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 |
|---|---|---|---|---|
2,068,643 | <p>I'm solving a problem about a <a href="https://en.wikipedia.org/wiki/Chemical_reactor#PFR_.28Plug_Flow_Reactor.29" rel="nofollow noreferrer">plug flow reactor</a> and I have this limit to compute. Just to control my result I asked <a href="http://www.wolframalpha.com/input/?i=lim_%7BR%20%5Cto%20%2B%5Cinfty%7D%20(1-e... | egreg | 62,967 | <p>Set $x/(R+1)=t$, so $R+1=x/t$ and
$$
\frac{R}{R+1}=\frac{x/t-1}{x/t}=\frac{x-t}{x}
$$
Then you have, depending on whether $x>0$ or $x<0$, the limit for $t\to0^+$ or the limit for $t\to0^-$. Let's compute the two-sided limit:
$$
\lim_{t\to0}\frac{1-e^t}{\frac{x-t}{x}-e^t}=
\lim_{t\to0}x\frac{1-e^t}{x-t-xe^t}=
\... |
87,869 | <p>Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.</p>
<p>In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.</p>
<p>My question is: Is there a Functional equation for this function? I mean a relationship of the form $ Z(s,N)=G(s) Z(1-s,N)$.</p>
| Gerry Myerson | 8,269 | <p>There is something like a functional equation in Section 3 of <a href="http://www.stanford.edu/group/journal/2005/pdfs/Carl.pdf" rel="nofollow">http://www.stanford.edu/group/journal/2005/pdfs/Carl.pdf</a>, although on my screen a lot of characters are missing - maybe they print correctly but just don't show up on my... |
87,869 | <p>Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.</p>
<p>In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.</p>
<p>My question is: Is there a Functional equation for this function? I mean a relationship of the form $ Z(s,N)=G(s) Z(1-s,N)$.</p>
| reuns | 241,370 | <p>there is the "approximate functional equation" <a href="https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula" rel="nofollow">https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula</a></p>
<p>which is described in the Tichmarsh and on the french wikipedia of zeta-Riemann function but not on the engli... |
349,680 | <p>The particular solution $Y_p(t)$ of this problem is actually in the form of $Ae^tt$, but isn't it supposed to be $Ae^t$ ? Since there is no homogenous root = 0, why do we need to multiply $t$.</p>
| Ron Gordon | 53,268 | <p>You need to multiply by $t$ because $e^t$ is already a solution of the homogeneous equation. </p>
<p>$$r^2-4 r+3 = 0 \implies (r-1) (r-3) = 0 \implies y^{(H)} = A e^t + B e^{3 t}$$</p>
<p>Substituting $e^t$ as the particular solution will produce zero, which will be no help. The next logical thing to do, then, i... |
4,509,183 | <p>The theorem says "<strong>If the function <span class="math-container">$f:\Bbb{R}^n→\Bbb{R}$</span> has a local extremum at <span class="math-container">$α∈\Bbb{R}^n$</span>, then <span class="math-container">$α$</span> is a critical point</strong>".<br />
For <span class="math-container">$n=1$</span>, its... | M A Pelto | 171,159 | <p><strong>Note:</strong> We need to assume that <span class="math-container">$f: \mathbb{R}^n \to \mathbb{R}$</span> is differentiable at a local extremum <span class="math-container">$\mathbf{a} \in \mathbb{R}^n$</span> in order to have
<span class="math-container">$$\lim_{h \to 0} \frac{f(\mathbf{a} + h\mathbf{u} ) ... |
4,509,183 | <p>The theorem says "<strong>If the function <span class="math-container">$f:\Bbb{R}^n→\Bbb{R}$</span> has a local extremum at <span class="math-container">$α∈\Bbb{R}^n$</span>, then <span class="math-container">$α$</span> is a critical point</strong>".<br />
For <span class="math-container">$n=1$</span>, its... | Matthew H. | 801,306 | <p>Here is how I would have proceeded with the proof.</p>
<p>Let <span class="math-container">$\delta>0$</span> so that <span class="math-container">$f(\alpha)\geq f(v)$</span> for all <span class="math-container">$v\in \mathbb{R}^n$</span> with <span class="math-container">$\|v-\alpha\|<\delta$</span>.</p>
<p>Ch... |
51,596 | <p>Some time ago, I asked <a href="https://math.stackexchange.com/q/42276/8271">this</a> here. A restricted form of the second question could be this:</p>
<blockquote>
<p>If $f$ is a function with continuous first derivative in $\mathbb{R}$ and such that $$\lim_{x\to \infty} f'(x) =a,$$ with $a\gt 0$, then $$\lim_{x... | Amitesh Datta | 10,467 | <p>There is a simpler way to prove this. (I apologize but I have not looked at your solution; perhaps another user will comment in this regard.) Let $M$ be a positive integer. Choose a positive integer $N$ such that $x>N$ implies $f'(x)>\frac{a}{2}$. If $y>\text{max}\{N,\frac{2(M-f(N))}{a}+N\}$, then $y>... |
4,601,103 | <p>Let <span class="math-container">$X_i$</span> be <span class="math-container">$i.i.d$</span> integrable random variables with bounded variance and <span class="math-container">$f$</span> be a continuous function with compact support. I want to prove <span class="math-container">$$
\lim_{N\to\infty} \frac{1}{N} \sum_... | Balaji sb | 213,498 | <p>By chebyshev inequality,
<span class="math-container">$P(|\frac{1}{N} \sum_i f(X_i)-E(f(X))|^2 \geq \frac{1}{N}) \leq \frac{Var(\frac{1}{N} \sum_i f(x_i))}{\frac{1}{N}} = \frac{Var(f(X))}{N} \rightarrow 0 \ as \ N \rightarrow \infty.$</span></p>
<p>where we used the fact that <span class="math-container">$Var(f(X))$... |
69,760 | <p>The classical Schauder estimates (see the link)
<a href="http://en.wikipedia.org/wiki/Schauder_estimates" rel="nofollow">http://en.wikipedia.org/wiki/Schauder_estimates</a></p>
<p>Requires $f\in C^\alpha$ in order to get a solution $u\in C^{2+\alpha}$ of the equation</p>
<p>$$\Delta u=f$$</p>
<p>In fact, we can c... | Andrew | 14,551 | <p>For <span class="math-container">$u$</span> to be from <span class="math-container">$C^2$</span> it is enough that the modulus of continuity of <span class="math-container">$f$</span> satisfies the <a href="http://www.jstor.org/pss/2372534" rel="noreferrer">Dini condition</a>. For example, modulus of continuity <spa... |
2,765,102 | <p>I am given the equation of a surface:
$$x^3+y^3+z^3-3xyz =0$$
And I need to find the equation of the plane tangent to this surface at $(1,1,1).$ <br>
At first, this task did not look easy for me as we are not given an explicit equation of a surface, but I tried using implicit differentiation assuming that $z$ depend... | Cesareo | 397,348 | <p>The so called surface $f(x,y,z) = x^3+y^3+z^3-3 x y z= 0$ is the product of a plane and a null radius cylinder or</p>
<p>$$
f(x,y,z) = (x+y+z)(x^2+y^2+z^2-x y- y z - x z) = 0
$$</p>
<p>The plane doesn't contain the point $(1,1,1)$ and the cylinder contains it but it has null radius so it is reduced to a line (gene... |
1,924,259 | <p>That is possible?, can you show me some theorem and who worked on these.</p>
<p>If we have the sum of n cubes, can we express that like the sum of m squares?</p>
<p>Thanks!</p>
| b00n heT | 119,285 | <p>Remark: after reading @fleablood's comment, I realized that I am not answering the OP's question. I haven't decided yet if I'll keep this answer or not.</p>
<p>Something like
$$\sum_{k=1}^nk^3=\left(\sum_{k=1}^nk\right)^2=\left(\frac{n(n+1)}{2}\right)^2\quad ?$$</p>
|
1,924,259 | <p>That is possible?, can you show me some theorem and who worked on these.</p>
<p>If we have the sum of n cubes, can we express that like the sum of m squares?</p>
<p>Thanks!</p>
| Community | -1 | <p>+1 to @valfar's question-you are correct my friend,as I have found through calculation that not only the numbers -23064014558266183195584063166306441237779125 ,290509918128065397278363068942306635399616 and 22773504640138117798305700097364134602379583,sum up to 74,but also 64,1 & 9 are the numbers that sum up to... |
1,407,053 | <p>Show using logarithms that if $y^k = (1-k)zx^k(a)^{-1}$ then $y = (1-k)^{(1/k)}z^{(1/k)}x(a)^{(-1/k)}$.</p>
| Emilio Novati | 187,568 | <p>The properties that you needs are:
$$
\log ab=\log a \log b \qquad and \qquad\log a^b=b \log a
$$
Using these properties your expression become:
$$
\log y^k=k \log y= \log (1-k)zx^ka^{-1}=\log(1-k)+\log z+k\log x-1 \log a
$$
Now you can find $\log y$ as:</p>
<p>$$
\log y=\dfrac{1}{k}\left[\log(1-k)+\log z+k\log x... |
4,548,865 | <p>I wish to determine whether the limit <span class="math-container">$L = \lim_{z \rightarrow i} \frac{z^3 + i}{|z| - 1}$</span> exists. Noticing it to be of the form <span class="math-container">$0/0$</span>, I separate the expression into its real and imaginary parts:
<span class="math-container">$$L = \lim_{(r, \th... | 311411 | 688,046 | <p>If the limit existed, it would need to be the same via any path of <span class="math-container">$z$</span> to <span class="math-container">$i$</span>.</p>
<p>Using the path <span class="math-container">$z=it,\;t\in(0,1)$</span>, and using the difference of two cubes formula, we find (as <span class="math-container">... |
1,510,626 | <p><a href="http://i949.photobucket.com/albums/ad332/Fractur65/DSCN0311.jpg" rel="nofollow">This is my work on the problem,</a> not sure if I did this wrong or I'm missing some way to simplify this and continue from here.</p>
| André Nicolas | 6,312 | <p>Let $u=6x^2-8x+3$. Then $(3x-2)\,dx$ is $\frac{du}{4}$. So you end up calculating $\int \frac{u^3}{4}\,du$.</p>
|
1,998,810 | <p>There exists $x \in \mathbb{R}$ such that the number $f(x)=x^2 +5x +4$ is prime.</p>
<p>I can't understand where to start. </p>
<p>This is what I have so far: </p>
<p>Let P(x) be the statement "$x^2 + 5x +4$ is prime".
Then we have $\exists x \in \mathbb{R}, P(x)$.</p>
<p>I built a table and I suspect that this ... | Arthur | 15,500 | <p>You could solve the equation
$$
x^2 + 5x + 4 = 3
$$</p>
|
1,198,627 | <p>I have the following approximation: </p>
<p>$$f(x) \simeq f(a) + f^{'}(a)(x - a)$$</p>
<p>Letting $a = \mu_{x}$, the mean of $X$, a Taylor seties expansion of $y=f(x)$ about $\mu_{x}$ gives the approximation:
$$y=f(x) \simeq f(\mu_{x}) +f^{'}(\mu_{x})(x - \mu_{x})$$
Taking the variance of both sides yields:</p>
... | Demosthene | 163,662 | <p>\begin{align}Var(Y)= Var[f(X)]&\simeq Var[f(\mu_x)+f'(\mu_x)(X-\mu_x)]\\&= Var(X-\mu_x)[f'(\mu_x)]^2\\
&=Var(X)[f'(\mu_x)]^2\end{align}
where we have used the following property of the variance:
$$Var(aX+b)=a^2Var(X)$$</p>
|
872,693 | <p>Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?</p>
| PhoemueX | 151,552 | <p>Yes, because by definition of the support (depending on your definition), you have $\mu ( \Bbb{R} \setminus \Bbb{Q}) \leq \mu(\Bbb{R} \setminus {\rm supp}(mu)) = 0$ and thus for every measurable set $M$:</p>
<p>$$
\mu(M) = \mu(M \cap \Bbb{Q}) = \sum_{q \in \Bbb{Q} \cap M} \mu({q}) = (\sum_{q \in \Bbb{Q}} \mu({q}) \... |
2,427,747 | <p>I want to prove that $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$ using proof by induction.</p>
<p>Attempt</p>
<p>Prove true for $n = 1$</p>
<p>$2^{1+2} + 3^{2(1) +1} = 35$</p>
<p>35 is divisible by 7 so true for $n =1$</p>
<p><em>Induction step</em>: Assume true for $n = k$ and prove true... | Arthur | 15,500 | <p>You're almost there. What is $2^{k+3} + 3^{2k + 3} - 2(2^{k+2} +3^{2k+1})$, according to your rewritings?</p>
|
1,543,542 | <p><strong>Proposition</strong> If $x \in \mathrm{int}(S \cap T)$, then $x \in \mathrm{int}(S) \cap \mathrm{int}(T)$, where $\mathrm{int}(S)$ is the set of interior points of the set $S$.</p>
<p>I have proven this proposition by demonstrating that $\mathrm{int}(S \cap T) \subseteq \mathrm{int}(S) \cap \mathrm{int}(T)$... | jose_castro_arnaud | 292,886 | <p>Without the definition of interior points, no proof.</p>
<p>Since you already did one side of the proof, here is the converse: if x ∈ int(S) ∩ int(T), then x ∈ int(S ∩ T).</p>
<p>x ∈ int(S) ⊂ S, and x ∈ int(T) ⊂ T, so x ∈ S ∩ T. And the intersection of a finite number of open sets is open, too, so int(S) ∩ int(T) ... |
1,543,542 | <p><strong>Proposition</strong> If $x \in \mathrm{int}(S \cap T)$, then $x \in \mathrm{int}(S) \cap \mathrm{int}(T)$, where $\mathrm{int}(S)$ is the set of interior points of the set $S$.</p>
<p>I have proven this proposition by demonstrating that $\mathrm{int}(S \cap T) \subseteq \mathrm{int}(S) \cap \mathrm{int}(T)$... | Zoe H | 168,114 | <p>If a point $x$ is not in a set $S$ then we will say $x$ is a non-point of $S$, and we will write $\mathrm{non}(S)$ for the set of non-points of $S$.</p>
<p>I claim that $\mathrm{non}(S \cap T) \subseteq \mathrm{non}(S) \cap \mathrm{int}(T)$. </p>
<p><strong>"Proof"</strong></p>
<p>Since $x \in \mathrm{non}(S \cap... |
69,680 | <p>Evaluating the following expression results in a non-zero output</p>
<pre><code>FullSimplify[Integrate[2 f[x], {x,0,1}] - 2 Integrate[f[x], {x,0,1}]]
</code></pre>
<p>I think the output should be zero, but do not know how to simplify this expression. Any suggestions ?</p>
| Basheer Algohi | 13,548 | <p>For this particular case:</p>
<pre><code>(Integrate[2 f[x], {x,0,1}] - 2 Integrate[f[x], {x,0,1}]) /. (Integrate[x_*f[h_], z_]) :> x* Integrate[f[h], z]
</code></pre>
|
1,026,725 | <p>How many even 3 digit numbers contain at least one 7.
I got 126, but it was not an answer choice for the problem. Can anyone help?</p>
| Bill | 193,474 | <p>There are $10$ digits, $0$ to $9$.
The probability of a $3$ digit even number without a $7$ is $$8/9 \times 9/10 \times 5/10 = 360/900$$
Which is $360$ even numbers without a $7$.
So the number with at least one $7$ is $450 - 360 = 90$.</p>
|
1,236,572 | <p>I've been stuck on this question for a long time now and was wondering if anyone could show me how it's done. So far I have done the following:
Since $\lim_{x\to \infty}f'(x)=M$ then $\forall \epsilon ,\exists A >0$ s.t. if $x>A$ then $|f'(x)-M|<\epsilon$. So we have that $M-\epsilon<f'(x)<M+\epsilon... | Elaqqad | 204,937 | <p>After this you have $|f(x+1)-f(x)-M|<\epsilon$ which signifies that the limit exists and equals to $M$, or you can do better, for every $x$ there exists $c_x\in[x+1,x]$ such that:
$$f(x+1)-f(x)=f'(c_x) $$
now because $\lim_{x\to \infty} c_x=+\infty$ (why? because $c_x\geq x+1$)and $f$ continue gives you:
$$\lim_{... |
3,313,216 | <p>How to prove that vectors are parallel iff their unit vectors are equal?</p>
<p><span class="math-container">$$\mathbf{u} \parallel \mathbf{v} \iff \hat{\mathbf{u}} = \hat{\mathbf{v}}$$</span></p>
<p>A vector can be written as a scalar multiple of its magnitude and unit vector in its direction: <span class="math-c... | Mohammad Riazi-Kermani | 514,496 | <p>The question should be rephrased as </p>
<p>Prove that for two non zero vectors <span class="math-container">$u$</span> and <span class="math-container">$v$</span> , <span class="math-container">$$u=\lambda v \iff \frac {u}{||u||}=\frac {v}{||v||}$$</span> </p>
<p>The proof is straightforward. </p>
|
2,340,231 | <p>I have the following ODE for complex $z$:</p>
<p>$$\dot{z}=i|z|^2z$$</p>
<p>and I would like to find the most general exact solution possible. It is easy to see that $z=0$ and $z=e^{it}$ are two solutions, but I am hoping for a way to see if these are the only ones or if more may be found.</p>
<p>Thanks</p>
| Robert Lewis | 67,071 | <p>Assume</p>
<p>$z \ne 0; \tag{1}$</p>
<p>then we may write</p>
<p>$z = r e^{i \theta}, \tag{2}$</p>
<p>so</p>
<p>$\dot z = \dot r e^{i \theta} + ir \dot \theta e^{i \theta}; \tag{3}$</p>
<p>$\vert z \vert^2 = r^2. \tag{4}$</p>
<p>We thus assemble the equation</p>
<p>$\dot r e^{i \theta} + i r \dot \theta e^{... |
3,289,305 | <p>I was trying to the following theorem:</p>
<blockquote>
<p>let <span class="math-container">$G=\textrm{GL}(n,\mathbb{R})$</span> and <span class="math-container">$N=\{A\in G: \, \det(A)>0\}$</span>. Prove that <span class="math-container">$G/N \cong \mathbb{Z}_{2}$</span> .</p>
</blockquote>
<p>In the solutio... | Mike Pierce | 167,197 | <p>It's not <em>trivial</em>, but it's not too tough. You've gotta prove that besides <span class="math-container">$N$</span>, there is a single other coset in <span class="math-container">$G/N$</span>. This'll be the coset consisting of matrices with negative determinant. So you've gotta pick some matrix <span class="... |
176,714 | <p>For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, and $\Omega(n)$ the total number of prime factors of $n$ counted with multiplicity. Obviously $Om_{0}(x)$ is just the ... | so-called friend Don | 16,510 | <p>This problem was studied by Renyi, in</p>
<p><em>On the density of certain sequences of integers</em><br>
Publ. Inst. Math. Belgrade <strong>8</strong> (1955) 157-162.<br>
<a href="http://elib.mi.sanu.ac.rs/files/journals/publ/14/13.pdf">http://elib.mi.sanu.ac.rs/files/journals/publ/14/13.pdf</a></p>
<p>Let $d_k =... |
142,481 | <ol>
<li><p>Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?</p></li>
<li><p>I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M ... | David Eppstein | 440 | <p>As well as the use of this by Lovász for perfect graphs (mentioned by earlier answers), this has been used by Häggkvist to find high-degree four-chromatic triangle-free graphs. See</p>
<p>Häggkvist, R. (1981), "Odd cycles of specified length in nonbipartite graphs", Graph Theory (Cambridge, 1981), pp. 89–99, <a hre... |
1,597,186 | <blockquote>
<p>In $a^n \equiv b^n \pmod m$, does $n$ have to be an integer?</p>
</blockquote>
<p>I just came upon this rule and am wondering its limits. Thank you</p>
| Community | -1 | <p>As long as $a^n$ and $b^n$ are integers, there is nothing new.</p>
<p>I don't think that anything forbids to extend the definition of the modulo to non-integer numbers and say</p>
<p>$$a\equiv b\mod m\iff \frac{a-b}m\in\mathbb Z.$$</p>
<p>Then, </p>
<p>$$2^{1/3}\equiv(15\sqrt[3]4+75\sqrt[3]2+127)^{1/3}\mod 5,$$
... |
200,723 | <p>Trying to get the modulus of the five numbers immediately before a prime, added together in there factorial form; I'll call this operation $S(p)$. For example,</p>
<p>$$S(p) = ((p-1)! + (p-2)! + (p-3)! + (p-4)! + (p-5)!) \bmod p$$</p>
<p>$$S(5) = (4!+3!+2!+1!+0!) \bmod 5$$</p>
<p>$$S(5) = 4$$</p>
<p>However, I h... | Michael Albanese | 39,599 | <p>This isn't a complete answer, but it may be helpful. Note the following:</p>
<p>$(p-1)! + (p-2)! + (p-3)! + (p-4)! + (p-5)!$ </p>
<p>$= (p-5)![(p-1)(p-2)(p-3)(p-4) + (p-2)(p-3)(p-4) + (p-3)(p-4) + (p-4) + 1]$ </p>
<p>$\equiv (p-5)![(-1)(-2)(-3)(-4) + (-2)(-3)(-4) + (-3)(-4) + (-4) + 1]\ \textrm{mod}\ p$ </p>
... |
2,660,076 | <p>Okay, so as stated here: <a href="https://math.stackexchange.com/questions/2209755/issue-with-proof-in-follands-real-analysis-theorem-6-18">Issue with proof in Folland's 'Real Analysis' (Theorem 6.18)</a>, the text says: </p>
<blockquote>
<p>(Theorem 6.18.) Let $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N... | Changlele | 443,993 | <p>Because the integral is just the limit of serious of sums, so that it must be measurable due to that the limit of a serious of measurable function is measurable.</p>
<p>For Fubini's Theorem. it states that integrable on product space imply a.e. integrable (as well as measurable) on every original space. So in your... |
144,644 | <p>I am a beginner in Mathematica, </p>
<p>Suppose I want to write a series of five vectors $p_1,...,p_5$ in terms of an arbitrary basis spanned by $\left\{p,n\right\}$ with each coefficient in such a decomposition a function of scalar variables. </p>
<p>e.g $$p_i = f_i p + g_i n,$$
with $f_i$ and $g_i$ labelling the... | webcpu | 43,670 | <p>Mathematica 11.
The result of f[x] is a list but not a number. </p>
<pre><code>{InterpolatingFunction[{{0., 1.}}, <>][x]}
</code></pre>
<p>In:</p>
<pre><code>Clear[s, x, y, g, f]
s = NDSolve[{y'[x] == g[x], g'[x] == -60 (1 - x) Cos[y[x]], y[0] == 0,
g[0] == 1.2}, {y, g}, {x, 0, 1}];
(*f[x_] = Evaluate... |
144,644 | <p>I am a beginner in Mathematica, </p>
<p>Suppose I want to write a series of five vectors $p_1,...,p_5$ in terms of an arbitrary basis spanned by $\left\{p,n\right\}$ with each coefficient in such a decomposition a function of scalar variables. </p>
<p>e.g $$p_i = f_i p + g_i n,$$
with $f_i$ and $g_i$ labelling the... | Pillsy | 531 | <p>OK, this is actually much simpler than it looks. Let's just take the first part, where you solve the diffy queues:</p>
<pre><code>ClearAll[f, intcos, intsin, s]
s = NDSolve[{y'[x] == g[x], g'[x] == -60 (1 - x) Cos[y[x]], y[0] == 0,
g[0] == 1.2}, {y, g}, {x, 0, 1}];
f[x_] = Evaluate[y[x] /. s];
</code></pre>
... |
967,198 | <p>Given a string consisting of lower-case characters from English alphabets, we want to reverse a substring from the string such that the string becomes a palindrome.</p>
<p><strong>Note :</strong> A Palindrome is a string which equals its reverse.</p>
<p>We need to tell if some substring exists which could be rever... | Sebastian Negraszus | 103,176 | <p><strong>Quick heuristic:</strong> For very large strings, we can count the frequency of all letters ("a", "b", ...) to quickly weed out strings that cannot possibly be turned into a palindrome.</p>
<p>Let $n$ be the length of the string $S$.</p>
<p>If $n$ is even, then each letter must occur an even number of time... |
744,309 | <p>Prove or disprove the following assertion. </p>
<blockquote>
<p>The set of all nonzero scalars matrices is a normal subgroup of $GL_2(\mathbb{R})$.</p>
</blockquote>
<p>Proof: </p>
<p>Let $I$ be the identity matrix. Consider the scalar matrix $sI$ where $s$ is some scalar.<br>
Then let $A$ be any other matrix i... | Kaj Hansen | 138,538 | <p>Note that scalar matrices commute with all other matrices, so if we can show that such matrices form a subgroup, then it is automatically a normal subgroup. It looks like you've gotten this far already. </p>
<p>First, note that the identity is a scalar matrix. Next, we want to show the following:</p>
<ol>
<li>The... |
202,235 | <p>I'm not quite sure how to do this problem: </p>
<p>Calculate the the proportion of values that is 3 in the following data set:</p>
<p>2, 3, 3, 6, 9</p>
| preferred_anon | 27,150 | <p>You have 5 values, and 2 of these are 3. Therefore, the proportion of the data that is 3 is given by
$$\frac{\text{number of 3s}}{\text{number of data}}=\frac{2}{5}$$ </p>
|
3,205,317 | <p>(a) <span class="math-container">$x(v)= 3, y(v)= 4, z(v)= v$</span> for <span class="math-container">$−\infty < v < \infty$</span>,</p>
<p>(b) <span class="math-container">$x(t)= 3\cos(t), y(t)= 2\sin(t), z(t)= 3t−1$</span> for <span class="math-container">$0 \leq t < 2\pi$</span>.</p>
<p>I have no idea w... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
S=& \sum_{k=0}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right) \cos((k+1) x)+i \sum_{k=0}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right) \sin (k+1) x \\
=& \sum_{k=0}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right) e^{(k+1) x i}=e^{x i} \sum_{k=0}^... |
142,220 | <p>Fermat proved that <span class="math-container">$x^3-y^2=2$</span> has only one solution <span class="math-container">$(x,y)=(3,5)$</span>.</p>
<p>After some search, I only found proofs using factorization over the ring <span class="math-container">$Z[\sqrt{-2}]$</span>.</p>
<p>My question is:</p>
<p>Is this Fermat'... | Kieren MacMillan | 19,844 | <p>A completely elementary proof can be found on <a href="https://www.jstor.org/stable/24497284" rel="noreferrer">page 561 of the Nov 2012 edition of <em>The Mathematical Gazette</em></a>, where a descent mechanism first used by Stan Dolan in <a href="https://www.jstor.org/stable/23249523?seq=1" rel="noreferrer">the Ma... |
4,578,252 | <p>On the MIT <span class="math-container">$2021$</span> <em>Integration Bee Qualifying Exam</em>, it asked to approximate
<span class="math-container">$$
\int_{0}^{\pi}{\rm e}^{{\rm e}^{x}}\,{\rm d}x
$$</span>
I got <span class="math-container">$\displaystyle{\rm e} + {\rm e}^{\rm e} + {\rm e}^{{\rm e}^{2}} + {\rm e}^... | Travis Willse | 155,629 | <p>The substitution <span class="math-container">$$t = \exp \exp x, \qquad dt = \exp (x + \exp x) \,dx ,$$</span> gives <span class="math-container">$$\int_0^\pi \exp \exp x \,dx = \int_e^{\exp\exp\pi} \frac{dt}{\log t} ,$$</span> and then applying integration by parts yields <span class="math-container">$$\int_e^{\exp... |
774,043 | <p>How many natural numbers less than ${10^8}$ are there,whose sum of digits equals ${7}$?</p>
<p>My Try: I used multinomial theorem to solve it and I am getting an answer of 1716. I want to know whether I am correct or not. Please help me as I have no way other than this to check my answer. Thank you! :))</p>
| Just_a_fool | 114,428 | <p>As Lucian commented, this question can be approached with partitions:</p>
<p>Note:</p>
<p>The partitions (order immaterial) of 7 number 15.</p>
<p>Also, at most, 7 can be partitioned into 7 integers (1+1+1+…). Also, fortunately, $n<10^8$ has 8 "slots" (the question would be a bit different if, say, the cap was... |
2,650,913 | <p>I was looking at the solid of revolution generated by revolving $\cos(x)$ about the $x$-axis on the interval $[0, 2\pi]$, and I noticed that when the volume of the solid was approximated with $3$ or more cylinders via the disk method the approximation would equal the true volume. To prove this, I deduced that it suf... | OnoL | 65,018 | <p>It is clearly false. Let $k=1$. Then the RHS equals $1/2$, while the LHS is reduced to $\cos^2(2\pi)=1.$</p>
|
2,773,457 | <p>I was doing a sample question and came across this question.</p>
<p>A given surface is defined by the equation:
$3x^2+2y^2-z=0$. Describe the normal vector at a point (x, y, z) on the surface. Calculate the normal vector at the point $(1,-1,5)$ on the surface. </p>
<p>The normal vector is
$(6x, 4y, -1)$</p>
<p>Ho... | José Carlos Santos | 446,262 | <p>Because that's the gradient of $3x^2+2y^2-z$.</p>
|
2,409,744 | <p>Given,
$|\frac{1}{x}-2|<4$, I can solve this via the theorem approach $|x-a|<b\Rightarrow-b<x-a<b$..... but in the above question, there comes a possibility in $-4<\frac{1}{x}-2$ where the solution for it is less than $-\frac{1}{2}$ but it can change if I assume $\frac{1}{x}$ to be a negative value. H... | N. F. Taussig | 173,070 | <p>First, note that $x \neq 0$. You can avoid casework by squaring both sides, then solving the resulting quadratic.
\begin{align*}
\left|\frac{1}{x} - 2\right| & < 4\\
\left|\frac{1 - 2x}{x}\right| & < 4\\
\left(\frac{1 - 2x}{x}\right)^2 & < 16\\
\frac{1 - 4x + 4x^2}{x^2} & < 16\\
1 - 4x ... |
2,780,281 | <p>Hey this is my first time using this website so please fix my formatting if it is bad.</p>
<p>Can someone please help me compute this$$ \prod_{n=1}^\infty\bigg(1+\frac{(-1)^n}{n+1}\bigg) $$</p>
| Thomas Andrews | 7,933 | <p>Write out some terms:</p>
<p>$$\frac{1}{2}\cdot \frac{4}{3}\cdot \frac{3}{4}\cdot \frac{6}{5}\cdot \frac{5}{6}\cdots$$</p>
<p>What does that look like?</p>
|
162,725 | <p>I am working with a very large sparse matrix (for example) given in what follows:</p>
<pre><code>m = 50; n = 40; o = 30; size = m*n*o;
B = SparseArray[{
{i_, i_} -> RandomReal[], {size, size - 1} ->
2., {i_, j_} /; Abs[i - j] == 5 ->
1., {i_, j_} /; Abs[3 i - j] == 2 -> 2.
}, {size, size... | george2079 | 2,079 | <p>Edit no this is not good. It seems the matrix multiplication expands the sparse arrays ( evidenced by memory usage). </p>
<p>--</p>
<p>I dont know if this is good performance wise or not, but here is a simple idea:</p>
<p>construct a 1-base sparse array with a zero row and multiply:</p>
<pre><code>B = B SparseA... |
2,724,754 | <p>I have a set of data of size $X$, say $X = 7$. I want to find all of the unique ways that the data can be grouped into two bins of a minimum size of two. For the example where $X = 7$, I have:</p>
<pre><code> Bin A: 5 4
Bin B: 2 3
</code></pre>
<p>As the possible arrangements. What I need is what the act... | 5xum | 112,884 | <p><strong>Hint</strong>:</p>
<p>If $f$ is continuous at $x$ and $g$ is continuous at $f(x)$, then $g\circ f$ is continuous at $x$.</p>
|
294,946 | <p>Are there any groups (besides $\mathbb{Z}$ itself) which have $\mathbb{Z}$ for their abelianization?</p>
<p>That is, is there any non-abelian group $G$ such that $G/[G,G] \cong \mathbb{Z}$?</p>
<p>I would appreciate some examples (if yes) or some hint as to why not (if no).</p>
| Jason DeVito | 331 | <p>Take your favorite group $G$ whose abelianization is trivial. For example, $G=A_5$ works. Then $\mathbb{Z}\times G$ has abelianization $\mathbb{Z}$.</p>
|
100,551 | <p>I need to get a matrix $\{a(x_i-x_j)\}$, where $x_i$ form a partition of an interval, $a(x)$ is a given function. I use </p>
<pre><code>In[67]:= a[x_?NumericQ] := N[Exp[-Abs[x]]];
x = Table[-10 + 0.02 (j - 1), {j, 1, 1001}];
A = Outer[a[#1 - #2] &, x, x]; // AbsoluteTiming
Out[69]= {2.99032, ... | eldo | 14,254 | <pre><code>Exp@-Abs@Outer[#1 - #2 &, #, #] &[Range[-10, 10, 0.02]]; //
AbsoluteTiming // First
</code></pre>
<blockquote>
<p>0.950001</p>
</blockquote>
|
3,799,157 | <p>If we consider a sequence of functions <span class="math-container">$f_n$</span> there are theorems relating the properties of the functions <span class="math-container">$f_n$</span> with the properties of the function <span class="math-container">$\displaystyle F(x) = \sum_{n=0}^\infty f_n$</span> (when the series ... | markvs | 454,915 | <p>The standard analysis theorem says that <span class="math-container">$f(t,x)$</span> should be uniformly (on <span class="math-container">$t$</span>) continuous in <span class="math-container">$x$</span>.</p>
|
1,436,655 | <p>I was reading my course notes and I came across this statement:</p>
<blockquote>
<p>If we are given a set of moments, we can identify the distribution that they came from.</p>
</blockquote>
<p>My question is: how do we identify the distribution when its moments are specified?</p>
| user459282 | 459,282 | <p>This is in no way a trivial question. So I will do my best in answering and filling the gap left by the previous comments. EDIT: I've had to remove links as I don't have the reputation to post them, please just search in google what is written.</p>
<p>It is firstly worth stating that the comment in your course note... |
26,256 | <p>One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an <span class="math-container">$n$</span>-cycle has <span class="math-container">$\chi = 0$</span> and <span class="math-container">$K_4$</span> has <span class="math-container">$\chi=-2$</span>.
Is t... | Benoît Kloeckner | 4,961 | <p>One can do the following : given a graph with $n$ vertices and $m$ edges, define the scalar curvature of a vertex $x$ of valency $v(x)$ by $S(x)=2-v(x)$. Isolated and pendant vertices have positive scalar curvature, $S$ vanishes precisely on degree two vertices (wich are those one want to call flat), and is negative... |
139,125 | <p>This is a variation on an earlier question resolved by <em>user35353</em>: <a href="https://mathoverflow.net/questions/139105/can-a-tangle-of-arcs-interlock">Can a tangle of arcs interlock?</a> In that question, the arcs were restricted to circular arcs, and <em>user35353</em>'s proof that one arc can be removed wit... | simeon2 | 38,476 | <p>This is not an answer, but I cannot write short comments.</p>
<p>Here I'll try to describe a counterexample, i.e. a construction of elliptical
arcs that cannot be separated mechanically from each other. Take 12 equal
ellipses with eccentricity close to 1, so the ellipses may be considered
"close" to a segment. Now ... |
2,757,687 | <p>Can you make the claim that for any ordinal, its cardinality equals it's least upper bound.</p>
<p>This is motivated by:</p>
<blockquote>
<blockquote>
<p>$\bigcup\omega+1=\omega$ and $|\omega+1|=\omega$</p>
</blockquote>
</blockquote>
<p>where $\bigcup\omega+1$ is also the $\text{sup}(\omega+1)$</p>
<p>T... | C Monsour | 552,399 | <p>If $+$ denotes ordinal arithmetic, then $\cup\omega + 1$ is already not $\omega$ but its successor (since $\cup\omega = \omega$).</p>
<p>If $+$ denotes something else, please tell us what you mean by it </p>
|
13,460 | <p>there're some students, who belive that <span class="math-container">$$\frac10 = \infty $$</span></p>
<p>I need to teach them that this is not true and <span class="math-container">$\frac10 $</span> is undefined, mathematically and give a good picture (for their minds)</p>
<p>what is the proper way to teach them wit... | Nicola Ciccoli | 465 | <p>What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever is $b$. This is about not being defined. Still... why is $\frac 1 0=\infty$ not so completely wrong? Because they ca... |
631,586 | <p>$$1.\quad p\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$2.\quad p\quad only\quad if\quad q\\ \equiv if\quad p\quad then\quad q\\ \equiv p\rightarrow q\\ \\$$$$3.\quad p\quad only\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$4.\quad p\quad i... | user119908 | 119,908 | <p>"$p$, only if $q$" means "if $p$, then $q$." It's infrequently used except as a component of the phrase "if and only if."</p>
|
631,586 | <p>$$1.\quad p\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$2.\quad p\quad only\quad if\quad q\\ \equiv if\quad p\quad then\quad q\\ \equiv p\rightarrow q\\ \\$$$$3.\quad p\quad only\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$4.\quad p\quad i... | rwolst | 76,789 | <p>A nice way to think about this is with sets. </p>
<p><b>If:</b></p>
<p>Consider outcome sets $P \subseteq Q$, then if an outcome in $P$ occurs, that outcome will also be in $Q$ i.e. $Q$ if $P$. </p>
<p><b>Only If:</b></p>
<p>Now consider $Q \subseteq P$, in this case the outcome is in $Q$ only if it is also in $... |
125,369 | <p>Probably a simple question, but I can't find an answer anywhere, not even in the suggested questions with similar titles. It might also be that I just don't get the correct terminology. This is not really a field with which I'm familiar.</p>
<p>I have a system of equations of the form $aw+bx+cy+dz=e$ and I want to ... | Robert Israel | 8,508 | <p>If you just have one matrix $A$ and one $b$, the simplest thing to do is Gaussian elimination to reduce the augmented matrix to echelon form.
You can use a fraction-free version of Gaussian elimination to take advantage of the entries being integers. The system is inconsistent iff the echelon form has a leading ent... |
2,112,802 | <p>If $A \subseteq \mathbb{R} $
$$
\exists p \in A, \forall q \in A , q \leq p $$</p>
<hr>
<p>Can I just use a specific value for $p$ and arbritary value for $q$ to disprove this?</p>
<p>$p = 3$ and $q = p + 1$, hence $q > p$</p>
<hr>
<p>Also, how would should one go about this one: </p>
<p>If .. $\exists p \... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Using the distance formula, a point $(x,y)$ will satisfy a) exactly when
$$
\sqrt{(x-4)^2 + y^2} = 2\sqrt{(x-1)^2 + y^2} = 1
$$
and it will satisfy b) if
$$
|x+1| = \sqrt{(x-2)^2 + (y-4)^2}
$$
However, both of these equations can be simplified. </p>
|
2,430,729 | <p>How do you calculate the zeros of $f(x) = x^4-x^2-6$ in best way?
Here are my attempts:</p>
<p>Factorize, but the problem is the $6$:</p>
<p>$$0=x^4-x^2-6 \iff 6 = x^2(x^2-1)$$</p>
<p>This doesn't lead to any good solution...</p>
<p>Here is another attempt, but I don't know if this is allowed:</p>
<p>$$0= x^4-x... | Dr. Sonnhard Graubner | 175,066 | <p>after my hint above we have to solve $$t^2-t-6=0$$ this is $$t_{1,2}=\frac{1}{2}\pm\sqrt{\frac{1}{4}+\frac{24}{4}}$$</p>
|
753,077 | <p>First of all, sorry for the crappy title, I have no idea how to ask this question.</p>
<p>I have a formula:
$$S = \left(\frac{T\cdot D\cdot C}{1000}+V^{\frac13}\right)^3$$</p>
<p>I now need to turn it around to find $D$, when I have $S$. I don't even know where to start. Any pointers in the right direction, or eve... | Thomas Russell | 32,374 | <p>You begin with the following equation:</p>
<p>$$S=\left(\frac{TDC}{1000}+\sqrt[3]{V}\right)^{3}$$</p>
<p>You then move from the last-operation on the left-hand side inwards and perform the inverse operation on both sides, for instance, the last operation to be performed is to cube everything in the brackets, so yo... |
4,621,925 | <p>I'm reading Lang's Complex Analysis (GTM 103) as an introduction to complex analysis. I came across the theorem which states that a set of complex numbers is compact if and only if it's closed and bounded.</p>
<p>The definitions used in the book is presented below:</p>
<p><strong>Accumulation points</strong>
Let <sp... | Zoe Allen | 1,107,685 | <p>If <span class="math-container">$S$</span> <em>isn't</em> closed, intuitively it means there is some gap in it, <span class="math-container">$v$</span>, such that <span class="math-container">$S$</span> gets arbitrarily close to <span class="math-container">$v$</span> but doesn't contain it. That means for any dista... |
1,557,097 | <p>Show that a finite domain $F$ is a field.</p>
<p>Let $I$ a proper ideal of $F$ and let $a\in I$. In particular, $a$ is not invertible, otherwise $I$ wouldn't be proper. </p>
<p>I would like to show that $I=(a)=(0)$, but without success. </p>
| Tsemo Aristide | 280,301 | <p>Hint: Use the fact that $f_a(x)=ax$ is bijective</p>
|
2,714,190 | <p>Find the last three digits of $6^{2002}$. I did some work and figured out that the last two digits is 36. Can anyone help me with the hundredth digit? By the way, I used modular arithmetic and the recursion method for the tens digit, but it fell short when I attempted to do the hundreds digit. Thank you in advance!<... | lab bhattacharjee | 33,337 | <p>$$6=1+5,6^{25n}=(1+5)^{25n}\equiv1\pmod{5^3}$$</p>
<p>$$\implies6^{25n-1}\equiv6^{-1}\equiv21$$</p>
<p>$$6^{25n+2}\equiv6^3(21)\pmod{5^36^3}$$</p>
<p>$$\equiv216\cdot21\pmod{2^35^3}\equiv?$$</p>
|
2,714,190 | <p>Find the last three digits of $6^{2002}$. I did some work and figured out that the last two digits is 36. Can anyone help me with the hundredth digit? By the way, I used modular arithmetic and the recursion method for the tens digit, but it fell short when I attempted to do the hundreds digit. Thank you in advance!<... | Jack D'Aurizio | 44,121 | <p>In order to find the last three digits of $6^{2002}$ it is enough to compute the remainders $\!\!\pmod{8}$ and $\!\!\pmod{125}$, where the former is clearly zero. About the latter, the binomial theorem grants</p>
<p>$$6^{2002} = (5+1)^{2002} = \sum_{k=0}^{2002}\binom{2002}{k}5^k \equiv \sum_{k=0}^{2}\binom{2002}{k... |
1,386,677 | <p>Proving that $$\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}, \qquad\forall x>\pi$$</p>
| Adelafif | 229,367 | <pre><code>Let f(x)=sinx-(((π²x-x³))/(π²+x²)),x≥π. Then f(π)=0,f(2π) is positive. Now if f(x)<0 for some x>π then by Rolle's theorem f'(y)=0 in the domain. But then
0= f ′(y)=cos y-1+((2π²(π²-y²))/((y²+π²)²)) implies that cos y=1-((2π²(π²-y²))/((y²+π²)²))>1, a contradiction.
</code></pre>
|
29,181 | <p>I ran across this infinite product:</p>
<p>$$\lim_{n\to\infty}\prod_{k=2}^n\left(1-\frac1{\binom{k+1}{2}}\right)$$</p>
<p>I easily found that it converges to 1/3. Using my calculator, I found that</p>
<p>$$1-\frac1{\binom{k+1}{2}}=\frac{(k-1)(k+2)}{k(k+1)}$$</p>
<p>Then, here is my question</p>
<p>$$\prod_{k=2}... | GeoffDS | 8,671 | <p>A much simpler example would be $\prod_{k=2}^n \frac{k}{k+1} = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n}{n+1} = \frac{2}{n+1}$. All the other terms here, obviously, cancel. In your case, much cancels but it's not quite as obvious.</p>
|
353,327 | <p>Can someone please help me out with this question? I have been at it for hours and I can't wrap my head around this one.</p>
<blockquote>
<p>Karen and Kurt's backyard has a width of $20$ meters and a length of $30$ meters. They want to put a rectangular flower garden in the middle of the backyard,leaving a strip... | Metin Y. | 49,793 | <p>$x^2-25x+66 = (x-22)(x-3) = 0$</p>
<p>I have found these $22$ and $3$ just by quickly checking their divisors (and their suitable sums, products). Infact, there is a method for finding roots (that are possibly hard to guess): For any quadratic equation $ax^2+bx+c=0$, its roots are of the form: $x_{1,2}=\dfrac{-b \p... |
2,090,885 | <p>I need to calculate the value of the integral:
$$\int_T\frac 1 {\sqrt{x^2+y^2}} \, dx \, dy$$ where $T=\{(x,y) : x\in[-2,2], x^2<y<4\}$.</p>
<p>Specifically, I need to know how to set integration extremes.</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>The integration limits are
$$\int_{-2}^2\int_{x^2}^4\cdots dy\,dx.$$
But <em>maybe</em> changing to polar coordinates is convenient.</p>
|
534,802 | <p>I need to find both a convergent and divergent sequence of positive numbers such that $$\lim \frac{s_{n+1}}{s_n}=1$$</p>
<p>I think the question is asking me to play with the ratio test. </p>
<p>Just when I was about to write down the answer, I realize that my answer was for series. (The ratio test is for series, ... | amWhy | 9,003 | <p>Convergent sequence: $s_n = \dfrac 1n$. </p>
<p>Divergent sequence $s_n = n$.</p>
|
534,802 | <p>I need to find both a convergent and divergent sequence of positive numbers such that $$\lim \frac{s_{n+1}}{s_n}=1$$</p>
<p>I think the question is asking me to play with the ratio test. </p>
<p>Just when I was about to write down the answer, I realize that my answer was for series. (The ratio test is for series, ... | DonAntonio | 31,254 | <p>You've been answered, but if you want to tighten up the conditions and that both sequences are convergent to zero but with one we have a convergent series and the other one a divergent one, then take</p>
<p>$$\begin{align*}a_n&=\frac1n\;\;\;\text{divergent series}\\
a_n&=\frac1{n^2}\;\;\;\text{convergent se... |
265,494 | <p>I have two lists given by:</p>
<pre><code>t1 = {{1, 2}, {3, 4}, {5, 6}};
t2 = {a, b, c};
</code></pre>
<p>and want to replace the second parts of <code>t1</code> with <code>t2</code> to get</p>
<pre><code>{{1,a},(3,b},{5,c}}
</code></pre>
<p>I tried</p>
<pre><code>t1 /. {u_, v_} -> {u, #} & /@ t2
</code></pre... | lericr | 84,894 | <p>"Correct form" would depend on how the lists are "more complicated". ReplaceAll works best for patterns, but you don't need patterns. I'd probably start with something like this:</p>
<pre><code>Transpose[{t1[[All, 1]], t2}]
</code></pre>
|
2,743,266 | <p>I am able to find the sixth derivative of $\cos(x^2)$ by simply replacing the $x$ in the Taylor series for $\cos(x)$ with $x^2$ but beyond simple substitutions, I am struggling... </p>
<p>Thanks for any help!</p>
| CY Aries | 268,334 | <p>$\displaystyle f(x)=\frac{1}{2}+\frac{1}{2}\cos 2x$.</p>
<p>$\displaystyle f^{(6)}(x)=\frac{1}{2}(2)^6(-\cos 2x)=-32\cos2x$.</p>
|
385,887 | <p>I'm looking for an example of a mathematical relation that is symmetric but not reflexive. A standard non-mathematical example is siblinghood. </p>
| André Nicolas | 6,312 | <p>Over say the integers: $x$ is not equal to $y$.</p>
<p>You can play this game with most mathematical notions: not congruent, if you like geometry, not of the same cardinality, if you want set theory, and so on.</p>
|
1,967,928 | <p>I came across a question that interested me recently. It asked the following:</p>
<blockquote>
<p>Prove that if <span class="math-container">$\mathbb R$</span> is homeomorphic to <span class="math-container">$X \times Y$</span>, then <span class="math-container">$X$</span> or <span class="math-container">$Y$</span> ... | Logician6 | 306,688 | <p>UPDATE 2: Ok, I realize critical assertion that $\pi_X \circ h \colon \mathbb{R} \to X$ is a homeomorphism was wrong. Going to see if I can stab at it without connectedness when I wake up.</p>
<p>UPDATE 1: Without loss of generality, suppose that $|X|=\mathbb{R}$. Let $h \colon \mathbb{R} \to X \times Y$ be a homeo... |
3,126,345 | <p><span class="math-container">$$x= \int_{0}^{f(x)} \frac{du}{\sqrt{1-u^2}\sqrt{1-k^2u^2}\sqrt{1-l^2u^2}}$$</span></p>
<p><span class="math-container">$$f(0)=0$$</span></p>
<p>If we apply derivative operation for both sides, we get:</p>
<p><span class="math-container">$$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(... | D.Matthew | 469,027 | <p><span class="math-container">\begin{align*}
w&=\sqrt{\frac{a^2 b^2 c^2 u^2 v^2 \left(u^2-v^2\right)^2+P^2}{4 a^2b^2c^2u^2 v^2 \left(u^2+v^2-a^2u^2v^2-b^2u^2v^2-c^2u^2v^2\right)+Q^2}}\\
\\
P&=u \sqrt{\left(1-a^2 v^2\right) \left(1-b^2 v^2\right) \left(1-c^2 v^2\right)}+v\sqrt{\left(1-a^2 u^2\right) \left(1-b^... |
1,365,882 | <p>The theorem statement is "if $f$ is continuous on $[a,b]$, $f$ is bounded on $[a,b]$". This is proven in the textbook Calculus by the author Apostol by the "method of successive bisection", which I'm sure many are familiar with. The proof is done by contradiction. </p>
<p>Here is my concern with this proof: we take... | Ethan Bolker | 72,858 | <p>Your question is interesting and perceptive - and hard to answer. I think you are really asking about how a proof can make "infinitely many statements". Once you are satisfied with that, you'll manage the rest of the argument just fine.</p>
<p>Well, that is in fact something mathematicians have struggled with for c... |
745,095 | <p>For a second order ODE </p>
<p>y''+10y'+ 21y=0</p>
<p>which was reduced to this quadratic expression
x^2+10x+21=0</p>
<ul>
<li>is there any way to tell whether the expression is bounded that is y(x) is either periodic or has a limit 0 as x tends to infinity?</li>
</ul>
<p>*Does periodic means having only comple... | Alessandro Codenotti | 136,041 | <p>You only need to calculate the discriminant, that is $\Delta=b^2-4ac$, if $\Delta >0$ the equation has two dinstinct real solutions, if $\Delta=0$ then it has a repeated real solution and if $\Delta<0$ then it has no real solutions.
Note that a quadratic with real coefficents cannot have a complex and a real s... |
104,492 | <p>Three people A,B,C attend the following game: from 0~100, the Host will come up a number with Uniform, but he doesn't tell them the number, the attendee will guess a number and the closes one will win. A choose the number first and tell the number, B will tell another different number based on A's number, C choose a... | TMM | 11,176 | <p>Building upon Greg's answer, we can analyze what's best for $B$ then. </p>
<p><strong>C's strategy</strong></p>
<p>As pointed out, depending on $M = \max\{a, (b-a)/2, 1 - b\}$, $C$ will choose $c = a - \gamma$ if $a = M$, $c \in (a,b)$ if $(b-a)/2 = M$ and $c = 1 - b + \gamma$ if $1 - b = M$, for some small value ... |
1,124 | <p>Recently, Oleksandr kindly showed a <a href="https://mathematica.stackexchange.com/questions/1096/list-of-compilable-functions">list of Mathematica commands that can be compiled</a>.
RandomVariate was part of that list. However, whether this can be compiled depends upon the distribution that is being sampled. </p>
... | Andy Ross | 43 | <p>To my knowledge <code>UniformDistribution</code> and <code>NormalDistribution</code> are the only distributions that are directly compilable for <code>RandomVariate</code>.</p>
<p>Consider that sampling from a <code>UniformDistribution</code> is what <code>RandomReal</code> was originally designed to do. This code... |
573,637 | <p>I'm trying to solve this problem.
A = {3,14}</p>
<p>What is the number of elements in this set?</p>
<p>I am thinking about the answer is 1. Because the priority of comma's mathematical decimal function is more than math's grammar in my opinion. I inspired computer languages' mechanism.</p>
<p>If there is a wrong ... | Michael Hoppe | 93,935 | <p>An answer appears here: <a href="http://en.wikipedia.org/wiki/Decimal_mark#Hindu.E2.80.93Arabic_numeral_system" rel="nofollow">http://en.wikipedia.org/wiki/Decimal_mark#Hindu.E2.80.93Arabic_numeral_system</a>
So if you live in Zimbabwe the answer is 2, would you live in Kirgistan the answer was 1.</p>
|
573,637 | <p>I'm trying to solve this problem.
A = {3,14}</p>
<p>What is the number of elements in this set?</p>
<p>I am thinking about the answer is 1. Because the priority of comma's mathematical decimal function is more than math's grammar in my opinion. I inspired computer languages' mechanism.</p>
<p>If there is a wrong ... | John Bentin | 875 | <p>You have written {3,14} with no space. If that is correct, then the set has one element. If the original was {3, 14} (note the space!), then the set has two elements.</p>
|
267,045 | <p>The Fibonacci recurrence $F_n=F_{n-1}+F_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question might even be known. Yet, if true, I like to ask for alternative proofs.</p>
<blockquote>
<p><strong>Question.</str... | Carlo Beenakker | 11,260 | <p><em>Mathematica</em> tells me it's a consequence of the two series (distinguished by $\pm$):
$$\sum_{k=0}^\infty\frac{F_{n\pm k}}{k!}=\frac{e^{\sqrt{5}} \phi^n-(1-\phi)^n}{\sqrt{5}\, \exp(\phi^{\mp 1})},$$
with $\phi$ the golden ratio (the positive solution of $1+1/\phi=\phi$). So the ratio of the $\pm$ series equal... |
3,365,569 | <p>I want to calculate the value of √2 but using any common probability distribution, preferably based on Bernoulli Trials.</p>
<p>I will perform a test in real and observe the output of the test and the output of this test should lead me to the value of √2 <strong>like as we can compute the value of π using <a href="... | pico | 666,807 | <p>The expectation of <span class="math-container">$X_1^2$</span> for dirichlet is correct:</p>
<p><span class="math-container">$$E[x_1^2] = \frac{(a_1)(a_1-1)}{(a_0)(a_0-1)}$$</span></p>
<p>The Variance for Dirichlet is:</p>
<p><span class="math-container">$$Var[x_1] = \frac{a_1(a_0-a_1)}{(a_0^2)(a_0+1)}$$</span></... |
934,051 | <p>Let $A$ and $B$ be sets. Show that $A = \bigcup C$ for some $C \subset B$ iff for every $a \in A$ there exists $b \in B$ such that $a\in b$ and $b \subset A$ </p>
<p>I am having trouble proving the right-to-left implication, but have proved the left-to-right implication. </p>
<p>Could anyone provide guidance? </p>... | André Nicolas | 6,312 | <p>Assume that $n\ge 1$. Let $d$ be a common divisor of $a$ and $x$. </p>
<p>(i) Since $a$ divides $x^n-1$, it follows that $d$ divides $x^n-1$. </p>
<p>(ii) Since $d$ divides $x$, it follows that $d$ divides $x^n$. </p>
<p>From (i) and (ii), we conclude that $d$ divides $x^n-(x^n-1)$, and therefore $d$ divides $1$... |
2,117,481 | <p>what must be added to $x^3-6x^2+11x-8$ to make a polynomial having factor $x-3$?</p>
<p>If the required expression to be added be $K$ then $x^3-6x^2+11x-8+K$ is exactly divisible by $x-3$ but how do I find $K$??</p>
| Community | -1 | <p>We need to add some constant $k $ such that the polynomial $p (x)+k $ is divisible by $(x-3) $, that is, having $3$ as one of its factors. Thus, $$p (3)+k =0$$ $$\Rightarrow (3)^3-6 (3)^2+11 (3) -8 +k =0$$ $$\Rightarrow \boxed {k = 2} $$ Hope it helps. </p>
|
3,420,053 | <p>It is a pigeonhole problem.</p>
<p>I have already known that there are <span class="math-container">$1972$</span> remainders in total and the two numbers which have the same remainder can be subtracted and the difference between the two numbers is divisible by <span class="math-container">$1973$</span>.</p>
<p>BUT... | J. W. Tanner | 615,567 | <p>As an alternative solution, <span class="math-container">$1973$</span> is prime, so <span class="math-container">$10^{1972}\equiv1\mod1973$</span> by Fermat's little theorem, </p>
<p>so <span class="math-container">$1973$</span> divides <span class="math-container">$
\dfrac{10^{1972}-1}9.$</span></p>
|
2,246,777 | <p>I was curious as to whether $$\lim_{x \to 0}\frac{1}{x^2}=\infty $$
Or the limit does not exist? Because doesn't a limit exist if and only if the limit tends to a finite number? </p>
| fleablood | 280,126 | <p>"Because doesn't a limit exist if and only if the limit tends to a finite number? "</p>
<p>It's a matter of notation and/or definition. </p>
<p>I'd argue "The limit is infity" and "the limit does not exist" are <em>NOT</em> contradictory statements:</p>
<p>.....</p>
<p>The symbols $\lim\limits_{x \rightarrow a}... |
277,331 | <p>One can write functions which depend on the type of actual parameter before they are actually called. E.g.:</p>
<pre><code>Clear[f,g,DsQ];
DsQ[x_]:=MatchQ[x,{String__}];
f[i_Integer, ds_?DsQ] :=Print["called with integer i and DsQ[ds]==True"];
f[i_String, ds_?DsQ] :=Print["called with String i and Ds... | Daniel Huber | 46,318 | <p>Here is a solution that minimizes the distance using "Minimize" with conditions.</p>
<p>E.g. for the 2 ellipse:</p>
<pre><code>G[x_, y_, X_, Y_, major_, minor_] = ((x - X)/major)^2 + ((y - Y)/minor)^2;
e1 = G[x1, y1, -1, 1, 2, 1];
e2 = G[x2, y2, 2, 2, 1, 2];
</code></pre>
<p>The expression to minimize and ... |
277,331 | <p>One can write functions which depend on the type of actual parameter before they are actually called. E.g.:</p>
<pre><code>Clear[f,g,DsQ];
DsQ[x_]:=MatchQ[x,{String__}];
f[i_Integer, ds_?DsQ] :=Print["called with integer i and DsQ[ds]==True"];
f[i_String, ds_?DsQ] :=Print["called with String i and Ds... | cvgmt | 72,111 | <ul>
<li><p>We write the parametric equation of the outside ellipsoid by parametric <code>s</code>, that is <code>pt2[s]</code>, and calculate the intersection points <code>pt1[s]</code> of the line <code>{X1,Y1}</code>,<code>pt2[s]</code> and the inside ellipsoid.</p>
</li>
<li><p>Here we use <code>Maximize</code> an... |
1,596,297 | <p>I must prove that $7^n-1$ $(n \in \mathbb{N})$ is divisible by $6$.</p>
<p>My "inductive step" is as follows:</p>
<p>$7^{n+1}-1 = 7\times 7^n-1 = (6+1)\times 7^n-1 = 6\times 7^n+7^n-1$</p>
<p>So now, $6\times7^n$ is divisible by 6, that's obvious. But what about the other part, the $7^n-1$ ? How do we know that i... | AnotherPerson | 185,237 | <p>To assume your induction hypothesis you should use the base case that $7^0-1=0$ is divisible by $6$. Then we can assume that $7^n-1$ is divisible by $6$.</p>
|
14,761 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://mathematica.stackexchange.com/questions/2157/customize-front-end-to-add-notifications-when-evaluation-finishes">Customize front end to add notifications when evaluation finishes?</a> </p>
</blockquote>
<p>How to setup sound alert notifica... | Rojo | 109 | <p>You could set the notebook, global, or cell level option "EvaluationCompletionAction" to "Beep"</p>
|
14,761 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://mathematica.stackexchange.com/questions/2157/customize-front-end-to-add-notifications-when-evaluation-finishes">Customize front end to add notifications when evaluation finishes?</a> </p>
</blockquote>
<p>How to setup sound alert notifica... | Mr.Wizard | 121 | <p>I'd be inclined to use something like this:</p>
<pre><code>Monitor[
Do[Pause[0.2], {i, 50}],
If[Mod[i, 10] == 0, Speak[ToString[2 i] <> " percent done"]]; i
]
</code></pre>
|
125,517 | <p>I'm plotting an ellipse <code>1/(1 - e Cos[x])</code> with <code>e=0.9</code> using <code>PolarPlot</code>. Taking $x\in[0,2\pi)$, i.e. one revolution, everything looks fine. But taking <code>n=500</code> revolutions produces something very vaguely resembling an ellipse:</p>
<pre><code>e = 0.9;
n = 500;
PolarPlot[1... | Feyre | 7,312 | <p>Note that with <code>n=1000</code> you have <code>500</code> revolutions.
<code>MaxRecursion</code> is the go to here, I tend to use <code>PlotPoints</code> only when there are tiny features like narrow peaks which are missing.</p>
<pre><code>AbsoluteTiming[
PolarPlot[1/(1 - e Cos[x]), {x, 0, n π}, Frame -> Tru... |
1,192,137 | <p>Define a linear transformation $T\colon \mathbb{R}^3\to\mathbb{R}^3$, such that $T(x) = [x]_B$ ($B$-coordinate vector of $x$). </p>
<p>$B = \{b_1, b_2, b_3\}$, which is a basis for $\mathbb{R}^3$.</p>
<p>$b_1 = (1, 1, 0)$
$b_2 = (0, 1, 1)$
$b_3 = (1, 1, 1)$</p>
<p>$T$ is a matrix transformation $T(x) = Ax$ for e... | HK Lee | 37,116 | <p>$T : V\rightarrow W$ is a linear transformation. So in $V=W={\bf R}^3$ we choose bases so that we can write $T$ into a matrix form $A$. As you said, in $V$ we have a canonical basis $E:=\{e_i\}$ and in $W$ we choose $B:=\{ b_i\}$. So in fact, $T(x)=[x]_B$ means $$ A[x]_E=[x]_B $$</p>
<p>For instance $$[e_1]_E=(1,0,... |
1,600,332 | <p>I find nice that $$ 1+5=2 \cdot 3 \qquad 1 \cdot 5=2 + 3 .$$</p>
<blockquote>
<p>Do you know if there are other integer solutions to
$$ a+b=c \cdot d \quad \text{ and } \quad a \cdot b=c+d$$
besides the trivial solutions $a=b=c=d=0$ and $a=b=c=d=2$?</p>
</blockquote>
| Olimjon | 300,977 | <p>I found that, through solving the equations, we will have following (if a>b):</p>
<p>$$a=\frac{cd+\sqrt{c^2d^2-4(c+d)}}{2}$$
and
$$b=\frac{cd-\sqrt{c^2d^2-4(c+d)}}{2}$$
I think it will be useful for you to find all of the solutions.
Only you should find the solutions for:
$$c^2d^2-4(c+d)=m^2$$
where $m$ is any pos... |
1,600,332 | <p>I find nice that $$ 1+5=2 \cdot 3 \qquad 1 \cdot 5=2 + 3 .$$</p>
<blockquote>
<p>Do you know if there are other integer solutions to
$$ a+b=c \cdot d \quad \text{ and } \quad a \cdot b=c+d$$
besides the trivial solutions $a=b=c=d=0$ and $a=b=c=d=2$?</p>
</blockquote>
| Travis Willse | 155,629 | <p>First, note that if $(a, b, c, d)$ is a solution, so are $(a, b, d, c)$, $(c, d, a, b)$ and the five other reorderings these permutations generate.</p>
<p>We can quickly dispense with the case that all of $a, b, c, d$ are positive using an argument of @dREaM: If none of the numbers is $1$, we have
$ab \geq a + b = ... |
13,635 | <p>I want to define</p>
<pre><code>isGood[___] = False;
isGood[#] = True & /@ list
</code></pre>
<p>where <code>list</code> is a list of several million integers. What's the fastest way of doing this?</p>
| kirma | 3,056 | <p>My solution is ugly, but task-specific. It builds a bitmap out of machine-sized integers in imperative fashion and uses Compile. This works reasonably in memory usage for ranges that have at least couple percent of True values.</p>
<p>A million integers:</p>
<pre><code>n = 6;
list = RandomInteger[{0, 10^(n + 1)}, ... |
483,392 | <p>For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are inside hyperbolic regions of one of the bulbs on the real axis. What is the probability that a random point on the real a... | njguliyev | 90,209 | <p>Hint: $$((b-1)+1)^k = 1 \quad (\operatorname{mod}\ b-1)$$</p>
|
1,408,893 | <p>How can I find the derivative of $\sqrt{x}$ using first principle. Specifically I'm having difficulty expanding $\sqrt{x + h}$ or rather $(x + h)^.5$. </p>
<p>Is there any generalized formula for the expansion of non integer exponents less than 1? </p>
| frog | 84,997 | <p>You can use the following
$$\begin{aligned}
\frac{\sqrt{x+h}-\sqrt{x}}{h}& =\frac{\sqrt{x+h}-\sqrt{x}}{h}\cdot\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\\& =\frac{1}{\sqrt{x+h}+\sqrt x}.
\end{aligned}$$</p>
|
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