<p>See</p>
<p><a href="https://e... |
4,060,755 | <p><a href="https://i.stack.imgur.com/xJH5E.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xJH5E.png" alt="![enter image description here" /></a></p>
<p>I'm having trouble understanding how to do the step highlighted in red if anyone could help</p>
| John Bentin | 875 | <p>The reason is that, if a function <span class="math-container">$f:\Bbb C\to \Bbb C$</span> has a power-series representation <span class="math-container">$$f(z)=\sum_{k=0}^\infty a_kx^k$$</span> that fails to converge at a point <span class="math-container">$z=c\in\Bbb C$</span>, then it cannot converge at any point... |
4,060,755 | <p><a href="https://i.stack.imgur.com/xJH5E.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xJH5E.png" alt="![enter image description here" /></a></p>
<p>I'm having trouble understanding how to do the step highlighted in red if anyone could help</p>
| Vercassivelaunos | 803,179 | <p>While <span class="math-container">$\tan(z)=\frac{\sin(z)}{\cos(z)}$</span>, it is <em>not</em> true that the Taylor series of <span class="math-container">$\tan$</span> is just the Taylor series of <span class="math-container">$\sin$</span> divided by the Taylor series of <span class="math-container">$\cos$</span>!... |
4,345,516 | <p>Let's say we have the polynomial, <span class="math-container">$x^{4}+x^{3}+x^{2}+x$</span>. It's derivative is, <span class="math-container">$4x^{3}+3x^{2}+2x+1$</span>. The solutions to the original polynomial are, <span class="math-container">$-1$</span>, <span class="math-container">$0$</span>, <span class="math... | Mike | 544,150 | <p>On can see the following:</p>
<blockquote>
<p>Claim 1: Iff a polynomial <span class="math-container">$P(x)$</span> has a root <span class="math-container">$r$</span> with multiplicity <span class="math-container">$k>0$</span> i.e., <span class="math-container">$(x-r)^k$</span> divides <span class="math-container"... |
2,508,614 | <p>In $\mathbb{Z}_7[x]$ ,write $f(x)=2x^3+2x+3$ as product of irreducibles .
make all your coefficients either $0,1,2,3,4,5$ or $6$</p>
<p>My attempt: as $f(x)=2x^3+2x+3\rightarrow f(1)=2+2+3=7=0 $</p>
<p>so we can write $f(x)=(x-1)g(x)$</p>
<p>but how to find $g(x)?$</p>
| The Question | 472,057 | <p>As I mentioned in the comments, you can just apply long division and obtain $g(x) = 2 x^2 + 2 x + 4 = 2(x^2 + x + 2)$. Again when you sub in $x = 3$, $g(3) = 0$. One method is to differentiate $g(x)$ which gives you $g(x) = 2x + 1$, so $x = 3$ is a double root. Otherwise you can just simply apply long division again... |
966,504 | <p>I had been following all the blogs, but I would like to understand, whether an attempt has been made to understand how many cycles are possible apart from the 1-4-2-1 cycle in collatz problem</p>
| Zubin Mukerjee | 111,946 | <p>A cycle other than the $1-4-2-1$ cycle has not been found. If such a cycle was found, then the conjecture would be disproved. </p>
<p>If it was proven that no such cycles exist, then the conjecture would still not be solved, since there could be initial values of $n$ for which the recursion diverges.</p>
<hr>
<p>... |
187,147 | <p>In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed. </p>
<p>for (1), $f*f=f$ gives $\widehat{f*f}=\hat{f}$, which is equal to $\hat{f}\cdot\hat{f}=\hat{f}$, but this does not guarantee t... | N. S. | 9,176 | <p>For $(i)$, $\hat{f}\cdot\hat{f}=\hat{f}$ implies that $\hat{f}(\xi) \in \{ 0,1 \}$ for all $\xi$.</p>
<p>Now use the fact that $\hat{f}$ is continuous. </p>
<p>For $2$ try to solve the problem backwards. Try to find some $g \in L^2(\mathbb{R})$ so that
$g(x) \in \{ 0,1 \}$ and whose FT is real valued. </p>
|
7,492 | <p>Inspired by <a href="https://mathoverflow.net/questions/7439/algebraic-varieties-which-are-also-manifolds">this thread</a>, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space underlying a complex varie... | JME | 4,046 | <ul>
<li><p>The simplest example of a singular
algebraic variety which is a
topological manifold is given by the
<strong>cusp</strong> <span class="math-container">$$z_1^2-z_0^3=0.$$</span> The cusp is a topological manifold
homeomorphic to a real plane
<span class="math-container">$\mathbb{R}^2$</span> as can ... |
7,492 | <p>Inspired by <a href="https://mathoverflow.net/questions/7439/algebraic-varieties-which-are-also-manifolds">this thread</a>, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space underlying a complex varie... | Greg Kuperberg | 1,450 | <p>The answer from Dmitri motivates this partial answer from the topological side of the question.</p>
<p>It is <a href="http://www.math.ias.edu/~goresky/pdf/triangulations.pdf">a theorem of Mark Goresky</a> and others that every stratified space, and in particular every complex variety $V$, has a smooth triangulation... |
2,194,490 | <blockquote>
<p>If $n$ is a natural number such that $ n \geq 2$, then the numbers $n! + 2, n! + 3, n! + 4... n! + n$ are all composite.
(Thus, for any n greater than or equal to 2, one can find n consecutive composite numbers)</p>
</blockquote>
<p>I started with just plugging in numbers to see if they were compos... | user402085 | 402,085 | <p>$n ! + k = k\cdot(n\cdot(n - 1)\cdot...\cdot (k + 1)\cdot(k - 1)\cdot... \cdot 2\cdot 1 + 1), 2\leq k \leq n.$</p>
|
2,194,490 | <blockquote>
<p>If $n$ is a natural number such that $ n \geq 2$, then the numbers $n! + 2, n! + 3, n! + 4... n! + n$ are all composite.
(Thus, for any n greater than or equal to 2, one can find n consecutive composite numbers)</p>
</blockquote>
<p>I started with just plugging in numbers to see if they were compos... | victoria | 412,473 | <p>For $ 2 \leq i \leq n, \ \ \ n! + i = n(n-1)(n-2) \ldots i \ldots (3)(2)(1) + i $</p>
<p>$ = i( \ (n)(n-1)\ldots (i+1)(1)(i-1)\ldots (3)(2)(1) + 1 \ )$</p>
<p>which is clearly composite</p>
|
2,194,490 | <blockquote>
<p>If $n$ is a natural number such that $ n \geq 2$, then the numbers $n! + 2, n! + 3, n! + 4... n! + n$ are all composite.
(Thus, for any n greater than or equal to 2, one can find n consecutive composite numbers)</p>
</blockquote>
<p>I started with just plugging in numbers to see if they were compos... | fleablood | 280,126 | <p>Notice that if $k \le n $ then $k $ divides $n! $. So $k $ divides $n! +k $. So $n! +k $ is composite if $k >1$.</p>
<p>let $\overline n_k=n!/k = 1*2*.... (k-1)*(k+l)*...*n $. Then $n!+k =k (\overline n_k +1)$.</p>
|
720,971 | <p>What do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! <span class="math-container">$$w + x + y + z = 6 \qquad w + y + z = 4 \qquad w + y = 2$$</span></p>
| Gerry Myerson | 8,269 | <p>The terms "leading variable" and "free variable" are usually defined for the matrix representing a system, and only when the matrix is in row-echelon form.</p>
<p>The augmented matrix for your system is
<span class="math-container">$$
\left( \begin{array}{c c c c|c}
1&1&1&1&6\\ 1&... |
720,971 | <p>What do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! <span class="math-container">$$w + x + y + z = 6 \qquad w + y + z = 4 \qquad w + y = 2$$</span></p>
| Praveen Sripati | 365,092 | <p>If a set of linear equations can be expressed as let's say</p>
<p>a = 3x + 4y + 5z - 12</p>
<p>b = 2x + 8y + z - 11</p>
<p>c = 9x + 7y -z - 15</p>
<p>where</p>
<ol>
<li>The left hand variables don't appear on the right side and vice versa.</li>
<li>On the left side, there is only one variable.</li>
</ol>
<p>Th... |
864,568 | <p>I am trying to figure out how to take the modulo of a fraction. </p>
<p>For example: 1/2 mod 3. </p>
<p>When I type it in google calculator I get 1/2. Can anyone explain to me how to do the calculation?</p>
| Bill Dubuque | 242 | <p>One can perform arithmetic of fractions mod $\,m\,$ as long as all fractions $\,a/b\,$ have denominator $\,b\,$ <em>coprime</em> to $\,m,\,$ since then, by Bezout, $\,b\,$ is invertible mod $\,m\,$ so the fraction has the unique denotation $\,x = a/b = ab^{-1}$ (the unique solution of $\,bx = a).\,$ The usual rule... |
2,163,306 | <p>Find the following limit $$I = \lim_{n \to\infty} \int_{n}^{e^n} xe^{-x^{2016}} dx$$</p>
<p>My attempt </p>
<p>Assumption: as $n \to \infty$ we can assume and interval on the positive real axis $[n,e^n]$</p>
<p>Here the function $e^{-x^{2016}}$ is a decreasing function, using this fact we use the sandwich lemma t... | dxiv | 291,201 | <p>Hint (without L'Hôpital): assume WLOG $Q$ is monic (otherwise cancel out the leading term between the two sides). Then $Q(z)=\prod_{k=1}^n(z-\alpha_k)$ and, by the product rule of differentiation, $\,Q'(z)=\sum_{k=1}^n \prod_{j \ne k}(z-\alpha_j)\,$ so in particular $\,Q'(\alpha_k)=\prod_{j \ne k}(\alpha_k-\al... |
2,842,177 | <blockquote>
<p>$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... | Will Jagy | 10,400 | <p>I have a factoring program that accepts a bound; it checks the target for divisibility by all primes up to the bound, then stops. In this way I can give a little bit of information about some very large numbers.</p>
<p>So, for example, $3^{10} \equiv -1 \pmod {1181}.$ It follows that, whenever $k$ is <strong>odd</s... |
756,111 | <p>Does anyone have any intuition on remembering or very quickly deriving that</p>
<p>$$\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r}) = \frac{1}{r} \frac{\partial ^2 }{\partial r^2}(r \phi )$$</p>
<p>holds for the Laplacian in spherical coordinates? Doing the IBP is too long and slow... | GregVoit | 143,014 | <p>You can use the series expansion for the exponential function:</p>
<p>$\exp(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}$</p>
<p>Then we get</p>
<p>$\lim_{n\rightarrow\infty}(n^{2}\exp(-1/n)+n\exp(-1/n)-n^{2})=
\\
\lim_{n\rightarrow\infty}\left(n^{2}\left(1-\frac{1}{n}+\frac{1}{2n^{2}}-\frac{1}{6n^{3}}\pm…\right)+n\lef... |
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... | Dr. Sonnhard Graubner | 175,066 | <p>we have $8(x^4+y^4)-(x+y)^4=7x^4-4x^3y-6x^2y^2-4xy^3+7y^4=(7x^2+10xy+7y^2)(x-y)^2\geq 0$
this is true.</p>
|
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... | GEO | 75,928 | <p>Regarding your edit and the question in the comment under OC-Sansoo's answer: (If I understand your issue right, you want reasoning for the choice of vectors?)</p>
<p>Start with the RHS of the inequality we want to show. </p>
<p>$$ 8\left(x^4+y^4\right) = \left(x^4+y^4\right)\left(2^2+2^2\right)$$
On the RHS we ... |
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>$xabx^{-1}x = bax$.<... | Golu Singh | 690,438 | <p>If I put <span class="math-container">$~X = b~$</span> or <span class="math-container">$~X = a^{-1} ~$</span>(consider as self inverse)than what we get</p>
<p>I get that the term <span class="math-container">$~xabx^{-1} = ba ~$</span>, become <span class="math-container">$~ ba = ba~$</span></p>
<p>Which give... |
2,990,614 | <p>Let <span class="math-container">$g: \mathbb{R^2} \to \mathbb{R}$</span>.</p>
<p><a href="https://i.stack.imgur.com/kIi5N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kIi5N.png" alt="Function g"></a></p>
<p><strong>How can I prove that <span class="math-container">$g$</span> is continuous in... | Mark Viola | 218,419 | <p>Note that <span class="math-container">$x^2+y^2\ge 2|xy|$</span>. Hence, we have </p>
<p><span class="math-container">$$\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \le \frac{\sqrt{|xy|}}{\sqrt 2}$$</span></p>
|
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... | clementine1001 | 991,725 | <p>Claim: Statement is false</p>
<p>Counterexample:</p>
<p>Let be <span class="math-container">$\sigma_1 = (1 2 34)(56) $</span> and <span class="math-container">$\sigma_2=(156)(234) $</span> <span class="math-container">$\in S_6$</span> and
<span class="math-container">$\sigma_1 \circ \sigma_2 = (16243)(5) \in S_6$</s... |
3,059,444 | <p>Exercise asks to verify that the sum of three quantities x, y, z, whose product is a constant k, is minimum when these three quantities are equal.</p>
<p>This is my amateurish attempt:</p>
<ol>
<li><span class="math-container">$x + y + z = S$</span>;</li>
<li><span class="math-container">$x*y*z=k$</span>;</li>
<li... | Wuestenfux | 417,848 | <p>The point is that complex conjugation <span class="math-container">${\Bbb C}\rightarrow {\Bbb C}:z\mapsto \bar z$</span>, where <span class="math-container">$\bar z = a-ib$</span> if <span class="math-container">$z=a+ib$</span>, is a (ring) automorphism. It follows that if you have a polynomial (with real coefficien... |
2,825,115 | <p>We have an urn with $n_1$ red balls and $n_2$ black balls (total number of balls is $n = n_1+n_2$). We will draw a total of $k$ balls with replacement. </p>
<p>1) What's the probability that in the $k$ draws, we end up getting at least $m_1$ distinct red and $m_2$ "distinct" black balls? Note that unlike most argum... | Federico Fallucca | 531,470 | <p>Sure because your binary operation on $A$
$\cdot: AxA\to A$ is a balanced product and so, by universal property of tensor product, there exists a unique map $\phi:A\otimes A\to A$ such that $\cdot=\phi\circ\otimes$ and so if you have that $u_1\otimes v_1=u_2\otimes v_2$ than $u_1v_1=\phi\circ\otimes (u_1,v_1)=\phi... |
2,825,115 | <p>We have an urn with $n_1$ red balls and $n_2$ black balls (total number of balls is $n = n_1+n_2$). We will draw a total of $k$ balls with replacement. </p>
<p>1) What's the probability that in the $k$ draws, we end up getting at least $m_1$ distinct red and $m_2$ "distinct" black balls? Note that unlike most argum... | Jendrik Stelzner | 300,783 | <p>I assume that $\otimes$ denotes the tensor product over $\mathbf{k}$, so that $\otimes = \otimes_\mathbf{k}$.</p>
<p><strong>To answer the question as stated:</strong>
The map
$$
A \times A
\to A,
\quad (u,v)
\mapsto uxv
$$
is $\mathbf{k}$-bilinear, and thus induces a $\mathbf{k}$-linear map
$... |
160,737 | <p>I try to solve the following sum:</p>
<p>$$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$$</p>
<p>I'm very curious about the possible approaching ways that lead us to solve it. I'm not experienced with these sums, and any hint, suggestion is very welcome. Thanks.</p>
| newguy | 34,079 | <p>I think one way of approaching this sum would be to use the partial fraction
$$ \frac{1}{k^2n+2nk+n^2k} = \frac{1}{kn(k+n+2)} = \frac{1}{2}\Big(\frac{1}{k} + \frac{1}{n}\Big)\Big(\frac{1}{k+n} - \frac{1}{n+k+2}\Big)$$
to rewrite you sum in the form
$$\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} \frac{1}{k^2n+n^2k+2kn}... |
160,737 | <p>I try to solve the following sum:</p>
<p>$$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$$</p>
<p>I'm very curious about the possible approaching ways that lead us to solve it. I'm not experienced with these sums, and any hint, suggestion is very welcome. Thanks.</p>
| qoqosz | 31,604 | <p>Let's write:
$$f(x) = \sum_{k = 1}^{+\infty}\sum_{n = 1}^{+\infty} \frac{x^{n+k+2}}{nk (n+k+2)}$$
then:
$$f'(x) = \sum_{k = 1}^{+\infty}\sum_{n = 1}^{+\infty} \frac{x^{n+k+1}}{nk} = - \sum_{k = 1}^{+\infty}\frac{x^{k+1} \ln (1-x)}{k}$$
We want to know the value of $f(1)$, so we integrate:
$$f(1) = - \sum_{k = 1}^{+... |
160,737 | <p>I try to solve the following sum:</p>
<p>$$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$$</p>
<p>I'm very curious about the possible approaching ways that lead us to solve it. I'm not experienced with these sums, and any hint, suggestion is very welcome. Thanks.</p>
| user26872 | 26,872 | <p>Here's another approach.
It depends primarily on the properties of <a href="http://en.wikipedia.org/wiki/Telescoping_series" rel="noreferrer">telescoping series</a>, <a href="http://en.wikipedia.org/wiki/Partial_fraction" rel="noreferrer">partial fraction expansion</a>, and the following identity for the <a href="h... |
754,057 | <p>I have the problem: $\sin(2x)=\tan(x)$</p>
<p>I used the double angle formula to get $2\sin(x)\cos(x)=\tan(x)$.</p>
<p>But after that step, I do not know whether or not to subtract $\tan(x)$ or to set $2\sin(x)\cos(x)$ to $u$ and solve for $U$.</p>
| Jay | 115,937 | <p>Write $\tan{(x)}=\frac{\sin (x)}{\cos(x)}$
allowing you to write it as:
$$ 2\sin{(x)}\cos{(x)}=\frac{\sin{(x)}}{\cos(x)} $$</p>
<p>After a little manipulation you can get it into this form:
$$ \sin{(x)}(2\cos^2(x)-1) =0$$
So you are then solving:
$$ \sin(x)=0 \;\;\; \text{ and } \;\;\; 2\cos^2(x)-1=0 $$
Can you see... |
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>
| JEET TRIVEDI | 115,676 | <p>$$\dfrac{dy}{dx}=8(2x-6)^3=8(2(x-3))^3=8\cdot(2^3(x-3)^3)=64(x-3)^3$$</p>
<p>As demostrated above, from the result you obtained to that obtained by Wolfram Alpha. As you can see, they are both equal.</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... | Amy B | 5,321 | <p>If you have gaps, Kahn Academy is a great way to go. It has videos teaching topics that you've forgotten, lots of practice material, and it will keep track of what you've mastered and what you need more help with. It will assign problems for your weaknesses.</p>
|
587,635 | <p>Differentiate with respect to $x$ </p>
<p>$$f(x)=\sqrt[3]{x^2}-4+\dfrac{8}{x^{2/3}}$$</p>
<p>Solution(is it correct having difficulties with the fractions):</p>
<p>$$=x^{2/3}-4+8x^{-2/3}$$</p>
<p>$$=\dfrac{2}{3}x^{-1/3} - \dfrac{16}{3}x^{-5/3}$$</p>
| Adriano | 76,987 | <p>If the question is:</p>
<blockquote>
<p>Differentiate the following function with respect to $x$:
$$f(x) = \sqrt[3]{x^2} - 4 + \frac{8}{x^{2/3}}$$</p>
</blockquote>
<p>Then the answer is indeed:
$$
f'(x) = \frac{2}{3}x^{-1/3} - \frac{16}{3}x^{-5/3}
$$</p>
|
3,560,742 | <p><span class="math-container">$X_1, \ldots , X_n$</span>, <span class="math-container">$n \ge 4$</span> are independent random variables with exponential distribution: <span class="math-container">$f\left(x\right) = \mathrm{e}^{-x}, \ x\ge 0$</span>. We define <span class="math-container">$$R= \max \left( X_1, \ldots... | user8675309 | 735,806 | <p>a clever probabilistic approach is one that takes advantage of the homogenous parameter <span class="math-container">$\lambda_i =1$</span> for all, and the memorylessness of the exponential distribution (and the fact that there is zero probability for any <span class="math-container">$X_i = X_j$</span> for <span cla... |
4,628,391 | <p>I came across the following statement which is supposedly true:</p>
<blockquote>
<p>There exists an infinite set of regular languages, such that their union is not a CFL</p>
</blockquote>
<p>it is explained this way:
we'll define <span class="math-container">$L_k = \{ 0^k1^k0^k \}$</span></p>
<p><span class="math-co... | J.-E. Pin | 89,374 | <p>Actually, <em>every</em> language <span class="math-container">$L$</span> on a finite alphabet is a countable union of regular languages. Just observe that all finite languages are regular and that
<span class="math-container">$$
L = \bigcup_{u \in L}\ \{u\}
$$</span>
Now, you can choose for <span class="math-contai... |
813,209 | <p>Can Someone help me solve this</p>
<p>$$
\int\frac{19\tan^{-1}x}{x^{2}}\,dx
$$</p>
<p>We have been told to use $\ln|u|$ and $C$.</p>
<p>Thanks!</p>
| Tunk-Fey | 123,277 | <p>Let $y=\arctan x\;\Rightarrow\;\tan y=x\;\Rightarrow\;\sec^2y\ dy=dx$, then the integral turns out to be
\begin{align}
\int\frac{\arctan x}{x^2}\ dx&=\int\frac{y}{\tan^2y}\cdot\sec^2y\ dy\\
&=\int\frac{y}{\sin^2y}\ dy.
\end{align}
The last integral can be solved by using IBP. Taking $u=y$ and $dv=\dfrac1{\si... |
2,398,193 | <blockquote>
<p>Let $f: \Bbb N \to \Bbb N$ via $f(n) = n^3+1$. Is the function bijective?</p>
</blockquote>
<p>Injective: Suppose $f(a) = f(b) $</p>
<p>$$\implies a^3 +1 = b^3 +1 \implies a^3 =b^3\implies a=b$$
Hence, $f$ is injective.</p>
<p>Surjective: We know that $n \ge 1 \implies n^3 \ge 1$
How do I prove it... | Furrane | 373,901 | <p>A function $f:E\mapsto F$ is surjective, if and only if :</p>
<p>$$\forall y\in F, \exists x \in E : y=f(x)$$</p>
<p>Here, $E=F=\mathbb{N}$ and $f(x) = x^3+1$</p>
<p>You can easily see that $\nexists x \in \mathbb{N} : f(x) = 0 $ since $\forall x \in \mathbb{N}, x^3+1 \ge 1$</p>
<p>Hence $f$ is not surjective.</... |
4,156,469 | <p>Let's say a random variable is supported on a semi-infinite interval (say <span class="math-container">$(0, \infty)$</span> or all real numbers). We take a finite interval within the support. We then consider the distribution of this random variable conditional on it lying within the finite interval. Without loss of... | A learner | 737,804 | <p>No , you are wrong.</p>
<p>In both case , answer will be same , i.e, there are <span class="math-container">$m$</span> elements of order <span class="math-container">$2$</span> in <span class="math-container">$D_{n}/Z(D_n)$</span></p>
<p>Edit : If <span class="math-container">$|D_n|=2n $</span> and <span class="mat... |
2,308,430 | <p>I have information about 2 points and an arc. In this example, point 1 (x1,y1) and point 2(x2,y2) and I know the arc for example 90 degree or 180 degrees.</p>
<p>From this information, I want to calculate the center of the circle. Which is (x,y) in this case.</p>
<p><a href="https://i.stack.imgur.com/JQ5sA.png" re... | FWE | 170,600 | <p>Say $\alpha$ is the angle of the arc. Then $$r := \sqrt{cos^2 \alpha + sin^2 \alpha}$$ is the radius of the arc. Then you have </p>
<p>$$(x-x_1)^2 + (y-y_1)^2 = r^2$$
$$(x-x_2)^2 + (y-y_2)^2 = r^2$$
so
$$(x-x_1)^2 + (y-y_1)^2 = (x-x_2)^2 + (y-y_2)^2$$
Assumed $y_1\neq y_2$ (otherwise $x_1\neq x_2$ should hold) you ... |
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... | Justin Palumbo | 2,436 | <p>One way to use forcing to prove actual ZFC results is by using absoluteness. I do not know of a forcing proof of Erdös-Rado, but there is a forcing proof of the somewhat similar Baumgartner-Hajnal theorem that $\omega_1\rightarrow(\alpha)^2_2$ for any countable ordinal $\alpha<\omega_1$. The only absoluteness fac... |
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... | Haim | 3,532 | <p>There are also some model-theoretic examples of such proofs (in ZFC). Once you prove that for a given logic L, the notions of consistency and completeness (for theories in L) are absolute, it's possible to prove (in ZFC) the existence of some interesting models (described by an L-theory) just by showing that such mo... |
461,305 | <p>For example, in computer science, there can be zero, one, two, etc. parameters to a computer program, and this is called its "arity". Sets can be countable or uncountable. Is there some word I can use to say "this set's <em>_</em>" is continuous/discrete, or "this set has a continuous/discrete __". For example, alth... | Christian Blatter | 1,303 | <p>In computer science a set $S$ is <em>discrete</em> if it is just a (finite or infinite) set, like a set of people, a set of colors used in coloring a graph, the sets ${\mathbb N}$ or ${\mathbb Z}^d$ or suitable parts of them. Variables taking values in such sets are called <em>discrete variables</em>. In dynamics we... |
2,059,571 | <blockquote>
<p>Let $$V=\{(x_1,x_2,x_3,\dots,x_{100})\in\mathbb{R}^{100}\,|\, x_1=x_2=x_3 \text{ and } x_{51}=x_{52}=x_{53}= \dots=x_{100}\}$$ What is $\dim V$?</p>
</blockquote>
<p>If W is a subspace of vector space $V$ then $$\dim W = \dim V - \text{number of linearly independent restrictions}$$
In our case $\dim ... | Andrei | 331,661 | <p>You have more than two restrictions:
$$
x_2=x_1\\
x_3=x_1\\
x_{52}=x_{51}\\
x_{53}=x_{51}\\
...\\
x_{100}=x_{51}$$</p>
|
4,174,111 | <p>In the following is Theorem 13.6 from Bruckner's Real Analysis which I don't understand some claims on it :</p>
<p>Question <em>in Blue:</em> <span class="math-container">$\mu (|f_j(x)| > \|f_j\|_∞)=0$</span> and <span class="math-container">$\mu (|f_k(x)| > \|f_k\|_∞)=0$</span>. But how that implies <span cla... | Danny Pak-Keung Chan | 374,270 | <p><span class="math-container">$B_{j,k}$</span> needs not have measure zero.
For, if <span class="math-container">$f_k$</span> is a constant function with <span class="math-container">$f_k =-1$</span>,
then <span class="math-container">$||f_k||_\infty = 1$</span>.
If <span class="math-container">$f_j$</span> is anothe... |
335,577 | <p>could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area of unit square on a sphere. </p>
| pritam | 33,736 | <p>Note that the parametrization of sphere is given by $$r(u,v)=(r\sin u\cos v,r\sin u\sin v,r\cos u)$$ where $0\leq u\leq \pi$ and $0\leq v\leq2\pi.$ So the surface area is given by the formula $$\int_0^{2\pi}\int_0^{\pi}||r_u\times r_v|| du\hspace{1mm} dv$$ Now calculate the integral to get the formula.</p>
|
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>
| André Nicolas | 6,312 | <p>You can even do it by using techniques from first-year calculus, and without using polar coordinates. Rotate the half-circle $y=\sqrt{r^2-x^2}$, from $x=-r$ to $x=r$, about the $x$-axis. Better, rotate the quarter circle, $0$ to $r$ about the $x$-axis, and then double the answer. So by the usual formula for the surf... |
3,533,260 | <p>find three distinct nonzero vectors a,b,c in three dimension such that
span(a,b)=span(b,c)=span(a,b,c)
but span(a,c) is not equal to span(a,b,c)</p>
| Berci | 41,488 | <p><strong>Trick:</strong> We must have <span class="math-container">$c\parallel a$</span> in this case. </p>
|
4,622,026 | <p>I am a bit confused about <span class="math-container">$\int e^{e^x+x}dx$</span>. If we made a <span class="math-container">$u$</span>-sub of <span class="math-container">$e^x$</span> then the derivative is <span class="math-container">$e^x$</span> and so we have <span class="math-container">$\int e^udu$</span>. But... | Minecraft dirt block | 1,080,850 | <p>I am pretty sure it is (12 * 11 * 10)/(12 * 12 * 12). My reasoning is below.</p>
<p>Note that there are a total of 12 * 12 * 12 different possible birthday month configurations of the three friends. We can see this by noting that one friend has 12 different possible birthday months, with each of those possible birth... |
2,917,858 | <p>Consider the vector space of all functions $f: \mathbb{R} \rightarrow \mathbb{C}$ over $\mathbb{C}$. If $W$ is a subspace spanned by $\beta$ = $\{1, e^{ix}, e^{-ix}\}$, show that $\beta$ is a basis for $W$.</p>
<p>I think I am very confused - I know I just have to show that $\beta$ is linearly independent, which me... | mechanodroid | 144,766 | <p>The Wronskian of $\{1, e^{ix}, e^{-ix}\}$ is</p>
<p>$$\begin{vmatrix} 1 & e^{ix} & e^{-ix} \\ 0 & ie^{ix} & -ie^{-ix} \\ 0 & -e^{ix} & -e^{-ix}\end{vmatrix} = -2i \ne 0$$</p>
<p>so $\{1, e^{ix}, e^{-ix}\}$ is linearly independent.</p>
|
2,016,540 | <p>Can someone help me prove this transition? </p>
<p><a href="https://i.stack.imgur.com/u4YVA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/u4YVA.png" alt="enter image description here"></a></p>
| Sarvesh Ravichandran Iyer | 316,409 | <p>Simply change the index. Let $j'=n-j$. Then, as $j$ varies from $1 \to n$, it follows that $j'$ varies from $0$ to $n-1$, because $j' =n-j$. When $j=1$, $j'=n-1$, and when $j=n$, then $j'=0$, so then $j'$ must vary between $0$ and $n-1$.</p>
<p>So, $$
\sum_{j=1}^n ((n-j) + (n-j)^2) = \sum_{j'=0}^{n-1} (j' + j'^2) =... |
1,229,729 | <p>I know that because both $a$ and $b$ are positive it is in the first quadrant and hence $\arg z$ should just equal to $\arctan(b/a)$, but I've been told that the answer is $\arg z= \arctan \sqrt5 $???</p>
| 2ndYearFreshman | 230,490 | <p>Let $(X,Y,Z)$ be the basis of $U$, compute $T(X)$, $T(Y)$, $T(Z)$ and express them as a linear combination of $X$, $Y$ and $Z$.</p>
|
27,307 | <p>Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between them. Now suppose we have $S\subset\mathbb{N}$. If $\lim_{n\rightarrow\infty} \frac{|S\cap A_n|}{|A_n|}=0$, does i... | Alex Becker | 8,173 | <p>I don't think you can simplify it any further. Your simplification does not work because $\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{n}|^{2}|b_{m}|^{2}$ is actually an upper bound for your expression. To see this, test the example $1*1 + 1*(-1) + \cdots = 0$.</p>
|
3,212,117 | <p>My question is how to calculate the following formula without iteration:</p>
<p><span class="math-container">$$ \max \{A,B,C,D\} \tag 1 $$</span></p>
<p>suppose <span class="math-container">$A,B,C,D$</span> are normal and independent:
I know (1) can be rewritten as</p>
<p><span class="math-container">$$\max(\max(... | Community | -1 | <p><span class="math-container">$$\frac{\max(A, B)}{F(A)\cdot F(B)}= \frac{ f(A)}{F(A)} + \frac{ f(B)}{F(B)}$$</span></p>
<p>seems to generalize nicely.</p>
|
974,270 | <p>Here's a homework problem I'm having some trouble with:</p>
<blockquote>
<p>Show that $$ \int_{|z|=3} \frac{1}{z^2-1} dz = 0$$</p>
</blockquote>
<p>So far, I've shown using Cauchy's Integral Formula that $$ \int_{|z-1|=1} \frac{1}{z^2-1} dz = \pi i$$ and $$\int_{|z+1|=1} \frac{1}{z^2-1} dz = - \pi i$$ where $|z-... | Jetsetter | 184,455 | <p>Write $\frac{1}{z^2-1}= \frac{1}{2}\left(\frac{1}{z-1}-\frac{1}{z+1}\right)$ then use Cauchy's integral formula</p>
|
3,055,677 | <p>I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?</p>
| mathcounterexamples.net | 187,663 | <p>Suppose that your knot points (not to be confused with the B-spline knots!) are ordered in a sequence <span class="math-container">$(x_1, \dots, x_m)$</span>. Then you can compute a B-spline of order <span class="math-container">$n$</span> between each ordered pair <span class="math-container">$(x_i, x_{i+1})$</span... |
3,055,677 | <p>I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?</p>
| Cesareo | 397,348 | <p>Follows a MATHEMATICA script showing how to construct a special spline type. (Bézier curves) In the script <span class="math-container">$BL, BQ,BC$</span> are Bézier curves of first, (linear), second (quadratic) and third (cubic) order and also how them are recursively built. The points <span class="math-container">... |
1,629,394 | <p>Hi I have recently started studying propositional logic and am finally understanding the truth tables and how to use them. I came across this formula which is confusing me.</p>
<blockquote>
<p>Use truth tables to establish the following:</p>
<p>(a) <span class="math-container">$p ∧ q,\, p ⇒ r\models r$</span>.</p>
<... | YoTengoUnLCD | 193,752 | <p>The statement</p>
<p>$$(p\wedge q),(p\implies r)\models r$$</p>
<p>Means that whenever the formulas $(p\wedge q)$ <strong>and</strong> $(p\implies r)$ are (both simultaneously) true, then $r$ must be true.</p>
<p>So you job is just to write the truth tables of those 3 formulae and verify that in the lines where $... |
301,393 | <p>The question I am working on is:</p>
<blockquote>
<p>Prove that if $m+n$ and $n+p$ are even integers, where
$m$, $n$,and $p$ are integers, then $m+p$ is even. What kind
of proof did you use?</p>
</blockquote>
<p>I was thinking--and I aware that this may not be the most efficient method--of proving four diffe... | lab bhattacharjee | 33,337 | <p>Let $m+n=2a,n+p=2b$ where $m,n,a,b$ are integers</p>
<p>$\implies m+n+n+p=2(a+b)\implies m+p=2(a+b-n)$</p>
|
643,505 | <p>I am struggling to solve the following equation numerically: $x'^2 - xx''=W(t)$, where W(t) is a sinusoidal function, only known by its samples (i.e. no analytic form is known).
Up until now I tried to write it as two first-order ODEs: $\begin{cases}v'=u \\ u'=\displaystyle\frac{u^2-W(t)}{v}\end{cases}$ and find a s... | Constructor | 114,355 | <p>$$x'^2-xx''=-x^2\left(\frac{x'}{x}\right)'$$</p>
<p>Hence we have another system of first order ODE:$$\left\{\begin{array}{l}x'=xy\\y'=-\frac{W(t)}{x^2}\end{array}\right.$$</p>
<p>Try to find a solution for it.</p>
|
507,657 | <p>I tried many approaches but none of them really worked I treated $p^b-1$ as a Geometric progression but it didn't work and that is about as far as I have been able to go I have no clue how to move forward</p>
| Arash | 92,185 | <p>Let $(a,m)=1$ and not necessarily prime. Then by <a href="http://en.wikipedia.org/wiki/Euler%27s_theorem">Euler theorem</a> you get :
$$
m^{\phi(a)}\equiv 1 \mod a\implies a\mid m^{\phi(a)}- 1
$$
where $\phi(a)$ is <a href="http://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler phi function</a>.
When $p$ is ... |
4,322,437 | <p>I'm attempting some questions from Abstract Algebra Theory and Applications by Judson (found <a href="http://abstract.ups.edu/index.html" rel="nofollow noreferrer">here</a>) and this one is a bit problematic.</p>
<p>Let <span class="math-container">$S = R\setminus \{−1\}$</span> and define a binary operation on <spa... | Trevor Gunn | 437,127 | <p>Let <span class="math-container">$f(x) = x + 1$</span> and <span class="math-container">$g(x) = x - 1$</span>. Then what we have is <span class="math-container">$$x * y = (x + 1)(y + 1) - 1 = g(f(x)f(y))$$</span></p>
<p>Or, to write this in a way that might be easier to catch what is going on:</p>
<p><span class="ma... |
3,681,504 | <p>The specific question I have to work on is:</p>
<p><span class="math-container">$\sqrt{n}$</span> , <span class="math-container">$\log{n^{100}, }$</span> <span class="math-container">$\ n^{10}$</span>, <span class="math-container">$\log(10^n)$</span> , <span class="math-container">$\log(n^n)$</span>, <span class="m... | DanielWainfleet | 254,665 | <p>You have the 1st two in the wrong order. Since <span class="math-container">$(\log x)/x\to 0$</span> as <span class="math-container">$x\to \infty,$</span> we have <span class="math-container">$\lim_{x\to \infty}(\log (x^A))/x^B=0 $</span> for any positive <span class="math-container">$A,B.$</span> </p>
<p>Because <... |
725,547 | <p>This is from the Chapter 15 text of Gourieroux and Monfort's Statistics and Econometric Models II:</p>
<p><strong>Set Up</strong>: Suppose that there are 2 possible parameter values $\theta_0$ and $\theta_1$ from which there are 2 density functions $l_{\theta_0}(y)$ and $l_{\theta_1}(y)$ on the data. Define
$$
F(k)... | Bill Dubuque | 242 | <p>$\overbrace{(x\ {\rm mod}\ b)}^{\large r}\ {\rm mod}\ a\, =\, \overbrace{(x\!-\!bq)}^{\large r}\ {\rm mod}\ a\, =\, (x\!-\!apq)\ {\rm mod}\ a\, =\, x\ {\rm mod}\ a$</p>
<p><strong>Remark</strong> $\ $ Generally $\ x \equiv y\pmod{ap}\,\Rightarrow\,x\equiv y\pmod a\ $ by $\ a\mid ap\mid x-y$ </p>
<p>Yours is the s... |
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