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 |
|---|---|---|---|---|
628,409 | <p>$$f(x)=\max\{x,0\}$$</p>
<p>I want to check whether this function is continuous in its domain $\mathbb{R}$ or not, but unfortunately I have no idea how to start.</p>
| Edward ffitch | 26,243 | <p>Notice that $f(x) = x$ for $x \in [0,\infty)$ and $f(x)=0$ for $x \in (-\infty,0]$. Proving continuity of $f$ on $\mathbb{R}$ from this point should be straightforward.</p>
|
307,144 | <p>Are there irrational numbers for which we know that computing its nth digit would take (at least) linear/polynomial/exponential/superexponential time (wrt to length of n and with "big enough" n)?</p>
| icurays1 | 49,070 | <p>If $A$ is invertible, they will indeed have the same solution because you could go from the second equation to the first by multiplying by $(A^{T})^{-1}$. In general, however, the first may have infinitely many solutions, whereas the second will have a unique solution if you have only linearly independent columns o... |
307,144 | <p>Are there irrational numbers for which we know that computing its nth digit would take (at least) linear/polynomial/exponential/superexponential time (wrt to length of n and with "big enough" n)?</p>
| Countable | 677,593 | <p>Let A = 0. Then <span class="math-container">$A^T = 0$</span> and <span class="math-container">$A^TAx = A^Tb$</span> always and <span class="math-container">$Ax = b$</span> iff b = 0.</p>
|
1,534,693 | <p>I am a little stuck on coming up with geometrical explanation for why the following equalities are true. I tried arguing the $\cos(\theta)$ is the projection to the x-axis of a vector $r$ inside a unit circle, so as it goes around by $2 \pi$, the projections on both the positive and negative part of the x-axis cance... | Intelligenti pauca | 255,730 | <p>We have $\cos(\theta) + \cos(\theta + 2 \pi/3) + \cos(\theta + 4\pi/3) =
\operatorname{Re}(e^{i\theta}+e^{i\theta+i2\pi/3}+e^{i\theta+i4\pi/3}) $ and $\sin\theta+ \sin(\theta + 2 \pi/3) + \sin(\theta + 4\pi/3)=\operatorname{Im}(e^{i\theta}+e^{i\theta+i2\pi/3}+e^{i\theta+i4\pi/3})$, but:
$$
e^{i\theta}+e^{i\theta+i2\... |
4,612,472 | <blockquote>
<p>Five friends <span class="math-container">$-\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$</span> and <span class="math-container">$\mathrm{T}-$</span> go to a movie and sit next to each other in a row. <span class="math-container">$P$</span> can only sit in <span class="math-container">$k$</span> of t... | Átila Correia | 953,679 | <p><strong>HINT</strong></p>
<p>Notice that</p>
<p><span class="math-container">\begin{align*}
\sin(x) - \cos(x) = \frac{1}{\sqrt{2}} & \Longleftrightarrow \sin(x)\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\cos(x) = \frac{1}{2}\\\\
& \Longleftrightarrow \sin(x)\cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4... |
3,282,719 | <p>In Javascript, the largest integer that can be represented exactly is <code>Number.MAX_SAFE_INTEGER</code>, with a value of <span class="math-container">$ 2^{53} - 1$</span>. What is the largest prime value that fits under this value threshold? I cannot find a suitable reference for this value on the internet.</p>
| Wang Weixuan | 549,773 | <p>I believe the question asks for <strong>the largest prime representable in JavaScript</strong>, instead of <strong>the largest prime less than <span class="math-container">$2^{53}$</span></strong>, which is trivial to calculate.
From this perspective, I have to point out that the suggested method using <code>Number.... |
4,768 | <p>How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...</p>
<p>Thanks in advance...</p>
| 薛定谔 | 1,122,170 | <p>Fix some typos (note from Url Math007)</p>
<p>Every permutation is either odd or even but not both.</p>
<p>Proof.First we know from the previous proposition that every permutation can be written as a product of transpositions, so the only problem is to prove that it is not possible to find two expressions for a give... |
7,940 | <p>I have multiple plots of permittivity against frequency, but they're hard to read because the frequency labels go from 4x10^14 to 1x10^15 in Hz. I want to label the frequency with THz instead of Hz, so each value would just get divided by 10^12. How do I do this?</p>
| Simon Woods | 862 | <p>The easiest thing is often to scale the data itself rather than mess around with custom labels for the ticks. For example:</p>
<pre><code>f[x_] := Log[x]
(* unscaled plot *)
Plot[f[x], {x, 4*10^14, 10^15}]
(* scaled plot *)
Plot[f[10^12 x], {x, 400, 1000}]
</code></pre>
<p>Or if you are plotting numerical data:<... |
1,689,892 | <p>I understand how to check if something is a tensor (if it transforms like a tensor, it's a tensor).</p>
<p>I have this example of a rank 2 tensor: $$\tau_{ij}=r_if_j - r_jf_i$$
and I understand that it is a tensor because $$\sum_j\tau_{ij}e_j$$ is a vector (e is some vector). </p>
<p>What I'm trying to find is a m... | Christian Blatter | 1,303 | <p>If you have a manifold $M$ covered with local coordinate systems $(U_\alpha,\phi_\alpha)_{\alpha\in I}$ then the transition maps
$$\psi_{\beta\alpha}:=\phi_\beta\circ\phi_\alpha^{-1}$$
are identified by two indices, but have no tensorial structure.</p>
|
3,822,415 | <p><em>SECTION 2.4 A Deductive Calculus</em> In Enderton's <em>A Mathematical Introduction to Logic</em> divides the set of axioms into several groups. The first group is called "tautologies" on p114, which are obtained from tautologies in Sentential Logic, by <a href="https://math.stackexchange.com/questions... | Mauro ALLEGRANZA | 108,274 | <p><em>Long comment</em></p>
<p>It is not a typo.</p>
<p>Having said that, the Theorem is a sort of "preliminary" result, that will be superseded later by the Soundness and Completeness Th for FOL (<strong>Sect.2.5</strong>, page 131-on).</p>
<p>The key fact (stated in previous Enderton's comment) is that:</p... |
1,254,175 | <blockquote>
<p>Prove that $D_{12}\cong S_3 \times C_2$.</p>
</blockquote>
<p>I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how this will lead necessarily to a unique solutions.</p>
<p>No full solutions please hi... | Timbuc | 118,527 | <p>The usual well known presentation of dihedral groups gives us in this case:</p>
<p>$$D=\left\{\;s,\,t\;:\;\;s^2=t^6=1\,,\,\,sts=t^5 (=t^{-1})\right\}$$</p>
<p>Now, take </p>
<p>$$\;s:=((12)\,,\,1)\;,\;\;t:=((123)\,,\,c)\in S_3\times C_2\;,\;\;C_2=\{1,c\}\;,\;\;c^2=1$$</p>
<p>Observe that $\;s^2=t^6=1\;$ , and</p... |
1,112,926 | <p>Problem: For the sequence $r$ defined by </p>
<p>$$r_n = 3 \cdot 2^n - 4 \cdot 5^n, \ \ \ n \geq 0$$ <br></p>
<p>Prove that {$r_n$} satisfies <br></p>
<p>$$r_n = 7r_{n-1} - 10r_{n-2}, \ \ \ n \geq 2$$</p>
<p>Can this problem be explained and broken down and show the process? I'd like to follow your steps on my o... | Winther | 147,873 | <p>To show that $r_n = 3\cdot 2^n - 4\cdot 5^n$ satisfy the equation then start by writing the equation as</p>
<p>$$r_n - 7r_{n-1} + 10r_{n-2} = 0$$</p>
<p>Now calculate $r_n, r_{n-1}$ and $r_{n-2}$ from the formula. This gives us</p>
<p>$$r_n = \color{red}{3\cdot 2^n - 4\cdot 5^n}$$
$$r_{n-1} = 3\cdot 2^{n-1} - ... |
1,120,826 | <blockquote>
<p>Prove that $2^{3^n} + 1$ can be divided by $9$ for $n\ge 1$.</p>
</blockquote>
<p><strong>Work of OP:</strong> The thing is I have no idea, everything I tried ended up on nothing.</p>
<p><strong>Third party commentary:</strong> Standard ideas to attack such problems include induction and congruenc... | Paddling Ghost | 178,532 | <p>Now, since we are trying to prove something for all n, we should look immediately to proof by induction. For $n = 1$, the result is trivial as $2^{3^1}+1 = 9$. Next, we assume that $2^{3^{(n-1)}} + 1$ is divisible by $9$, say $2^{3^{(n-1)}} + 1=9\cdot k$. So, $2^{3^{(n-1)}}=9\cdot k -1$. Now, we examine $2^{3^n... |
368,117 | <blockquote>
<p>Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$.</p>
</blockquote>
<p>I think the two thing's I'm trying to show are:</p>
<ul>
<li>If sign$(\sigma)$ = sign$(\tau) = \pm 1 \implies$ sign$(\sigma \tau)$ = $1$</li>
<li>Wlog, if sign$(\sigma) = 1$,... | Mikasa | 8,581 | <ul>
<li>Let $\pi\in S_n$ and set $\Delta=\prod_{1\leq i\leq j\leq n}(x_i-x_j)$.</li>
<li>Set $\Delta^{\pi}=\prod_{1\leq i\leq j\leq n}(x_{(i)\pi}-x_{(j)\pi})$.</li>
<li>Show that $\Delta^{\pi}=\text{Sgn}(\pi)\Delta$.</li>
<li><p>Show that for $\pi,\phi\in S_n$, $\text{Sgn}(\pi\phi)=\text{Sgn}(\pi)\text{Sgn}(\phi)$ by ... |
368,117 | <blockquote>
<p>Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$.</p>
</blockquote>
<p>I think the two thing's I'm trying to show are:</p>
<ul>
<li>If sign$(\sigma)$ = sign$(\tau) = \pm 1 \implies$ sign$(\sigma \tau)$ = $1$</li>
<li>Wlog, if sign$(\sigma) = 1$,... | Community | -1 | <p>Given that you are asking this question, you may have not seen the following. But typically, $sgn$ is defined to be a homomorphism from any symmetric group to the multiplicative group $\{1,-1\}$, with the alternating group as kernel (i.e. the even permutations map to 1). So your formula is immediate by the definit... |
92,956 | <p>I am beginner in algebra. I want to know if every power of a prime ideal is a principal ideal. Is the statement correct or is there a counterexample?</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ In a domain D, every power of a prime ideal is principal iff D is a PID. Indeed, if every power of a prime ideal is principal then every prime ideal is principal, so D is a PID, by the proof below. Conversely, in a PID every ideal is principal, hence so is every power of a prime ideal.</p>... |
92,956 | <p>I am beginner in algebra. I want to know if every power of a prime ideal is a principal ideal. Is the statement correct or is there a counterexample?</p>
| Georges Elencwajg | 3,217 | <p>If you consider the ring $\mathbb R[X,Y]$ , its ideal ${\frak m}=(X,Y)\subset \mathbb R[X,Y]$ is prime (do you see why?).
However none of its powers ${\frak m}^n \quad (n\geq 1)$ is principal. </p>
<p>To get you started assume $n=1$ and suppose ${\frak m}$ is principal, that is ${\frak m}=(X,Y)=(f(X,Y))$ for... |
288,234 | <p>The inclusion $I\colon \mathbf{Grpd}\hookrightarrow\mathbf{Cat}$ of groupoids into categories has both a left and a right adjoint $L,R\colon \mathbf{Cat}\to \mathbf{Grpd}$, with $R(C)$ being largest groupoid contained in $C$ and $L(C) = C[C^{-1}]$ being $C$ with all morphisms brutally inverted. Going into $\infty$-c... | Valery Isaev | 62,782 | <p>The inclusion of $\infty$-groupoids into $(\infty,1)$-categories certainly has a left adjoint. The functor $\mathbf{Kan} \hookrightarrow \mathbf{WKan}$ does not have a left adjoint, but the higher left adjoint can be modeled by any <a href="https://ncatlab.org/nlab/show/Kan+fibrant+replacement" rel="noreferrer">fibr... |
456,961 | <p>What is the maximum of ${\frac{(1-\cos x)}{x}}$ in the interval $[0, \pi]$?</p>
<p>I can show that the maximum is less than 1, but I want an exact value.</p>
| ra1nmaster | 88,300 | <p>By the quotient rule, the derivative of your function is: </p>
<p>$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{1-cos(x)}{x} \right )=\frac{x\sin x+\cos x-1}{x^2}$</p>
<p>The maximum will occur when this is equal to 0: </p>
<p>$\frac{x\sin x+\cos x-1}{x^2}=0$</p>
<p>which yields, by multiplying through by $x^2$... |
1,868,755 | <p>I came across this in a set of notes. </p>
<blockquote>
<p>Let $K$ be a field of characteristic $p$ and let $\lambda\in K$. Then $$\lambda^{p-1}=1.$$ </p>
</blockquote>
<p>I've never seen this before. Is it correct?</p>
| Patrick Da Silva | 10,704 | <p>The statement you gave is "known" (i.e. when stated correctly, see the comments or the link) and is called <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow">Fermat's little theorem</a>.</p>
<p>Hope that helps,</p>
|
1,868,755 | <p>I came across this in a set of notes. </p>
<blockquote>
<p>Let $K$ be a field of characteristic $p$ and let $\lambda\in K$. Then $$\lambda^{p-1}=1.$$ </p>
</blockquote>
<p>I've never seen this before. Is it correct?</p>
| David Wheeler | 23,285 | <p>As stated, it isn't true-it is known (but I shall not prove it here) that in a field, $F$, that a polynomial of degree $n$ in $F[x]$ has at most $n$ roots.</p>
<p>If $\text{char}(F) = p$, we certainly have: $x^p - x$ is a polynomial of degree $p$ in $F[x]$, and this factors in the prime field of $F$ (which is isomo... |
2,741,162 | <p>I need to show that $$2B_{2m}\equiv1\pmod{4}$$ for $m\ge2.$ This should be something easy using (at most) Claussen and von Staudt's theorem, but I haven't been successful yet. Writing $B_{2m}=U_{2m}/V_{2m},(U_{2m},V_{2m})=1,V_{2m}>0$ I know $$V_{2m}\equiv2U_{2m}\pmod4,$$
but since $2\mid V_{2m},$ this is not very... | Robert Z | 299,698 | <p>Note that for $n\geq 2$
$$0=2\sum_{k=0}^{2n}\binom {2n+1}kB_k=1-2n+\binom{2n+1}{2}\frac{1}{3}+\sum_{k=2}^{n-1}\binom{2n+1}{2k}(2B_{2k})+(2n+1)(2B_{2n}).$$
Now if $2B_{2m}\equiv 1\pmod{4}$ for $m=2,\dots,n-1$ then
$$(2n+1)(2B_{2n})\equiv -1+2n-3\binom{2n+1}{2}
-\sum_{k=2}^{n-1}\binom{2n+1}{2k}\\=-4^n-4n^2+2n+1\equiv ... |
2,741,162 | <p>I need to show that $$2B_{2m}\equiv1\pmod{4}$$ for $m\ge2.$ This should be something easy using (at most) Claussen and von Staudt's theorem, but I haven't been successful yet. Writing $B_{2m}=U_{2m}/V_{2m},(U_{2m},V_{2m})=1,V_{2m}>0$ I know $$V_{2m}\equiv2U_{2m}\pmod4,$$
but since $2\mid V_{2m},$ this is not very... | Angina Seng | 436,618 | <p>By <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula" rel="nofollow noreferrer">Faulhaber's formula</a> for the $2m$-th powers,$$2^{2m}+1=\frac1{2m+1}\left(2^{2m+1}+\frac{2^{2m}}2+\cdots
+(2m+1)2B_{2m}\right).$$
All terms but the term are multiples of $4$ so we get $1\equiv 2B_{2m}
\pmod4$.</p>
|
424,978 | <blockquote>
<p><img src="https://i.stack.imgur.com/VMq02.png" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/eEZlE.png" alt="enter image description here"></p>
</blockquote>
<p>How does he know that $f^{-1}$ is one-one? Doesn't he have to prove that? Or is he applying his first ... | gvdr | 70,245 | <p>It seems to me you may retrieve the one-to-one property from a similar reasoning as the one used in the proof.</p>
<p>Suppose that $f^{-1}$ is not one-to-one. This means there exist $a < b$ such that $f^{-1}(a) = f^{-1}(b) = k$. Hence $f(k) = f(f^{-1}(a)) = a$ and $f(k) = f(f^{-1}(b)) = b$, that is $a = b$. But ... |
2,274,514 | <p>I want to know how to evaluate $\int \frac{\log x}{x^2}$.
Using by parts, and after moving terms, we get something like
$$2 \int\frac{\log x}{x^2} = \frac{(\log x)^2}{x} + \int \frac{(\log x)^2}{x^2}$$
Using by parts again gives
$$\int \frac{(\log x)^2}{x^2} = \frac{2}{3}(\log x)^3 - \int \frac{(\log x)^3}{x^2}$$
At... | Claude Leibovici | 82,404 | <p>Use $$u=\log(x)\implies u'=\frac{dx}x$$ $$v'=\frac {dx}{x^2}\implies v=-\frac 1x$$</p>
|
1,826,711 | <p>How can I find the smallest positive integer $n$ such that
$$(1-0.03)^n<0.03$$</p>
<p>without the help of a computer? </p>
| Conrad Turner | 201,962 | <p>If $(1-0.03)^n<0.03$ we have taking logs: $n\ln(0.97)<\ln(0.03)$ and as both of these logs are negative that gives us: $n>\frac{\ln(0.03)}{\ln(0.97)}$ and so if we are asking for the smallest integer for which the inequalities hold that would be:
$$
n=\left\lceil \frac{\ln(0.03)}{\ln(0.97)}\right\rceil
$$
I... |
156,468 | <p>When I use <code>ListDensityPlot</code>to plot a matrix with dimension $21\times400$, I find the result has dimension $20\times399$.
When the dimension of the matrix is small, for example $5\times 5$, this problem still exists:</p>
<pre><code>mat = Table[Tan[x + y], {x, 1, 5}, {y, 1, 5}];
ListDensityPlot[mat, Inte... | Akku14 | 34,287 | <p>I think, this is near to what you want.</p>
<pre><code>mat = Flatten[Table[{x, y, Tan[x + y]}, {x, 1, 5}, {y, 1, 5}], 1];
ListDensityPlot[mat, InterpolationOrder -> 0]
</code></pre>
<p><a href="https://i.stack.imgur.com/218sG.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/218sG.jpg" alt="ent... |
378,028 | <p>Let us say that a polynomial with real coefficients is <strong>totally real</strong> if all its complex roots are real and distinct. Let <span class="math-container">$P \in \Bbb R [X]$</span> be totally real. Is it true that</p>
<p><span class="math-container">$$Q(X)=\int_0^XP(t)\,dt+aP(X)$$</span></p>
<p>is also to... | CHUAKS | 112,259 | <p>Not an answer but there are linear operators sending any totally real <span class="math-container">$p(x)$</span> of degree <span class="math-container">$n$</span> to another one with degree <span class="math-container">$n+1$</span> with interlacing roots. If <span class="math-container">$a,t$</span> are real number... |
2,182 | <p>I wondered if it is appropriate to ask on mathematica stack exchange a question about what they think about the ergonomy of mathematica in comparison to other softwars (matlab etc).</p>
<p>Because I find mathematica very unfriendly in comparison of everything I learnt but I would have to have other point of view to... | rcollyer | 52 | <p><strong>Partial merge option</strong>: </p>
<ul>
<li>Leave <a href="https://mathematica.stackexchange.com/questions/tagged/geographics" class="post-tag" title="show questions tagged 'geographics'" rel="tag">geographics</a> separate as while the creation of a <code>GeoGraphics</code> object, by hand, is very... |
3,767,129 | <p>Problem:</p>
<p>The vertex of a pyramid lies at the origin, and the base is perpendicular to the x-axis at <span class="math-container">$x = 4$</span>. The cross sections of the pyramid perpendicular to the x-axis are squares whose diagonals run from the curve <span class="math-container">$y = -5x^2$</span> to the c... | Narasimham | 95,860 | <p>Comment.. maybe did not follow?</p>
<p>radius <span class="math-container">$= 5 \cdot 4^2=80$</span></p>
<p>Square pyramid Vol
<span class="math-container">$$ (80 \sqrt 2)^2 \times( h=) 4 \times \frac13=51200/3$$</span>
does not tally</p>
|
1,492,350 | <blockquote>
<p><strong>Find $f(x)$ if $\Delta f(x)=e^x$, where $\Delta f(x)$ is the first order forward difference of $f(x)$, step size $=h=1$.</strong></p>
</blockquote>
<p>Attempt:
We have the definition $\Delta f(x)=f(x+h)-f(x)=f(x+1)-f(x)$</p>
<p>Given $\Delta f(x)=e^x$ i.e $f(x)=\Delta^{-1}e^x=(E-1)^{-1}f(x)$... | Zach466920 | 219,489 | <p>$$(1) \quad \Delta f(x)=e^x$$
Which is equivalent to,
$$(2) \quad f(x+1)=f(x)+e^x$$</p>
<p>Assume that an initial condition for $f(0)$ holds. We then have,</p>
<p>$$(3) \quad f(x)=g(x)+\sum_{n=0}^{x-1} e^n$$</p>
<p>Where $x \ge 1$. The nature of $g(x)$ will be shown momentarily. To prove $(3)$, we'll substitute b... |
1,492,350 | <blockquote>
<p><strong>Find $f(x)$ if $\Delta f(x)=e^x$, where $\Delta f(x)$ is the first order forward difference of $f(x)$, step size $=h=1$.</strong></p>
</blockquote>
<p>Attempt:
We have the definition $\Delta f(x)=f(x+h)-f(x)=f(x+1)-f(x)$</p>
<p>Given $\Delta f(x)=e^x$ i.e $f(x)=\Delta^{-1}e^x=(E-1)^{-1}f(x)$... | user247327 | 247,327 | <p>So you want to solve $$f(x+ 1)- f(x)= e^x.$$ It should be obvious that $f$ must be of the form $$f(x)= Ae^{bx}.$$ Then $$f(x+ 1)= Ae^{bx+ b}= (Ae^b)e^{bx}$$ so that $$f(x+ 1)- f(x)= (Ae^b)e^{bx}- Ae^{bx}= (Ae^b- A)e^{bx}= e^x.$$ We can take $b= 1$ and that reduces to $$Ae^b- A= A(e- 1)e^x= e^x$$ or $A(e- 1)= 1$, ... |
1,049,677 | <p>I'm trying with matrices over $\mathbb F_2$ and trying to have a look at the Jordan canonical forms of these matrices. If the size of the biggest Jordan block is the same with 1's in all diagonal entries, we do get non-similar invertible matrices with same minimal and characteristic polynomial. But what do I do for ... | coffeemath | 30,316 | <p>We cannot have $p=q$ else $2pe^p=1$ leading to $2p=e^{-p}$ which for nonconstant $p$ cannot hold. And neither can we have $p-q=k$ for a constant $k,$ otherwise from $pe^p+(p+k)e^{p+k}=1$ follows $p+(p+k)e^k=e^{-p},$ which again cannot hold for nonconstant $p.$</p>
<p>Now consider the relatin $e^pp'(p+1)+e^qq'(q+1)=... |
4,634,293 | <p>Given a set <span class="math-container">$A$</span> of cardinality <span class="math-container">$n$</span>, let <span class="math-container">$\mathbb{P}(A)$</span> be the power set of <span class="math-container">$A$</span>. What is the number of edges of the intersection graph of the powerset of <span class="math-c... | Misha Lavrov | 383,078 | <p>To specify an ordered pair <span class="math-container">$(x,y)$</span> with <span class="math-container">$x,y \in \mathbb P(A)$</span>, we can go through all <span class="math-container">$n$</span> elements <span class="math-container">$i \in A$</span> and decide whether <span class="math-container">$i \in x$</span>... |
610,029 | <p>The time between successive cars on a certain road is exponentially distributed and the probability is $1/2$ that the next car will arrive within two minutes. Assume the time between and particular pair of cars is independent of the times between all other pairs of cars. </p>
<ol>
<li><p>What is the probability the... | Bill Dubuque | 242 | <p>Another way is to specialize the ascent in the <a href="https://mathoverflow.net/questions/33697/assistance-with-understanding-parent-child-relationships-in-pythagorean-triples/33726#33726">ternary tree of Pythagorean triples</a> (see below). Specializing $\,z = y+3\,$ in the $\rm\color{#c00}{\,formula\,}$below yiel... |
1,236,753 | <p>I am trying to compute $\int x\ln (x+1)\, dx$. I tried integrating by parts and ended up with:
$$\int x\ln(x+1)\,dx = \frac{1}{2}x^2\ln(x+1) - \frac{1}{2}\int\frac{x^2}{x+1}\,dx$$ but I'm stuck here.</p>
| Adhvaitha | 228,265 | <p>$$\int \dfrac{x^2}{x+1}dx = \int \dfrac{x^2-1}{x+1}dx + \int \dfrac{dx}{x+1} = \int (x-1) dx + \int \dfrac{dx}{x+1} = \dfrac{x^2}2 - x + \ln(x+1)+ c$$</p>
|
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | vadim123 | 73,324 | <p>Your plan is exactly backwards.</p>
<p>All proofs should be readable as English prose, i.e. sentences arranged into paragraphs. Symbols may be used as needed, but they need to be human-readable. If you've defined enough symbols, you <em>can</em> write parts of the proof entirely in symbols, provided that they can... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Alex Jones | 350,433 | <p>I really like all the answers here but there's a point that I think is somewhat implied, but not explicitly stated in any of them. </p>
<p>The symbols in mathematics are there as an extension of the language (English in this case), not a replacement of it. A statement can always be written in just words, but this i... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Joonas Ilmavirta | 166,535 | <p><strong>Natural language contains more redundancy and is therefore more robust against errors.</strong>
When you write proofs, chances are that you will sometimes make mistakes.
This is not a feature of mathematics, but of life in general.
If you mistype a character in a symbolic proof, it can become utterly undecip... |
2,141,643 | <p>I am trying to solve the following differential equation:</p>
<p>$$yy''= 3(y')^{2} \\$$</p>
<p>I feel like there must be some substitution to turn this equation into an easier one, but I can not seem to find a substitution that works. Also, I have tried separating the equation into terms of a chain rule, but I can... | DeepSea | 101,504 | <p>We have the quotient rule for taking derivative: $\left(\dfrac{y}{y'}\right)'= \dfrac{(y')^2-yy''}{(y')^2}= 1-3 = -2\implies \dfrac{y}{y'} = -2x+C\implies \dfrac{y'}{y} = \dfrac{1}{-2x+C}$. Can you continue?</p>
|
3,205,693 | <p>Algebras are defined with respect to an underlying ring. Now it makes sense that the underlying ring is smaller. But I was wondering if it is possible to have an underlying ring which is larger. In this case the homomorphism from the ring <span class="math-container">$R$</span> to the algebra <span class="math-conta... | MathematicianByMistake | 237,785 | <p><span class="math-container">$$i)x+2x^2+3xy=6\\ii)2x+x^2+3xy+y=5\\iii)x-x^2+y=7$$</span></p>
<p>We add <span class="math-container">$i)$</span> and <span class="math-container">$iii)$</span> to obtain <span class="math-container">$$x+2x^2+3xy+x-x^2+y=2x+x^2+3xy+y=13$$</span>
But <span class="math-container">$ii)$</... |
3,205,693 | <p>Algebras are defined with respect to an underlying ring. Now it makes sense that the underlying ring is smaller. But I was wondering if it is possible to have an underlying ring which is larger. In this case the homomorphism from the ring <span class="math-container">$R$</span> to the algebra <span class="math-conta... | A.Γ. | 253,273 | <p>Hint: add equations -(1)+(2)-(3).</p>
|
2,405,575 | <p>I have a problem converting an equation. I want to flip an independent to dependent.</p>
<p>Volume of a spherical shell is:</p>
<p>$$V= \frac{4}{3}\pi\bigl[(R+t)^3-(R-t)^3\bigl]$$</p>
<p>Where R is radius, t is half of shell's thickness. It is important to me that it has a form of ± t rather than the most common ... | Claude Leibovici | 82,404 | <p>Starting from $$V= \frac{4}{3}\pi\bigl[(R+t)^3-(R-t)^3\bigl]=\frac{4}{3}\pi\left(6 R^2 t+2 t^3 \right)$$ and intoducing $$x=\frac t R \qquad\text{and}\qquad b=\frac{3 V}{8 \pi R^3}$$ we end with a cubic equation $$x^3+3x-b=0\tag 1$$ Now, let $x=y-\frac 1y$ to get $$y^6-b y^3-1=0 $$ which is a quadratic in $y^3$ mak... |
2,405,575 | <p>I have a problem converting an equation. I want to flip an independent to dependent.</p>
<p>Volume of a spherical shell is:</p>
<p>$$V= \frac{4}{3}\pi\bigl[(R+t)^3-(R-t)^3\bigl]$$</p>
<p>Where R is radius, t is half of shell's thickness. It is important to me that it has a form of ± t rather than the most common ... | dromastyx | 453,578 | <p>Mathematica gives:</p>
<p>$$t(V,R)=\frac{2\cdot(2\pi)^{1/3}R^2}{\left(-3V+\sqrt{256\pi^2R^6+9V^2}\right)^{1/3}}-\frac{\left(-3V+\sqrt{256\pi^2R^6+9V^2}\right)^{1/3}}{2\cdot(2\pi)^{1/3}}$$</p>
|
1,046,229 | <p>For this problem in proving that the cardinality of <span class="math-container">$(0,1)$</span> is equal to that of the set of real numbers, would I just prove that <span class="math-container">$(0,1)$</span> is uncountable, and then use the theorem that the subset of an uncountable set is uncountable, by saying <sp... | EZLearner | 91,223 | <p>Use the following bijection: $ f:(0,1) \to \mathbb{R},$ $ f(x) = \frac{x-\frac{1}{2}}{x(x-1)} $. </p>
<p>This will map $0 \rightarrow -\infty, \frac{1}{2} \rightarrow 0, 1 \rightarrow \infty$, and from the obvious continuity of the function, you could use the intermediate value theorem for the proof of bijection.<... |
1,302,271 | <p>I haven't been able to prove this statement from my Elementary Number course:</p>
<p><strong>There are infinitely many primes $p$ such that $p\equiv -1 \mod12$.</strong></p>
<p>From <a href="http://projecteuclid.org/download/pdf_1/euclid.facm/1229442627" rel="noreferrer">here</a> I know that there exists a "Eulcid... | Jef | 188,361 | <p>Suppose there are only finitely many primes of this form, say $\{p_1,\dots,p_n\}$. Let $P = p_1\dots p_n$ denote their product. Consider now the following expression:
$$Q = 12P^2-1$$
Then we observe the following:</p>
<p><strong>Every prime divisor of $Q$ is $\pm 1 \pmod{12}$</strong> </p>
<p><strong>Proof.</stro... |
3,137,295 | <p>I am doing tasks from Concrete Mathematics by Knuth, Graham, Patashnik for trainning, but there are a lot of really tricky sums like that: <br><br>
Calculate sum <span class="math-container">$$S_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{2^k} + \frac{1}{2} \right\rfloor $$</span></p>
<h2>My idea</h2>
<p>I had th... | Robert Z | 299,698 | <p>Since <span class="math-container">$S_1=1$</span> try to prove that <span class="math-container">$S_n=n$</span> by induction. Note that if <span class="math-container">$n=2m$</span> is even
<span class="math-container">$$\begin{align*}
S_n=\sum_{k=1}^{\infty}\left\lfloor\frac{n}{2^k}+\frac12\right\rfloor
&=\sum_... |
3,137,295 | <p>I am doing tasks from Concrete Mathematics by Knuth, Graham, Patashnik for trainning, but there are a lot of really tricky sums like that: <br><br>
Calculate sum <span class="math-container">$$S_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{2^k} + \frac{1}{2} \right\rfloor $$</span></p>
<h2>My idea</h2>
<p>I had th... | Mike Earnest | 177,399 | <p>Consider how <span class="math-container">$\left\lfloor \frac{n}{2^k} + \frac{1}{2} \right\rfloor$</span> depends on the binary representation of <span class="math-container">$n$</span>. Dividing by <span class="math-container">$2^k$</span> shifts the digits to the right by <span class="math-container">$k$</span>. T... |
278,669 | <p>$f:(0,\infty)\rightarrow\mathbb{R}$ is differentiable, $\lim_{x\rightarrow\infty}f(x)=1$, $\lim_{x\rightarrow\infty}f'(x)=c$ we need to show $c=0$</p>
<p>well, I tried like this $|f(x)-1|<\epsilon\forall x>M$, where $M$ is very large, $|f'(x)-c|<\epsilon\forall x>M$, what more I can say?thank you.</p... | N. S. | 9,176 | <p>Since $\lim_{x \to \infty} \frac{f'(x)}{1}$ exists, by L'Hospital</p>
<p>$$\lim_{x \to \infty} \frac{f(x)}{x}=\lim_{x \to \infty} \frac{f'(x)}{1}=c \,.$$</p>
<p>But the first limit is $0$.</p>
<p><strong>Alternately</strong> Use the MVT on the interval $[x,2x]$. You have</p>
<p>$$\frac{f(2x)-f(x)}{x}=f'(c_x)$$</... |
964,543 | <p>My question is what are the possible order of $A_6$? And how would I show I get $\frac{6!}{2}=360$. Any tips? I know that $A_6$ is the group of even permutations on six elements. I also know that $360=2^3*3^2*5^1$. How do I got about doing this?</p>
| Kaj Hansen | 138,538 | <p>Consider a homomorphism $S_n \rightarrow \{-1, 1\}$ defined such that $\phi(\pi) \mapsto \operatorname{sgn}(\pi)$. </p>
<p>What is the kernel of this homomorphism? From here, apply the fact that $S_n/\ker(\phi) \cong \operatorname{Im}(\phi)$. What must be the orders of each of these?</p>
<p>Alternately, if you'... |
9,758 | <p>What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.</p>
<p>The definition is:</p>
<p>A $n$ × $n$ Hermitian matrix M is called <em>positive-sem... | Qiaochu Yuan | 232 | <p>First I'll tell you how I think about Hermitian positive-definite matrices. A Hermitian positive-definite matrix $M$ defines a sesquilinear inner product $\langle Mv, w \rangle = \langle v, Mw \rangle$, and in fact every inner product on a finite-dimensional inner product space $V$ has this form. In other words it... |
9,758 | <p>What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.</p>
<p>The definition is:</p>
<p>A $n$ × $n$ Hermitian matrix M is called <em>positive-sem... | Andrey Rekalo | 723 | <p>Let's consider the set $E$ of all vectors $y=Mx$, where $x\in\mathbb R^n$ belongs to the unit sphere (i.e. $\|x\|=1$). In other words, $E$ is the image of the unit sphere under the linear transformation. If the matrix is non-degenerate, $E$ is an $n$-dimensional ellipsoid. But if we assume additionally that $M$ is s... |
2,943,037 | <p>Statement : Prove that <span class="math-container">$SL(n,\mathbb{Z})$</span> is generated by <span class="math-container">$(n^2-n)$</span> elements.</p>
<p>The determinant is a n linear function of the rows of the matrix. Given any matrix, if the determinant is nonzero, say <span class="math-container">$det(x_1, x... | Fred | 380,717 | <p>Let <span class="math-container">$S=L+L^T$</span>, then <span class="math-container">$S$</span> is symmetric, hence all eigenvalues of <span class="math-container">$S$</span> are real.</p>
<p>It should be clear, that <span class="math-container">$e^{-S}$</span> is symmetric. Now let <span class="math-container">$ \... |
11,154 | <p>I have an image of gold electrodes on a flat substrate obtained with a scanning electron microscope.
I would like to colour in all the gold electrodes and particles that appear in the center of the image and make the background a different colour.</p>
<p>I'm trying to use some straightforward processing; applying <... | cormullion | 61 | <p>You could try another binarizing function. For example:</p>
<pre><code>mask = MorphologicalBinarize[pq, {.5, 0.6}]
GraphicsRow[{mask,
ImageApply[# {1., 0.843104, 0.} &, q, Masking -> mask]},
ImageSize -> 500]
</code></pre>
<p><img src="https://i.stack.imgur.com/hcYEA.png" alt="image of two gold thing... |
3,667,131 | <p>I have this indefinite integral , with <span class="math-container">$a\in \Bbb R, \: a\neq 0$</span></p>
<p><span class="math-container">$$\int \frac{dx}{\sqrt{a^2+x^2}}, \tag 1$$</span></p>
<p>I solve the integral <span class="math-container">$(1)$</span> with <span class="math-container">$x=at$</span>, and using... | Vishu | 751,311 | <p>Hint: Substitute <span class="math-container">$x= a\tan\theta$</span> for <span class="math-container">$(1)$</span> and <span class="math-container">$x=\tan \theta$</span> for <span class="math-container">$(2)$</span>.</p>
|
1,224,085 | <p>A series is an expression of the form
$$
\sum_{n=k}^{\infty} a_n
$$
where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would <em>say that</em> the two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the <em>same... | TZakrevskiy | 77,314 | <p>Depends on your understanding of the term "series". If you are talking about the sequence of partial sums, then yes, two series are equal iff their respective terms are equal. If you are talking about the value of the series (suppose it converges), i.e. the limit of partial sums, then no, we can have two series with... |
446,386 | <blockquote>
<p>I we are given a vector space $(V,+,\cdot)$ over a field $\mathbb K$ (where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$), is there the finest topology $\mathcal T$, such that $(V,\mathcal T)$ is a <a href="http://en.wikipedia.org/wiki/Topological_vector_space" rel="nofollow noreferrer">topological ... | Eric Wofsey | 86,856 | <p>In fact, your idea of iteratively taking the topology generated by the structure maps continuous does work--you just have to use the joint addition and scalar multiplication maps rather than the single-variable coordinatewise maps that you propose.</p>
<p>To be precise, given a topology <span class="math-container">... |
2,871,589 | <p>This may be a stupid question, but I was learning some set and group theory and it just made me think. Clearly the continuum is an infinite quantity $\mathfrak{c}$, but the set of all reals is also infinitely long. Or is it that $\sup(\mathbb{R})=\mathfrak{c}$ and $\mathfrak{c}\notin\mathbb{R}$. Regardless this is a... | Henry | 6,460 | <ul>
<li>There are $xy$ tiles</li>
<li>$z$ tiles have mines</li>
<li>$xy-z$ do not have mines</li>
<li>Each tile without a mine has probability $\dfrac1{z+1}$ of being untouched when the last mine is found</li>
<li>So the expected number of tiles untouched when the last mine is found is$\dfrac{xy-z}{z+1}$ </li>
<li>An... |
3,267,311 | <p>The author in this example is trying to show that the norm of Hilbert matrix is less than or equal to <span class="math-container">$\pi$</span>. Hilbert matrix has entries
<span class="math-container">$$a_{ij}=\frac{1}{i+j+1};1\leq i,j \leq \infty$$</span> He used the fact that if <span class="math-container">$\exis... | postmortes | 65,078 | <p>The summation considers only positive integer values for <span class="math-container">$i$</span>, while the integral considers all positive real values of <span class="math-container">$i$</span>. Since all the other terms in the summation are positive, it follows that the summation can be no bigger than the integra... |
3,412,063 | <p>I'm studying the Lorenz dynamical system, and I'm asking myself if the critical points are unstable critical points. </p>
<p>Considering the theory they are unstable - one eigenvalue <span class="math-container">$\in \mathbb{R}$</span> which is negative and 2 complex eigenvalues with a negative real part. But when ... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$a_0v+a_1Tv+\cdots+c_{n-1}T^{n-1} v=0$</span>, Apply <span class="math-container">$T^{n-1}$</span> to both sides to see that <span class="math-container">$a_0=0$</span>. So we get <span class="math-container">$a_1Tv+\cdots+c_{n-1}T^{n-1} v=0$</span> and applying <span class="math-con... |
1,563,429 | <p>Working through some questions and I'm stuck on the following:</p>
<p>Find an example of vector spaces $V$ and $W$, and linear transformations $T:V \to W$
and $S: W \to V$, such that $T\circ S$ = $I_w$ but $S\circ T \neq I_V.$ </p>
| ChocolateAndCheese | 148,590 | <p>Well, first, if $T\circ S = I_W$, then we know that $S$ must be injective, since it has a left-inverse. We also see that $T$ is surjective for similar reasons. This helps us think of some examples. The easiest one I can think of is setting $W = \mathbb{R}$ and $V = \mathbb{R}^2$. We then let $S:V\to W$ be the in... |
587,950 | <p>There are 2 planes.</p>
<p>plane 1: $2x+3y-4z=15$
plane 2: $x+y-4z=17$</p>
<p>How can I find the acute angle between 2 planes with those 2 equations?
thanks ;)</p>
| GTX OC | 98,653 | <p>Angle between 2 planes is <em>defined</em> as the angle between their normals.</p>
|
3,123,120 | <p>Prove that given sequence <span class="math-container">$$\langle f_n\rangle =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{(-1)^{n-1}}{n}$$</span> </p>
<p>is a Cauchy sequence </p>
<p>My attempt :
<span class="math-container">$|f_{n}-f_{m}|=\Biggl|\dfrac{(-1)^{m}}{m+1}+\dfrac{(-1)^{m+1}}{m+2}\cdots\dots+\dfra... | Andrei | 331,661 | <p>You have <span class="math-container">$$w^Tx=\sum_{i=1}^D w_ix_i$$</span> For the derivative with respect to <span class="math-container">$w_i$</span> you can write the function as <span class="math-container">$$\frac 1{1+e^{-\sum_{j=1}^D w_jx_j}}=\frac 1{1+e^{-\sum_{j=1,j\ne i}^D w_jx_j}e^{-w_ix_i}}$$</span>
The te... |
92,654 | <p>Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view.</p>
<p>Suppose we have a sphere. Consider a grid of points on its surface, lying on longitude and latitude lines, equally divided around th... | Lubin | 17,760 | <p>First you solve the problem by any means at all, and then you see how to
optimize the solution for computation. I can do the first part, and I’ll
leave the second part to @occulus.</p>
<p>This is a simple problem in spherical trigonometry, once you look at it
close enough, and I managed to find a kind of a solution... |
4,666 | <p>It looks to me like a number in a base other than base 10 gets evaluated before the evaluator ever gets a chance to be tweaked.</p>
<p>For example, <code>FullForm[16^^abcdef]</code> or even <code>FullForm[HoldAll[16^^abcdef]]</code> both produce <code>11259375</code>.</p>
<p>Am I missing a trick that would get me ... | Rojo | 109 | <p>You can see that the base never survives to the evaluation stage by trying for example </p>
<pre><code>16^^98 // Unevaluated // AtomQ
</code></pre>
<blockquote>
<p>True</p>
</blockquote>
<pre><code>16^^98 // Unevaluated // Head
</code></pre>
<blockquote>
<p>Integer</p>
</blockquote>
<pre><code>Trace[16^^98,... |
415,597 | <p>I am trying to understand the concept of a natural transformation by considering the following example, an exercise from Mac Lane's <a href="http://books.google.com/books/about/Categories_for_the_Working_Mathematician.html?id=cUNdcgAACAAJ" rel="nofollow"><em>Categories for the working mathematician</em></a> (p. 18, ... | Pece | 73,610 | <p>Concerning the further question. You have now shown that $-^S$ determines a functor $T \colon \mathbf{Sets} \to \mathbf{Sets}$. As you might know, the application $X \mapsto X \times S, f \mapsto (f,1_S)$ determines also a functor $U \colon \mathbf{Sets} \to \mathbf{Sets}$.</p>
<p>The question is then to show that ... |
1,343,715 | <p>Convolution is associative on e.g. integrable function on $\mathbb{R},$ but not on distributions. </p>
<p>What about the convolution of measures on an unimodular group $G$?</p>
| Robert Israel | 8,508 | <p>Convolution of finite Borel measures on a topological group is always well-defined and associative. </p>
|
20,807 | <p>我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題，
我可以建置一個私人的社群，
並且讓他們可以在這裡用中文問問題，
並且讓我用中文回答他們嗎？</p>
<hr>
<p>Google translate produces:</p>
<blockquote>
<p>I want to let my abstract algebra class of students through the Internet to be able to ask me questions, I can build a private community, and so that they can ask questions here ... | wythagoras | 236,048 | <p>Vote here! [<strong>no downvotes</strong> please. Only upvotes count]</p>
<p>$$\Huge{\mathrm{Yes.}}$$</p>
|
20,807 | <p>我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題，
我可以建置一個私人的社群，
並且讓他們可以在這裡用中文問問題，
並且讓我用中文回答他們嗎？</p>
<hr>
<p>Google translate produces:</p>
<blockquote>
<p>I want to let my abstract algebra class of students through the Internet to be able to ask me questions, I can build a private community, and so that they can ask questions here ... | bfhaha | 128,942 | <p>This is my solution.
<a href="http://bfhaha.hostei.com/?p=5" rel="nofollow">bfhaha.hostei.com/?p=5</a></p>
<p>Anyone can reply the question using Latex code.
It use \( Latex Code \) instead of the dollar sign $.</p>
<p>But it has no live preview (preview the equation) like Math Stack Exchange :(
It is the most imp... |
1,860,782 | <p>I have the following recurrence relation that I'm trying to solve:</p>
<p>$$f(n)=2f(n-1)-f(n-2)-2$$</p>
<p>The homogeneous part is easy:</p>
<p>The characteristic polynomial $r^2-2r+r=0$ has root $r=1$ with multiplicity 2, so the general solution is:</p>
<p>$$f(n)=An+B$$</p>
<p>for some initial conditions.</p>
... | JMoravitz | 179,297 | <p>$f(n)=2f(n-1)-f(n-2)-2$, $f(0)=f_0$ $f(1)=f_1$</p>
<p>Associated characteristic polynomial for the homogeneous recurrence: $x^2-2x+1=0$ which factors as $(x-1)^2=0$ implying homogeneous part is of the form $An+B$ (<em>one can think of $1^n$'s being present in the same way they would have been had the roots been any... |
696,859 | <p>Is there a closed form for $k$ in the expression
$$am^k + bn^k = c$$
where $a, b, c, m, n$ are fixed real numbers?</p>
<p>If there is no closed form, what other ways are there of finding $k$?</p>
<p>Motivation: It came up when trying to apply an entropy model to allele distribution in genetics. The initial populat... | Joseph Zambrano | 63,654 | <p>In general, if we wish to find a solution to an equation $f(x)=c$ and we can rewrite said equation as $x=g(x)$ where $g(x)$ is some function involving $c$. Then, if possible, we may find an interval $I$ on which $g(x)$ is a contraction. By the Banach fixed-point theorem, the sequence $x_0$, $x_n+1=g(x_n)$ where $x_0... |
1,808,222 | <p>While solving PhD entrance exams I have faced the following problem:</p>
<blockquote>
<p>Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n x_i =1$ and $x_i \ge0$.</p>
</blockquote>
<p>I was trying to use <a href="https://en.wikipedia.org... | wdm81 | 347,292 | <p>The Lagrangian for the problem is
$$L(x,\lambda) = - \sum_{i=1}^n \ln (a_i + x_i) - \lambda \left(1- \sum_{i=1}^n x_i\right) = - \lambda + \sum_{i=1}^n \left( \lambda x_i - \ln(a_i + x_i)
\right).$$
According to the Lagrangian sufficiency theorem (see Theorem 2.1 in [1]), if there exist $\lambda^{\ast}$ and $x^{\ast... |
931,851 | <p>In trying to prove that every tree, <em>T</em>, has at most one perfect matching, I came across this idea:</p>
<blockquote>
<p>Since the matchings are perfect, each vertex has degree $0$ or $2$ in the symmetric difference, so every component is an isolated vertex or a cycle.</p>
</blockquote>
<p>Why is this true... | Palash Kumar Mallik | 295,741 | <p>Since every tree of two or more vertices is two chromatic. Tree with even no of vertices will have the perfect matching as all the vertices with same color can be grouped together and a matching can be established between two groups. But any tree with odd no of vertex will have no perfect matching for obvious reason... |
1,669,923 | <p>Are $\cos^2 \theta$ and $\cos \theta^2$ the same?
I mean be it $\sin,\cos, \tan ,\cot ,\sec,\csc$.
Are they same? Please help a maths noob here.</p>
| Onil90 | 271,255 | <p>No, they are not the same.</p>
<p>When you type $\cos^2 \theta$ you actually mean $(\cos \theta)^2$.</p>
<p>When you type $\cos \theta^2$ you mean $\cos(\theta ^2)$. </p>
|
1,669,923 | <p>Are $\cos^2 \theta$ and $\cos \theta^2$ the same?
I mean be it $\sin,\cos, \tan ,\cot ,\sec,\csc$.
Are they same? Please help a maths noob here.</p>
| Ng Chung Tak | 299,599 | <p>It's a matter of syntax.</p>
<p>In handwriting or literature: $\sin^{2} x \equiv (\sin x)^{2}$ and $\cos x^{2} \equiv \cos (x^{2})$;</p>
<p>however for computer software, say <strong><em>Mathematica</em></strong>:</p>
<p><strong>Cos[x]$^2$</strong> refers to $(\cos x)^2$ while <strong>Cos[x$^2$]</strong> refers t... |
1,672,509 | <p>I missed a couple of my Linear algebra classes, so I'm a little lost on this question...</p>
<p>Given $S_1$, $S_2$, $S_3 : \mathbf{R}^2\to \mathbf{R}^2$ are linear mappings defined by:</p>
<p>$S_1(x_1, x_2) = (x_1-x_2, -2x_1+x_2)$</p>
<p>$S_2(x_1,x_2) = (2x_1- x_2, -4x_1+ x_2)$</p>
<p>$S_3(x_1,x_2) = (-x_1+ x_2 ... | Ross Millikan | 1,827 | <p>Without a calculator, I would just use the fact that at the minimum you must have $\frac {\partial g(x,y)}{\partial x}=\frac {\partial g(x,y)}{\partial y}=0$ That will give you two equations in two unknowns. The symmetry indicates that $x=y$ at the minimum (though you should check that). </p>
<p>My graphing calc... |
2,891,334 | <p>Let $T$ be the smallest positive integer which, when divided by $11,13,15$ leaves remainders in the sets $\{7,8,9\},\{1,2,3\},\{4,5,6\}$ respectively. What is the sum of the squares of the digits of $T$?</p>
<p>My working </p>
<p>After applying CRT, I got </p>
<p>$T\equiv 469+1365a+495b+286c\,\, \left(\mod 2145\r... | Matt | 263,495 | <p>In answer to your specific question, <em>yes</em>. I thought that the simplest (although possibly not quickest) way to check would be to check all the consecutive triplets satisfying the condition for remainder upon division by $15$, up to $184$. For the sake of having a more complete answer, these are:</p>
<p>$\{1... |
2,485,482 | <p>$4*3^x - 9*2^x = 5* 3^\frac x2 * 2^\frac x2$ </p>
<p>I did not understand this equality how to solve it for $x$?</p>
| marty cohen | 13,079 | <p>$4*3^x - 9*2^x = 5* 3^\frac x2 * 2^\frac x2
$</p>
<p>The right side is
$5\cdot 6^\frac x2
$.</p>
<p>The left side can be factored as</p>
<p>$4*3^x - 9*2^x
=(2\cdot 3^\frac x2-3\cdot 2^\frac x2)(2\cdot 3^\frac x2+3\cdot 2^\frac x2)
$.</p>
<p>By inspection,
a solution is $x=4$.
($2\cdot 9= 18,
3\cdot 4 = 12,
(18... |
4,608,244 | <p>It is desirable that the volume of the three-dimensional object is located both inside the cylinder <span class="math-container">$x^2+y^2=1$</span> and inside the sphere <span class="math-container">$x^2+y^2+z^2=4$</span>.</p>
| Abezhiko | 1,133,926 | <p>Your question is not very detailed, nevertheless, I understand that you would like to integrate a function <span class="math-container">$f(\vec{r})$</span> over a volume <span class="math-container">$V$</span> being the cylindrical central part of a sphere of radius 2 centered at the origin, whose base is the unit c... |
338,090 | <p>A is a $100 \times 100$ matrix.</p>
<p>The element in the $i^{th}$ row and $j^{th}$ column is given by $i^2 + j^2$</p>
<p>Find the rank</p>
| Robert Israel | 8,508 | <p>Hint: rank of sum $\le$ sum of ranks.</p>
|
1,906,342 | <p>15 people are randomly assigned to three cars, each holding 4, 5, and 6 people respectively. The owners of the cars are among the 15, and will be randomly assigned to one of the cars. What is the probability that each owner gets assigned to his or her own car? </p>
<p>My thought process:
The total number of ways t... | true blue anil | 22,388 | <p>Nothing wrong in your argument, but the simplest way is to seat the owners properly first, it doesn't matter where the rest go</p>
<p>Thus $Pr = \frac4{15}\cdot\frac5{14}\cdot\frac6{13}$</p>
|
2,080,644 | <p>I know that a function is odd when
$$f(-x) = -f(x)$$
Therefore I can say that if for a function $$-f(x) + f(x) = f(-x) + f(x) = 0$$</p>
<p>Then the function is odd!</p>
<p>I tried to use this <em>trick</em> to prove that $f(x) = \ln\left(x+\sqrt{x^2 + 4}\right) - \ln2$ is odd.</p>
<p>However, I would want to prov... | Bernard | 202,857 | <p>It simply means
\begin{align}
&\ln(-x+\sqrt{x^2+4})-\ln2=-\ln(x+\sqrt{x^2+4})+\ln2\\
\iff &\ln(x+\sqrt{x^2+4})+\ln(-x+\sqrt{x^2+4})=2\ln 2\\
\iff&\ln[(\sqrt{x^2+4}+x)(\sqrt{x^2+4}-x)]=\ln4\\
\iff &(\sqrt{x^2+4}+x)(\sqrt{x^2+4}-x)=4.
\end{align}</p>
|
2,216,418 | <p>Liz and Sara start new jobs on the same day. Liz works three days in a row followed by $1$ rest day. Sara works $7$ days in a row followed by $3$ rest days. How many days between Day $1$ and Day $1000$ will they both have a rest day ? I know the answer is $100$, but how does this come about ?</p>
| Bérénice | 317,086 | <p>Day $x$ is a rest day for Liz if $x=0\pmod4$.<br>
Day $x$ is a rest day for Sara if $x=8 \text{ or } 9\text{ or } 0\pmod{10}$.</p>
<p>So you have to solve $3$ systems of congruence with the chinese remainder theorem. </p>
|
2,439,140 | <blockquote>
<p>Let $\{a,b,c\}\subset\mathbb R$ such that $a+b+c=3$ and $abc\ge -4$. Prove that: $$3(abc+4)\ge 5(ab+bc+ca).$$</p>
</blockquote>
<hr>
<p>*) $ab+bc+ca<0$ This ineq is right</p>
<p>*) $ab+bc+ca\ge 0$ then in $ab,bc, ca$ at least a non-negative number exists assume is $ab$</p>
<p>$\Rightarrow \dis... | Community | -1 | <p>We have <span class="math-container">$a+ b+ c= 3$</span>, in <span class="math-container">$a,\,b,\,c$</span> always at least a positive number exists. Assume <span class="math-container">$a> 0$</span>, hypothesis
<span class="math-container">$$abc\geqq -\,4\,\therefore\,bc\geqq -\,\dfrac{4}{a}$$</span>
On the oth... |
2,946,986 | <p>Suppose <span class="math-container">$(P, \leq)$</span> is a partially ordered set. </p>
<p>For <span class="math-container">$x \in P$</span>, define <span class="math-container">$U_x := \{ y \in P \ | \ y \geq x\}$</span>.</p>
<p>Is it true that for any <span class="math-container">$x,y \in P$</span>, either <spa... | Phil H | 554,494 | <p><span class="math-container">$27$</span> atoms would be arranged as <span class="math-container">$3\times 3\times 3$</span> so assuming atoms are positioned at the very edge of the cube the spacing would be <span class="math-container">$\frac{1}{3-1} = \frac{1}{2}$</span> m </p>
<p>So <span class="math-container">$... |
4,421,316 | <p>Consider the sum,
<span class="math-container">$$\sum_{k=0}^n F_{4k}$$</span>
I would like to find this sum,
<span class="math-container">$F_n$</span> being the <span class="math-container">$n$</span>-th Fibonacci number.</p>
| true blue anil | 22,388 | <p><strong>Corrected and simplified answer :28 April '22</strong></p>
<p><span class="math-container">$1-\left(1-\dfrac{\binom81}{\binom{16}2}\right)\left(1- \dfrac{\binom61}{\binom{14}2}\right)\left(1-\dfrac{x}{\binom{12}2}\right)$</span></p>
<p>The second big <span class="math-container">$(...)$</span> evaluates to <... |
3,020,674 | <p>I need to use Fermat's Theorem to prove that 10001 is not prime. I understand that I just need to find a counterexample where <span class="math-container">$a^{10000}$</span> mod 10001 = 1 mod 10001 does not hold true, but this seems kind of difficult with such large numbers. Any ideas?</p>
| LAAE | 359,940 | <p>For 5-digit size of numbers you can still use Fermat's factorisation method and a difference of two squares. If your number <span class="math-container">$10001$</span> has got minimum of two factors, they won't be in such a great distance between each other. In this case it happened that only <span class="math-conta... |
99,936 | <p>I am new to this and I've plotted this:Plot[3 ArcSin[x + 4] - 16, but I don't know what/how to specify the range?</p>
| Bob Hanlon | 9,362 | <p>Extended comment.</p>
<p>Amplifying on the comment by @J.M.</p>
<pre><code>Solve[y == 3 ArcSin[x + 4] - 16, x, Reals][[1]]
(* {x -> ConditionalExpression[-4 + Sin[(16 + y)/3],
1/2 (-32 - 3 π) <= y <= 1/2 (-32 + 3 π)]} *)
</code></pre>
<p>Or</p>
<pre><code>Reduce[y == 3 ArcSin[x + 4] - 16, x, Real... |
560,234 | <p>Given the series</p>
<blockquote>
<p>$$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$</p>
</blockquote>
<p>I need to test for convergence/divergence. I think the divergent test might work here. I could see that the $\lim_{n\rightarrow\infty}(-1)^n\sin(\frac{n}{\pi})$ might not exist, so the series is ... | Theon Alexander | 165,460 | <p>You are working modulo $2\pi$, so you need to think of $\frac{n}{\pi}$ modulo $2\pi \mathbb{Z}.$</p>
<p>Since $1/\pi^2$ is irrational, there are sequences of natural numbers $n_k$ such that the fractional part of $\frac{n_k}{2\pi^2}$ tends, for instance, to $1/3$ (any number such that $sin\ a\neq 0$ will do). This... |
241,586 | <p>It is perhaps well known that the sign function is discontinuous, if defined for $f:\mathbb{R}\rightarrow \mathbb{R}$. However, if we were to define the sign function for $f:\mathbb{R} \setminus \left \{ 0 \right \}\rightarrow \mathbb{R}$, would the sign function still remain discontinuous? </p>
<p>My belief is yes... | Community | -1 | <p>If we restrict the domain to exclude the point $0$, the function becomes continuous. This is easily verified using the definition of continuity. Let $\epsilon>0$ be given. For any $x\neq 0$, simply take $\delta=|x|$. Then any $y$ with $|y-x|<\delta$ will be such that $f(y)=f(x)$ so that $|f(y)-f(x)|=0<\epsi... |
3,744,801 | <p>I've been studying physics and I found this weird differentiation.</p>
<blockquote>
<blockquote>
<p><span class="math-container">$\ln x = \ln a + \ln b$</span></p>
</blockquote>
</blockquote>
<blockquote>
<p>Now differentiating both sides,</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\dfrac{dx}x ... | Peter Szilas | 408,605 | <p>Option:</p>
<p>MVT for integrals.</p>
<p><span class="math-container">$I(x) =(1/x)\int_{-2x}^{x}\ sin (e^t) dt=$</span></p>
<p><span class="math-container">$(1/x)\sin (e^s) \int_{-2x}^{x}dt=$</span></p>
<p><span class="math-container">$(1/x)\sin (e^s) 3x=$</span></p>
<p><span class="math-container">$3\sin (e^s),$</s... |
3,744,801 | <p>I've been studying physics and I found this weird differentiation.</p>
<blockquote>
<blockquote>
<p><span class="math-container">$\ln x = \ln a + \ln b$</span></p>
</blockquote>
</blockquote>
<blockquote>
<p>Now differentiating both sides,</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\dfrac{dx}x ... | Paramanand Singh | 72,031 | <p>Rewrite the expression under limit as <span class="math-container">$$\frac{1}{x}\int_{0}^{x}f(t)\,dt+2\cdot \frac{1}{(-2x)}\int_{0}^{-2x}f(t)\,dt$$</span> where <span class="math-container">$f(t) =\sin(e^t) $</span>. Using Fundamental Theorem of Calculus the above tends to <span class="math-container">$$f(0)+2f(0)=3... |
3,193,823 | <p>I am asked to evaluate: <span class="math-container">$\frac{4+i}{i}+\frac{3-4i}{1-i}$</span></p>
<p>The provided solution is: <span class="math-container">$\frac{9}{2}-\frac{9}{2}i$</span></p>
<p>I arrived at a divide by zero error which must be incorrect. My working:</p>
<p><span class="math-container">$\frac{4+... | Andronicus | 528,171 | <p>You've missed a minus sign there:</p>
<p><span class="math-container">$\frac{4+i}{i}=\frac{4i-1}{-1}=1-4i$</span></p>
<p>The second one is</p>
<p><span class="math-container">$\frac{3-4i}{1-i}=\frac{(3-4i)(1+i)}{(1-i)(1+i)}=\frac{7-i}{2}$</span></p>
<p>Hence the sum is <span class="math-container">$1-4i+\frac{7}... |
3,207,628 | <p>There is this following statement which I need to evaluate to be true or false: </p>
<blockquote>
<p>Let <span class="math-container">$f(x) $</span>, <span class="math-container">$f:\mathbb{R} \rightarrow \mathbb{R_+} $</span> be a continuous probability density function. Then <span class="math-container">$\lim_{... | David | 119,775 | <p>I'm not sure that "the limit is zero except for a countable set of points" makes any sense. It just says that the limit is not zero, and your friend is right.</p>
<p>Furthermore, it's not in fact true that "the limit is zero except for a countable set of points". To make this clear, let's write your friend's cons... |
2,282,578 | <p>Let $X$ have distribution function</p>
<p>$F_{X}(x)$ = \begin{cases}
0, & \text{if } a = 1, \\
\frac{1}{2}z^2, & \text{if 0 ≤ x ≤ 2}, \\
1, & \text{if } x > 2.
\end{cases}</p>
<p>Let $Y = X^2$. </p>
<p>Find $P(X + Y ≤\frac{3}{4})$.</p>
<p>All I have done is:</p>
<p>$P(X + Y ≤\frac{3}{4}) ... | Alex Shtof | 5,073 | <ol>
<li>Even for convex functions, the equation $\nabla f(x) = 0$ is, in most cases of interest, does not have a closed form solution. So you cannot solve it, and you must use an iterative scheme (like Gradient Descent)</li>
<li>The same idea holds for Neural Networks, except that the function is, in most cases of int... |
3,347,170 | <blockquote>
<p><span class="math-container">$$|x+1|+|x|>3$$</span></p>
</blockquote>
<p>I have one simple problem. When I break the modulus function, ie. when I get the following expressions</p>
<p><span class="math-container">$$-x-1-x$$</span> <span class="math-container">$$x+1-x$$</span> and <span class="math... | Milan | 503,397 | <p>Maybe you are troubled by the fact that when multiplied by a negative number inequality switches
<span class="math-container">$$ -2x>4$$</span>
We multiply both sides by <span class="math-container">$-1/2$</span>
<span class="math-container">$$ x<-2 $$</span></p>
|
2,394,613 | <p>Minimizing the following function</p>
<p>$f(x_1,x_2,\cdots,x_n)=\prod\limits_i^n x_i^{x_i}$ </p>
<p>such that </p>
<p>$x_1+x_2+\cdots+x_n=P, 2\le x_i$ and $x_i$ are integers.</p>
<p>My attempt: In my opinion we obtain the result when all $x_i's$ are almost equal i.e. $|x_i-x_j|\le 0$ for all $i$ and $j$. I am tr... | Francesco Alem. | 175,276 | <p>try as a logarithm:
$$
f(x_1,x_2,\cdots,x_n)=\prod_i^nx_i^{x_i}
$$
log:
$$
g(\vec x)=\log(f(\vec x))=\sum_i^n x_i \log(x_i)
$$
Then lagrange method:
$$
\mathcal L(\vec x)=g(\vec x)+\lambda(P-\sum_i^n x_i)=\sum_i^n x_i \log(x_i)+\lambda(P-\sum_i^n x_i)=
$$
$$
=\sum_i^n x_i( \log(x_i) -\lambda)+\lambda P
$$
now impos... |
3,395,177 | <p>I'm interested in knowing whether or not <span class="math-container">$\mathbb{C}[x,y]/\langle x^2+y^2\rangle$</span> is a field, where <span class="math-container">$\langle x^2+y^2\rangle$</span> denotes the ideal generated by the polynomial <span class="math-container">$x^2+y^2\in\mathbb{C}[x,y]$</span> and <span ... | Community | -1 | <p><span class="math-container">$R/I$</span> is a field <span class="math-container">$\iff$</span> <span class="math-container">$I$</span> is maximal.</p>
<p>So <span class="math-container">$R/I$</span> is not a field <span class="math-container">$\iff$</span> <span class="math-container">$I$</span> is not maximal.</p... |
4,274,526 | <p>When proving a limit at <span class="math-container">$a$</span> with value <span class="math-container">$L$</span> with the definition, we must show that for all <span class="math-container">$\epsilon >0$</span>, there is <span class="math-container">$\delta >0$</span> such that:</p>
<p><span class="math-conta... | David C. Ullrich | 248,223 | <p>In fact <span class="math-container">$|x-a|\le|f(x)-L|$</span> does not imply <span class="math-container">$\lim_{x\to a}f(x)=L$</span>. For example, let <span class="math-container">$a=L=0$</span> and <span class="math-container">$$f(x)=\begin{cases}1,&(|x|\le1),
\\|x|,&|x|>1.\end{cases}$$</span></p>
<p>... |
695,948 | <p>I don't have complex analysis at my beck and call, and I only have a low level of knowledge in topology, but I need to prove that this metric space (for any real $r$ and $R$ with $r < R$)$$ X = \{ (x, y) \in \mathbb{R}^2 \ | \ r \leq x^2 + y^2 \leq R \}$$</p>
<p>with the Manhattan metric $d((x_1, y_1), (x_2, y_2... | robjohn | 13,854 | <p><strong>Hint:</strong> Compute the path integral
$$
\int\frac{x\,\mathrm{d}y-y\,\mathrm{d}x}{x^2+y^2}\tag{1}
$$
along the two paths from $(\sqrt{rR},0)$ to $(-\sqrt{rR},0)$ parametrized by $(\sqrt{rR}\cos(\theta),\sqrt{rR}\sin(\theta))$ for $\theta\in[0,\pi]$ and $\theta\in[0,-\pi]$.</p>
<p>Use <a href="http://en.w... |
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