qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,143,200 | <p>The jump diffusion model is defined as
$$dS_t = \mu S_t dt + \sigma S_t dW_t + S_t d \left(\sum^{N_t}_{i=1}(V_i - 1)\right)\;\;\;\;\;\;\;(1)$$
, where ${V_i}$ is a sequence of iid non-negative random variables and it is independent of $W_t$. In the Merton's jump diffusion model ,
$log(V) \sim N(\mu_J, \sigma^2_J)$ ... | ki3i | 202,257 | <p>I suspect that your SDE defining the $S_t$ process is problematic in the term "$S_t\text d\Big(\sum\limits_{i=1}^{N_t}(V_i-1)\Big)$". Intuitively, this term should capture instantaneous random jumps when they occur in any given realisation of the process. That is, if a jump occurs at time $t$, then the process jumps... |
4,056,073 | <p>I need help with this task, if anyone had a similar problem it would help me !</p>
<p>The task is: Determine the type of interruption at the point x = 0 for the function</p>
<p><span class="math-container">$$f(x)=2^{-\frac{1}{x^{2}}}$$</span></p>
<p>I did:</p>
<p><span class="math-container">$$L=\lim_{x\to 0^{-}} 2^... | Community | -1 | <p>There is no paradox.</p>
<p>You cannot reason with real coefficients, because they convey infinite information and you can add them extra bits at no cost. You have to reason with coefficients having a finite representation, hence a finite number of polynomials. Let <span class="math-container">$p$</span> this number... |
48,629 | <p>Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and Riemannian geometry. Therefore, I'm looking for the relations between the two areas. I should also mention, that I'm interes... | Will Sawin | 18,060 | <p>While a real/complex variety, being a subset of a real/complex vector space, will have a Riemannian/Kahler metric, this metric will not be defined up to isomorphism of the algebraic variety. For instance, if the equation defining it is linear, then the variety is isomorphic to a scaled-up version of itself.</p>
<p>... |
88,199 | <p>Is there a function that would satisfy the following conditions?:</p>
<p>$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,</p>
<p>where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in \{0,1,\ldots,255\}$.</p>
<p>I would like to find a function that will have as an input RGB color values (tri... | Community | -1 | <p>I think $f(x,y,z) = (x + 1, y, z)$ if $x$ is even and $f(x,y,z) = (x-1, y, z)$ when $x$ is odd does the trick. Suppose $x$ is even. Then $x+1$ is odd so
$$f(f(x,y,z)) = f(x+1,y,z) = (x,yz)$$
and similarly when $x$ is odd. That the function has no fixed points is obvious because $x \neq x+1$ and $x \neq x-1$ for all ... |
88,199 | <p>Is there a function that would satisfy the following conditions?:</p>
<p>$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,</p>
<p>where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in \{0,1,\ldots,255\}$.</p>
<p>I would like to find a function that will have as an input RGB color values (tri... | Michael Hardy | 11,667 | <p>$$f(x) = 5-x$$</p>
<p>$$f(x) = \frac{16}{x}$$</p>
<p>There are tons, scads, bushels, truckloads, imponderable multitudes, of functions like this. You could fill a fathomless abyss up to <em>here</em> with them.</p>
<p>They're called "involutions".</p>
<p>(Except, technically, the ones that satisfy $f(x)=x$ are ... |
1,627,050 | <p>Let $f:[a,b]\rightarrow R$ be differentiable at $c\in [a,b]$. Show that for every $\epsilon >0$, there is a $\delta(\epsilon) >0$ s.t if $0<|x-y|<\delta(\epsilon)$ and $a\leq x \leq c\leq y \leq b$, then\
$$ |{\frac{f(x)-f(y)}{x-y}-f'(c)}|<\epsilon$$
I can only think of using triangle inequality, but ... | DanielWainfleet | 254,665 | <p>$$\text {Let } d>0 \text { where } 0<|z-c|<d \implies |\frac {f(z)-f(c)}{z-c}-f'(c)|<e/3.$$ For $c-d<x\leq c\leq y<c+d)$ with $x\ne y$ we have $$f(x)=f(c)+(x-c)(f'(c)+e_x),$$ $$ f(y)=f(c)+(y-c)(f'(c)+e_y),$$ $$\text {where }\; |e_x|<e/3 \;\text { and }\; |e_y|<e/3.$$ $$\text {Therefore }\;... |
516,544 | <p>The following is an <a href="http://placement.freshersworld.com/placement-papers/IBM/Placement-Paper-Whole-Testpaper-37851" rel="nofollow">aptitude problem (question no: 29-32)</a>, I am trying to solve:- </p>
<blockquote>
<p>Questions 29 - 32:</p>
<p>A, B, C, D, E and F are six positive integers such that</... | Mufasa | 49,003 | <p>F=2D implies F must be even.<br/>
E+F=2C+1 implies E+F must be odd, hence E must be odd (as F is even).<br/>
Let E=2q+1, hence 2q+1+F=2C+1, hence 2q+2D=2C, hence C=q+D<br/>
C+D+E=2F, hence q+D+D+2q+1=2(2D), hence 3q+1=2D, hence q must be odd.<br/>
Let q=2r+1, hence 6r+4=2D, hence D=3r+2.<br/></p>
<p>Summarising so ... |
1,677,035 | <p>I'm new to this website so I apologize in advance if what I'm going to ask isn't meant to be posted here.</p>
<p>A bit of background though: I haven't been to school in 6 years and the last level I've graduated was Grade 7 due to financial problems, as well as my mom frequently being in and out of the hospital. I a... | G987 | 686,625 | <p>Currently, these two websites are down for me. (If anyone can check, please let me know if it isn't.) I will remove the links in a few days if it is still down:
<a href="http://math30.ca" rel="nofollow noreferrer">math30.ca</a>
<a href="http://math20.ca" rel="nofollow noreferrer">math20.ca</a></p>
<p>The Math 30 we... |
1,643,201 | <p>The spectrum-functor
$$
\operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set}
$$
sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a prime ideal of R}\}$ and a morpshim $f:S\to R$ to the map $\operatorname{Spec}(R)\to \operatorname{Spec}(S)$ with $\m... | Joshua | 430,661 | <p>Find the lower bound and upper bound that is a multiple of 7! so for a three digit integer that is a multiple of 7, 100 is the lowest possible three digit integer and 999 is the highest, however, neither of these are a multiple of 7!
So starting at 100 find a multiple of 7! (105) is the product of (15*7), that is yo... |
1,386,682 | <p>How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$?</p>
<p>I tried $$\lim_{z\to0} \frac{\bar{z}^2}{z}=\lim_{\overset{x\to0}{y\to0}}\frac{(x-iy)^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}\cdot\frac{x-iy}{x-iy} \\ \\ =\lim_{\overset{x\to... | Eclipse Sun | 119,490 | <p>Since the modulus of $\dfrac{\bar z^2}{z}$ is $|z|$, the limit is $0$.</p>
|
1,386,682 | <p>How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$?</p>
<p>I tried $$\lim_{z\to0} \frac{\bar{z}^2}{z}=\lim_{\overset{x\to0}{y\to0}}\frac{(x-iy)^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}\cdot\frac{x-iy}{x-iy} \\ \\ =\lim_{\overset{x\to... | Community | -1 | <p>Let $z=re^{i\theta}$. Then this equals:</p>
<p>$$\lim_{r\to 0} \frac{(re^{-i\theta})^2}{re^{i\theta}}=\lim_{r\to 0} \frac{r^2e^{-2i\theta}}{re^{i\theta}}=\lim_{r\to 0} r e^{-3i\theta}=0$$</p>
|
2,113,777 | <p>I have the following IVP (Initial value problem, Cauchy-Problem), and I do not know how to solve this.</p>
<p>$$y'=e^{-x}-\frac{y}{x} \qquad \qquad y(1)=2$$</p>
<p>I hope you can help me, cause I really do not know how to start.</p>
<p>Thank you! :)</p>
| MatheMagic | 397,530 | <p>\begin{align}
\frac{dy}{dx}+\frac yx&=e^{-x}\\
&\text{IF:}e^{\int\frac1xdx}=x\\
&yx=\int xe^{-x}dx\\
&=-(x+1)e^{-x}+c\\
&\text{putting the initial value $x=1,y=2$}\\
&2=-2e^{-1}+c\\
&c=2+\frac2e\\
&\text{our Ans is: $yx=-(x+1)e^{-x}+2+\frac2e$}
\end{align}</p>
|
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | Brethlosze | 386,077 | <p>$$
6-2√3 \sim 3√2-2\\
8 \sim 3√2 +2√3 \\
64 \sim 30+12√6\\
34 \sim 12√6\\
17 \sim 6√6\\
289 \sim 36 \cdot 6\\
289 > 216
$$</p>
|
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | TonyK | 1,508 | <p>$6-2\sqrt 3 \gtrless 3\sqrt 2-2$ </p>
<p>Rearrange: $8 \gtrless 3\sqrt 2 + 2\sqrt 3$ </p>
<p>Square: $64 \gtrless 30+12\sqrt 6$</p>
<p>Rearrange: $34 \gtrless 12\sqrt 6$</p>
<p>Square: $1156 \gtrless 864$</p>
|
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | Oscar Lanzi | 248,217 | <p>Define</p>
<p>$a=6-2\sqrt 3>0$</p>
<p>$b=3\sqrt 2-2>0$</p>
<p>$a-b = 8 - (2\sqrt 3 + 3\sqrt 2)$</p>
<p>$(2\sqrt 3 + 3\sqrt 2)^2 = 30+12\sqrt 6 = 6×(5+2\sqrt 6)
< 60 < 64$
because $6=2×3 < (5/2)^2$</p>
<p>$a-b > 8-8=0, a>b$</p>
|
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | Will Jagy | 10,400 | <p>Using simple continued fractions for $\sqrt {12}$
and $\sqrt {18}.$ Worth learning the general technique...Method described by Prof. Lubin at <a href="https://math.stackexchange.com/questions/2215918/continued-fraction-of-sqrt67-4/2216011#2216011">Continued fraction of $\sqrt{67} - 4$</a> </p>
<p>$$ \sqrt { 12} ... |
192,125 | <p>Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$
Little help here? >.<</p>
| André Nicolas | 6,312 | <p>We will assume that $x$ ranges over the reals $\ge 4$, to make sure that the square roots are real. Note that
$$\sqrt{x+4}-\sqrt{x-4}=\frac{(\sqrt{x+4}-\sqrt{x-4})(\sqrt{x+4}+\sqrt{x-4})}{\sqrt{x+4}+\sqrt{x-4}} =\frac{8}{\sqrt{x+4}+\sqrt{x-4}} .$$
For $x\ge 4$, $\sqrt{x+4}+\sqrt{x-4}\ge 2\sqrt{2}$. It follows that $... |
351,846 | <p>The following problem was on a math competition that I participated in at my school about a month ago: </p>
<blockquote>
<p>Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.</p>
</blockquote>
<p>I will outline my proof below. I think it has some holes. My approach to the problem was to... | lab bhattacharjee | 33,337 | <p>$$\cos(\sin x)=\sin(\cos x)=\cos\left(\frac\pi2-\cos x\right)$$</p>
<p>$$\text{So}, \sin x=2n\pi\pm \left(\frac\pi2-\cos x\right)$$</p>
<p>$$\text{Taking the '+' sign,} \sin x=2n\pi+ \left(\frac\pi2-\cos x\right)\implies \sin x+\cos x=\frac{(4n+1)\pi}2$$ which can be $\cdots,-\frac{3\pi}2,\frac{\pi}2,\frac{5\pi}2,... |
56,481 | <p>I don't know if the question trivial or not (if so I will delete it), but here is what I have:</p>
<p>I know that </p>
<pre><code>x x
(* x^2 *)
</code></pre>
<p>Looking at the full form gives</p>
<pre><code>FullForm[Times[x, x]]
(* Power[x, 2] *)
</code></pre>
<p>It is clear that the <code>Times</code> head h... | Mr.Wizard | 121 | <p>We can use this function to see that the conversion is not made during parsing:</p>
<pre><code>parseString[s_String, prep : (True | False) : True] :=
FrontEndExecute[FrontEnd`UndocumentedTestFEParserPacket[s, prep]]
parseString["x x"]
</code></pre>
<blockquote>
<pre><code>{BoxData[RowBox[{"x", "x"}]], Standard... |
279,808 | <p>I was working on a way of calculating the square root of a number by the method of x/y → (x+4y)/(x+y) as shown by bobbym at <a href="https://math.stackexchange.com/questions/861509/">https://math.stackexchange.com/questions/861509/</a></p>
<p>I tried to do it via functions on mathematica, everything seems correct. W... | Syed | 81,355 | <p>Consider using <code>NestList</code> to see how the solution evolves with every step. The square root of <code>num</code> is being calculated starting at various initial values as shown on <a href="https://math.stackexchange.com/questions/861509/what-is-the-sum-that-the-square-root-button-on-calculator-does-so-i-can... |
3,561,664 | <p>I did part of this question but am stuck and don't know how to continue</p>
<p>I let <span class="math-container">$x= 2k +1$</span></p>
<p>Also noticed that <span class="math-container">$x^3+x = x(x^2+1)$</span></p>
<p>therefore
<span class="math-container">$4m+2 = 2k+1((2k+1)^2+1)$</span></p>
<p>I simplified th... | Toby Mak | 285,313 | <p>Consider <span class="math-container">$f(x) =x^3+x-2$</span> instead. This can be factorised as <span class="math-container">$(x-1)(x^2+x+2)$</span> instead using the remainder theorem (<span class="math-container">$1^3+1-2 = 0$</span>).</p>
<p>If <span class="math-container">$x = 1 \pmod 2$</span>, then <span clas... |
1,386,683 | <p>I posted early but got a very tough response.</p>
<p>Point $A = 2 + 0i$ and point $B = 2 + i2\sqrt{3}$ find the point $C$ $60$ degrees ($\pm$) such that Triangle $ABC$ is equilateral. </p>
<p>Okay, so I'll begin by converting into polar form:</p>
<p>$A = 2e^{2\pi i}$ and $B = 4e^{\frac{\pi}{3}i}$</p>
<p>$\overli... | mvw | 86,776 | <p>How about
$$
\lim_{z\to 0} \frac{\bar{z}^2}{z} =
\lim_{z\to 0} \frac{\bar{z}^2z^2}{z^3} =
\lim_{z\to 0} \frac{\lvert z\rvert^4}{z^3} =
\lim_{z\to 0} \frac{\lvert z\rvert^4}{\lvert z \rvert^3 e^{3i\phi(z)}}
= 0
$$</p>
|
170,240 | <p>I have the following function of ω</p>
<pre><code>f[ω_] := (2 Sqrt[Γ] (4*g2^2 + (κ1 - 2*I*ω) (κ2 - 2*I*ω)))/(4*g2^2 (Γ -
2*I*ω) + (4*
g1^2 + (Γ - 2*I*ω) (κ1 -
2*I*ω)) (κ2 - 2*I*ω))
</code></pre>
<p>And I wish to obtain the poles for the denominator of the function:</p>
<pre><code>wroots1 = x /. Solve[(Denomina... | ulvi | 1,714 | <p>You can use <code>ComplexExpand</code>, but the cube root term (what I define as <code>cRoot</code> below) interferes with its logic.
The following could be a work-around (suppressing all output):</p>
<p><code>wroots2 =</code>
<code>FullSimplify[Conjugate[wroots1],
Assumptions -> {Γ ∈
Reals, κ1 ∈ Reals, κ2 ∈... |
2,668,616 | <p>I am fairly new at MAPLE and I'm having some trouble solving this ODE. </p>
<p>$$(t+1)\frac{dy}{dt}-2(t^2+t)y=\frac{e^{t^2}}{t+1}$$</p>
<p>My initial value problem is $$t>-1, y(0)=5$$</p>
<p>I put the equation in standard form and typed into maple</p>
<p><a href="https://i.stack.imgur.com/rctFl.png" rel="nofo... | acer | 12,448 | <p>The image of your Maple code shows an italic <code>e</code>, which indicates that you may have tried to just type the letter "e" as the base of the natural logarithm. That would be committing a common usage mistake.</p>
<p>If entered correctly the terms like <code>exp(t^2)</code> would get displayed in 2D Output as... |
2,736,426 | <p>Let's imagine a point in 3D coordinate such that its distance to the origin is <span class="math-container">$1 \text{ unit}$</span>.</p>
<p>The coordinates of that point have been given as <span class="math-container">$x = a$</span>, <span class="math-container">$y = b$</span>, and <span class="math-container">$z = ... | Hope | 426,849 | <p>I used a way finding the angle between two vectors.</p>
<pre><code>angle = aCos ( (V1.V2) / (|V1|.|V2|) )
</code></pre>
<p>When I want to calculate the angle of a point to x axis.</p>
<p>x Axis is</p>
<pre><code>V1 = 1.i + 0.j + 0.k
</code></pre>
<p>The point is </p>
<pre><code>V2 = a.i + b.j + c.k
</code><... |
2,371,108 | <p>Cubic equations of the form $ax^3+bx^2+cx+d$ can be solved in various ways. Some are easy to easy to factor in a pair, for some the roots can be found out by trial-and-error, some are one-of-a-kind, some can be reduced to a quadratic equation. A compilation of all possible ways to solve cubic equations would be very... | J.G. | 56,861 | <p>As with quadratics, first we divide through to make the polynomial monic and then shift the unknown by a constant so the second highest power has zero coefficient, giving $x^3+px+q=0$. We now use Cardano's method. There exist complex numbers $u,\,v$ with $u+v=x,\,uv=-\frac{p}{3}$ (since if only you knew $x$ we'd jus... |
1,462,379 | <p>I have been given the task to compute $\int_{-1}^1 \sqrt{1-x^2} dx$ by means of calculus. We got the hint to substitute $x=\sin u$, but that only seems to make things more complicated:</p>
<p>$$\int_{-1}^1 \sqrt{1-x^2}dx = \int_{\arcsin-1}^{\arcsin1} \sqrt{1-\sin^2u} \frac{d\arcsin u}{du} du = \int_{\arcsin-1}^{\ar... | Surb | 154,545 | <p><strong>Hint</strong></p>
<p>$$\int_{-1}^1\sqrt{1-x^2}dx=\int_{\arcsin(-1)}^{\arcsin(1)}\sqrt{1-\sin^2(x)}\cos(x)dx=...$$</p>
|
3,382,464 | <p>Let <span class="math-container">$g$</span> be a <strong>smooth</strong> Riemannian metric on the closed <span class="math-container">$n$</span>-dimensional unit disk <span class="math-container">$\mathbb{D}^n$</span>. Let <span class="math-container">$f$</span> be a harmonic function w.r.t <span class="math-contain... | YiFan | 496,634 | <p>Per the suggestion of fleablood, I'm going to assume the question asks you to find the integer <span class="math-container">$n$</span> so that <span class="math-container">$n<a^6<n+1$</span>. You already obtained
<span class="math-container">$$a^7=2a^2-a+2,$$</span>
so we divide through on both sides by <span ... |
3,438,653 | <p>I have this thing written on my notes: let <span class="math-container">${x}, {y}\in\mathbb R^n$</span> be two distinct points, then the set <span class="math-container">$$\{ \lambda x + (1-\lambda){y}\;\lvert\; \lambda \in [0,1] \}$$</span> contains all the points on the line segment that connects <span class="math... | herb steinberg | 501,262 | <p>Any point on the line passing through <span class="math-container">$x$</span> and <span class="math-container">$y$</span> is expressed as a linear combination of these points. The requirement on <span class="math-container">$\lambda$</span> insures that the points are in between <span class="math-container">$x$</sp... |
2,386,689 | <p>I know I can find solution for These equations using subtraction</p>
<p>$3x+2y=14.....(\text{i})$</p>
<p>$4x+3y=20.....(\text{ii})$</p>
<p>My question is can I divide $(\text{i})$ by $(\text{ii})$ to get values of $x$ and $y$ ?</p>
| hamam_Abdallah | 369,188 | <p>Your equations can be written as
$$3x=14-2y $$
$$4x=20-3y $$</p>
<p>$x=0$ yields to a contradiction.
we can assume $x\ne 0$.</p>
<p>by division, we get
$$\frac {3x}{4x}=\frac {14-2y}{20-3y}=\frac {3}{4} $$</p>
<p>thus
$$4 (14-2y)=3 (20-3y )$$
or
$$y=60-56$$
You can finish.</p>
|
3,996,218 | <p>Well I wanted to know whether or not <span class="math-container">$y = x^2 + x + 7$</span> is a quadratic equation since the general form is <span class="math-container">$ax^2 + bx + c = 0$</span> here the equation
<span class="math-container">$y=x^2+x+7$</span>. Isn't equal to zero so I'm a bit confused</p>
| Teshan | 505,730 | <p>It could be taken as both a relationship between <span class="math-container">$y$</span> and <span class="math-container">$x$</span> or as a quadratic equation, depending on the context of the problem.</p>
<p>Most of the time the form <span class="math-container">$y=ax^2+bx+c$</span> is used to indicate the relation... |
92,983 | <p><strong>Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a <em>topological</em> triangulation with complexity $O(n)$?</strong></p>
<p>Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ triangular facets, possibly with positive genus. A <em>topological</em> triangulation of $P$... | Sam Nead | 1,650 | <p>The question in the comment was as follows.</p>
<blockquote>
<p>Is there an infinite family of (hyperbolic) stick knots whose crossing numbers are quadratic in the number of edges?</p>
</blockquote>
<p>The answer is yes. Suppose that $T$ is the $(p,q)$-torus knot, with $2 \leq p < q \leq 2p$. Wikipedia tells... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Robin Chapman | 4,213 | <p>One big book on distributions is the first volume
of Hormander's <em>The Analysis of Linear Partial Differential Operators</em>.
This may not be the easiest book to read, but it is comprehensive
and a definitive reference.</p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| John D. Cook | 136 | <p>Robert Adams' <em>Sobolev Spaces</em>. Maybe not the best first book, but a very good second book.</p>
|
618,339 | <p>Could somebody please check my solution?</p>
<p>I want to check, whether $\sum\limits_{n=1}^{\infty}\frac{(1+\frac{1}{n})^n}{n^2}$ converges or diverges.</p>
<p>Using the Comparison test:</p>
<p>Let $a_n = \frac{(1+\frac{1}{n})^n}{n^2},~ b_n=\frac{1}{n^2}$</p>
<p>Since $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$ c... | mathematics2x2life | 79,043 | <p>Your solution is correct. However, the simpler trick implied by the problem is to notice the numerator. Recall that
$$
\lim_{n \to \infty} \left(1+\frac{1}{n}\right)^n=e
$$
Since all the terms are positive, we have
$$
0\leq \sum_{n=1}^\infty \frac{\left(1+\frac{1}{n}\right)^n}{n^2} \leq \sum_{n=1}^\infty \frac{e}{n... |
3,752,455 | <blockquote>
<p><strong>Problem.</strong> Show that for <span class="math-container">$n\ge 2$</span> there are no solution <span class="math-container">$$x^n+y^n=z^n$$</span> such that <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> are prime num... | Sage Stark | 745,622 | <p>First, by Fermat's Last Theorem, this equation has no integral solutions, and thus no prime solutions, for all <span class="math-container">$n\geq 3$</span>, so we only need to focus on the <span class="math-container">$n=2$</span> case.</p>
<p><strong>Case 1: <span class="math-container">$x, y, z>2$</span></stro... |
3,752,455 | <blockquote>
<p><strong>Problem.</strong> Show that for <span class="math-container">$n\ge 2$</span> there are no solution <span class="math-container">$$x^n+y^n=z^n$$</span> such that <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> are prime num... | Wolfgang Kais | 640,973 | <p>If <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <span class="math-container">$z$</span> are primes with <span class="math-container">$x^n+y^n=z^n$</span> for an integer <span class="math-container">$n\ge 2$</span>, then they can't all be odd, so at least one of them must be ev... |
3,837,548 | <p>The triple integral
<span class="math-container">$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{1-xyz}=\zeta(3) \dots (1)$$</span>
is not separable in <span class="math-container">$x,y,z$</span> and the integral representation of reciprocal: <span class="math-container">$\frac{1}{1-xyz}=\int_{0}^{1} t^{-xy... | Angina Seng | 436,618 | <p>Expand out as a geometric progression. You get
<span class="math-container">$$\sum_{n=0}^\infty\int_0^1\int_0^1\int_0^1x^n y^nz^n\,dx\,dy\,dz.$$</span>
Now do the integral.</p>
|
3,837,548 | <p>The triple integral
<span class="math-container">$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{1-xyz}=\zeta(3) \dots (1)$$</span>
is not separable in <span class="math-container">$x,y,z$</span> and the integral representation of reciprocal: <span class="math-container">$\frac{1}{1-xyz}=\int_{0}^{1} t^{-xy... | Z Ahmed | 671,540 | <p><span class="math-container">$$I=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{1-xyz}=
\int_{0}^{1} \int_{0}^{1} \frac{ln(1-yz)}{yz} dy dz =\int_{0}^{1} \int_{0}^{z} \frac{\ln(1-u)}{u} \frac{du dz}{z}, u=yz$$</span>
<span class="math-container">$$ \implies I=\int_{0}^{1}\int_{0}^{z}[1+u/2+u^2/3+u^3/4+....]\... |
4,101,974 | <p>I'm trying to understand what does this matrix operator norm means and what it does to matrix A. <span class="math-container">$${{\left\| A \right\|}_{1,\,\infty }}:={{\max }_{{{\left\| x \right\|}_{\infty }}=1}}{{\left\| Ax \right\|}_{1}}$$</span> Can somebody help with the explanation and maybe an example?</p>
| Lee Mosher | 26,501 | <p>Let me reword that sufficient condition in order to make it more explicit:</p>
<blockquote>
<p>... it suffices to know that the representation of <span class="math-container">$1$</span> by the empty word <em>is the</em> unique <em>representation of <span class="math-container">$1$</span> by a reduced word</em>.</p>
... |
935,454 | <p>Suppose we have the integral operator $T$ defined by</p>
<p>$$Tf(y) = \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}f(xy)\,dx,$$</p>
<p>where $f$ is assumed to be continuous and of polynomial growth at most (just to guarantee the integral is well-defined). If we are to inspect that the kernel of the operator, we would... | Jack D'Aurizio | 44,121 | <p>If you are able to prove that, given your constraints, $Tf$ is an analytic function, we have:
$$(Tf)^{(2n+1)}(0) = 0,\qquad (Tf)^{(2n)}(0) = 2^{n+1/2}\cdot\Gamma(n+1/2)\cdot f^{(2n)}(0)$$
hence $Tf\equiv 0$ implies that all the derivatives of $f$ of even order in the origin must vanish. Assuming that $f$ can be well... |
3,491,816 | <p>Find the min and max values of the function <span class="math-container">$$f(x,y)=10y^2-4x^2$$</span> with the constraint <span class="math-container">$$g(x,y)=x^4+y^4=1$$</span>
I have done the following working;
<span class="math-container">$$\frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x... | Martund | 609,343 | <p>There are errors in both of your equations.
<span class="math-container">$$x(2+\lambda x^2)=0$$</span>
<span class="math-container">$$y(5-\lambda y^2)=0$$</span>
Now, if <span class="math-container">$\lambda\ge 0$</span>, from the first equation we get <span class="math-container">$x=0$</span>. If <span class="math... |
1,392,209 | <blockquote>
<p>Evaluate the limit $$\lim_{x \to 0}\left( \frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$$</p>
</blockquote>
<p>My attempt </p>
<p>So we have $$\frac{1}{x^{2}}-\frac{\cos^{2}x}{\sin^{2}x}$$</p>
<p>$$=\frac{\sin^2 x-x^2\cos^2 x}{x^2\sin^2 x}$$
$$=\frac{x^2}{\sin^2 x}\cdot\frac{\sin x+x\cos x}{x}\cdot\fr... | Paramanand Singh | 72,031 | <p>It is done much simpler in the following manner with just one application of LHR.
\begin{align}
L &= \lim_{x \to 0}\left(\frac{1}{x^{2}} - \frac{1}{\tan^{2}x}\right)\notag\\
&= \lim_{x \to 0}\frac{\tan^{2}x - x^{2}}{x^{2}\tan^{2}x}\notag\\
&= \lim_{x \to 0}\frac{\tan^{2}x - x^{2}}{x^{4}}\cdot\frac{x^{2}}... |
3,652,518 | <p>I am no mathematician but have studied mathematics some 20 years ago. So I know basics of number theory but have lost the skills to solve problems. </p>
<p>I was wondering if the equation <span class="math-container">$3ax^2 + (3a^2+6ac)x-c^3=0$</span> in which <span class="math-container">$a$</span> and <span cla... | achille hui | 59,379 | <p>Multiply <span class="math-container">$x + x^2 + \cdots + x^n$</span> by <span class="math-container">$1-x$</span> and rearrange terms, you get
<span class="math-container">$$\begin{array}{c}
x &+& \color{red}{x^2} &+& \color{green}{x^3} &+& \cdots &+&\color{blue}{x^n}\\
&-&... |
1,680,862 | <p>I am attempting to find the expected value and variance of the random variable $X$ analytically (in addition to a decimal answer). $X$ is the random variable <code>expression(100)[-1]</code> where <code>expression</code> is defined by:</p>
<pre><code>def meander(n):
x = [0]
for t in range(n):
... | Umang Gupta | 188,747 | <p>Clement C has already explained the theoritical part. Here is my programmatic explanation. </p>
<p>Let the variables be $X = \sum_1^k x_k$ where $x_k$ is $U(0,1)$</p>
<p>You can calculate mean & variance in two ways :
One is straight forward. Compute X and use following formulae.
$$mean(X) = \sum_1^nX$$
$$va... |
947,254 | <p>The problem is part (b):</p>
<p><b>1.4.7.</b> A pair of dice is cast until either the sum of seven or eigh appears.</p>
<p> <b>(a)</b> Show that the probability of a seven before an eight is 6/11.</p>
<p> <b>(b)</b> Next, this pair of dice is cast until a seven appears twice or until each of a six and e... | awkward | 76,172 | <p>Others have already given some excellent answers to this problem, and the original post was over two years ago. Nevertheless, I would like to show how the problem can be solved by using an exponential generating function.</p>
<p>We know the last number rolled must be a 6 or an 8, and by symmetry we know these two ... |
804,483 | <p>The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign?</p>
<p>$$I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$$
$$I_2 = \int_{-\infty }^{\infty } \frac{\sin ^2(x)}{x \sinh (... | xpaul | 66,420 | <p>For $I_2$, we can use a well-known result:
$$ \int_{-\infty }^{\infty } \frac{\sinh (ax)}{\sinh(bx)}dx=\frac{\pi}{b}\tan\frac{a\pi}{2b}. $$
Note $\sinh(ix)=\sin(x), \tanh(ix)=\tan(x)$. Thus
$$ \int_{-\infty }^{\infty } \frac{\sin (ax)}{\sinh(bx)}dx=\int_{-\infty }^{\infty } \frac{\sinh (iax)}{\sinh(bx)}dx=\frac{\pi}... |
3,689,513 | <p>I've been stuck on one of my homework numbers.
The number precise that the following equation is a non-linear equation of order 1 with x>0.</p>
<p><span class="math-container">$$y' + {{y\ln(y)}\over x}= xy$$</span></p>
<p>So far, I tried 2 different methods to solve them. As suggested by internet (link below): Be... | user577215664 | 475,762 | <p><span class="math-container">$$y' + {{y\ln(y)}\over x}= xy$$</span>
<span class="math-container">$$\dfrac {y'}{y}+\dfrac {\ln y}{x}=x$$</span>
<span class="math-container">$$(\ln y)'+\dfrac {\ln y}{x}=x$$</span>
It's a linear first order DE. Substitute <span class="math-container">$u=\ln y$</span>
<span class="math-... |
878,939 | <p>I have found the eigen vaues, I also know that you can find the eigenvectors through a Gausian Jordan.
-- x1, gauss jordan gives me rows(1 -1/3 ,, 0 0 ), so [a, b] = [1,3]
For vector x2, GJ gives (1 -2/5 ,, 0 0 ), I would assume [a,b] = [2,5], but why did they choose to go with [-2,-5]. I don't get it?</p>
<p... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ By the Division with Remainder Algorithm </p>
<p>$\qquad\qquad\qquad\qquad n\, =\, 4q + r,\ \ \ 0 \le r < 4$</p>
<p>Now notice that $\ 2\mid n\iff 2\mid 4q+r\iff 2\mid r$</p>
<p>That is: $\,n\,$ is even iff its remainder mod $4$ is even iff $\,r \in \{0,2\}$</p>
<p><strong>Remark</s... |
3,417,001 | <blockquote>
<p>I am wondering if the ring of polynomials <span class="math-container">$F[x]$</span> with coefficients in the field <span class="math-container">$F$</span> is ever isomorphic to <span class="math-container">$\mathbb{Z}$</span> for some field <span class="math-container">$F$</span>. </p>
</blockquote>
... | Mark | 470,733 | <p>If there was such an isomorphism of rings then the additive groups would be isomorphic. However, the additive group of <span class="math-container">$\mathbb{Z}$</span> is cyclic. Now, can the additive group of <span class="math-container">$F[x]$</span> be cyclic? </p>
|
127,808 | <p>I have <a href="https://math.stackexchange.com/questions/356925/a-basis-of-the-symmetric-power-consisting-of-powers">asked this question on math.se</a>, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:</p>
<p>Let $V$ be a complex vector spac... | Peter Michor | 26,935 | <p>Can you work with polarization?
<BR>
$2 x_1x_2 = (x_1-x_2)^2 -x_1^2-x_2^2$.
<BR>
If $k=2$ then $(x_i+x_j)^2$ for $i\le j$ is already a basis.
<BR>
In general,
$(x_{i_1}+\dots +x_{i_k})^k$ for $1\le i_1\le\dots\le i_k\le n$ might do the job. </p>
|
4,408,772 | <p>I'm read about a Lienard System in Perko books, but I don't understand how this applies
<a href="https://i.stack.imgur.com/PFVeR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PFVeR.png" alt="enter image description here" /></a>
<a href="https://i.stack.imgur.com/vgttf.png" rel="nofollow noreferr... | Futurologist | 357,211 | <p>To be honest, the theorem you quote is a bit out of context for my taste. For that reason, maybe one can just cook up their own argument (or maybe that argument is analogous to the theorem, I am not sure). So here is what I think, by assessing the situation quickly. I am going to assume we are dealing with the follo... |
46,837 | <p>I am looking for jokes which involve some serious mathematics. Sometimes, a totally absurd argument is surprisingly convincing and this makes you laugh. I am looking for jokes which make you laugh and think at the same time. </p>
<p>I know that a similar <a href="https://mathoverflow.net/questions/1083/do-good-math... | Michael Hardy | 6,316 | <p>Cosgrove's writings in the <em>Mathematical Intelligencer</em> about 20 or 30(?) years ago had lots of puns, many of which would be understood only by mathematicians. E.g. someone was even worse than an unprincipled infiltrator: he was a non-principal ultrafilter. The biographies of Victoria Cross (famous for Cros... |
2,113,413 | <p>A man on the top of the tower, standing in the seashore finds that a boat coming towards him makes 10 minutes to change the angle of depression from $30$ to $60$. How soon will the boat reach the seashore.
<img src="https://i.stack.imgur.com/iHWdl.jpg" alt="enter image description here"></p>
<p>My Attempt,
We kno... | Community | -1 | <p>If the side length of <span class="math-container">$CB $</span> is <span class="math-container">$x $</span>, then in <span class="math-container">$\triangle ABC $</span>, we have length of <span class="math-container">$AB =\sqrt {3}x $</span>. Now in <span class="math-container">$\triangle ABD $</span>, we have <sp... |
2,667,230 | <p>Let (X, d) be a complete metric space. Let$ f : X → X$ be a function such that for all distinct$ x, y ∈ X$ ,</p>
<p>$ d(f^ k (x), f^ k (y)) < c · d(x, y)$, for some real number $c < 1$ and an integer $k > 1$. Show that f has a unique fixed point. </p>
<p>my attempt : i take $f(x) = x $and $f(y) = y$ .now... | G Cab | 317,234 | <p>Supposing "chain" means string, i.e. word, then we are dealing with binary words of lenght $n$, where</p>
<p><em>a) all the ones (including the last) shall be followd by at least $d$ two's</em></p>
<p>In this case we can figure out that each one be followed by a "bumper" string of $d$ two's.<br>
Assuming that the... |
345,310 | <p>This is computed based on the following recursive formula <span class="math-container">$$w_n=\frac{\lambda_nw_{n+1}+\mu_nw_{n-1}+1}{\lambda_n+\mu_n}$$</span> where: <span class="math-container">$n$</span> is the inital state, State <span class="math-container">$0$</span> is absorbing, <span class="math-container">$\... | mike | 143,907 | <p>Here is a suggestion: <span class="math-container">$\omega_n - \omega_{n+1}$</span> is the time to move from state n+1 to n, and it can't be much worse that the return time to state n+1 (proof: every time you return you have a fixed probability of succeeding in getting to state n on next trial, so no worse than a ... |
17,134 | <p>On a very regular basis we see new users that are not accustomed with the use of MathJaX on MSE. Sometimes even some users that aren't that new to the site. Most of us, when this happens, kindly bring to this users attention that there is a <a href="http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tut... | copper.hat | 27,978 | <p>Yes, it should. It took me 5 mins. to locate a pointer to a tutorial (in fact, the one in the question) to add as a comment in response to a new user's MathJax-free question. And I am moderately familiar with the site.</p>
<p>Either the tour or the 'Ask Question' page should have a suitable pointer.</p>
|
151,864 | <p>I would like to generate a random password of a defined length which can easily be typed in with a standard keyboard.</p>
<p>As a start I tried the following:</p>
<pre><code>SeedRandom["pass"];
StringJoin[RandomChoice[CharacterRange[33, 126], 10]
(* "=IP@7mbYcB" *)
</code></pre>
<p>Do you know other solutions?</p... | Mr.Wizard | 121 | <p>A shorter formulation equivalent to your own code is:</p>
<pre><code>FromCharacterCode @ RandomInteger[{33, 126}, 10]
</code></pre>
<blockquote>
<pre><code>"+(pCT4W#;T"
</code></pre>
</blockquote>
<p>However quite a few places only accept alphanumeric passwords, and not all keyboards have the same easily accessib... |
771,959 | <p>Let $p$ be prime, $n \in \mathbb{N}$ and $p \nmid n$. </p>
<p>$\Phi_n$ is the $n$-th cyclotomic polynomial.</p>
<p>How can I find the maximum $n \in \mathbb{N}$ (with $p \nmid n)$ so that $\Phi_n$ splits into linear factors over $\mathbb{Z}/(p)$.</p>
| user88595 | 88,595 | <p>The answer by Kaj Hansen is very good. Here's a slightly different approach with simple equalities like we all know them :
$$a\equiv b \mod n \Longleftrightarrow a = b + kn\Longrightarrow ra = rb + (rk)n$$</p>
<p>Writing it back under modulo form you obtain $ra \equiv rb \mod n$</p>
|
2,138,009 | <p>Let $f(z)=(1+i)z+1$. Then $f(z)=\sqrt 2 e^{i\pi/4}z+1$ and thus $f=t\circ h\circ r$ where $t$ is the translation of vector $1$, $r$ the rotation of center $0$ and angle $\pi/4$ and $h$ the homothetic of parameter $\sqrt 2$. I found a fix point $z=\frac{-1}{2}+\frac{i}{2}$.</p>
<p>1) What if $f\circ f\circ f\circ f\... | Surb | 154,545 | <p>1) The linear part of $f^4$ is $-4z$, therefore, $f^4$ will be of the form $$f^4(z)=-4z+\tau,$$
therefore, it will be an homothetic transformation with scale factor $4$ composed by a point reflexion (you can also say that it's an homothetic transformation with scale factor $-4$) and a translation of vector $\tau$. T... |
1,920,994 | <p>My calculus teacher gave us this interesting problem: Calculate</p>
<p>$$ \int_{0}^{1}F(x)\,dx,\ $$ where $$F(x) = \int_{1}^{x}e^{-t^2}\,dt $$</p>
<p>The only thing I can think of is using the Taylor series for $e^{-t^2}$ and go from there, but since we've never talked about uniform convergence and term by term in... | Math1000 | 38,584 | <p>To rigorously justify the use of <em>Tonelli's</em> theorem, let $g:\mathbb R^2\to\mathbb R$ be defined by $g(x,t) = e^{-t^2}\mathsf 1_E$, where $$E = \{(x,t)\in\mathbb R^2 : 0 < x < t < 1\}. $$
Since $g(x,t)\geqslant 0$ for all $x,t$ and $g$ is measurable as the product of measurable functions, it follows... |
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | bluemaster | 460,565 | <p>Clearly impossible in base 10. Can it be done in another number base?</p>
<p>In base 5: $11+11+3=30$</p>
<p>Actually there are many other possibilities!</p>
<p>Another possibility: fill the boxes with $\binom{5}{3}$,15 and 5.</p>
<p>It holds that $\binom{5}{3}+15+5=30$ (base 10). Notice that $\binom{5}{3}=10$. ... |
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | Eric Towers | 123,905 | <ul>
<li><p>Change the base: $$11_9 + 11_9 + 11_9 = 10+10+10 = 30 \text{.}$$</p></li>
<li><p>Parentheses are in the list of usable box contents. Commas too. But you're upper limited to one pair of parens and to seven commas.
$$(15+15,15)+15 = 30 \\ (15+15,15+15) = 30 \text{.}$$ Here, the parens represent the <a ... |
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | Andrew | 259,577 | <p>What about
6+9+15=30?</p>
<p>they don't stay you have to fill the boxes with the numbers in their usual orientation, after all. :-)</p>
|
1,415,505 | <p>I was trying to find out how to prove </p>
<p>$$ \sin(A-\arcsin(0.3 \ \sin \ A)) \ \cdot \ \sin(A+\arcsin(0.3 \ \sin \ A)) \ = \ 0.91 \ \sin^2 \ A \ \ . $$
When I put this equation into my calculator both sides appear to be exactly the same, but I have no idea how to prove it.</p>
| colormegone | 71,645 | <p>[I thought of another way since I left my comment above.]</p>
<p>Applying the "product-to-sum" formula for two sine factors,</p>
<p>$$ \sin \ \alpha \ \cdot \ \sin \ \beta \ = \ \frac{1}{2} \ [ \ \cos(\alpha \ - \ \beta) \ - \ \cos(\alpha \ + \ \beta) \ ] \ \ , $$</p>
<p>with $ \ \alpha \ \ \text{and} \ \ \beta ... |
539,027 | <p>How many ways are there to distribute $18$ different toys among $4$ children?</p>
<ol>
<li><p>without restrictions</p></li>
<li><p>if $2$ children get $7$ toys each and $2$ children get $2$ toys each.</p></li>
</ol>
<p>For $1$ since toys are different, then there are $4^{18}$ ways to distribute .</p>
<p>for $2$, ... | André Nicolas | 6,312 | <p>Choose the $2$ children who will get $7$ toys. Then choose the $7$ toys the older of these will get. Then choose the $7$ for the younger one. Then choose the $2$ toys for the older of the remaining children. That gives a total of $\binom{4}{2}\binom{18}{7}\binom{11}{7}\binom{4}{2}$. </p>
|
154,722 | <p>Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter.</p>
<p>Now consider the family of representations $r_{\alpha}$ of the free group on two generators $F_2 = \langle a,b\rangle$ in $\mathrm{SL}(2, \mathbb{C})$ setting ... | Danny Ruberman | 3,460 | <p>If you want to know if your group is not discrete, apply Jorgensen's inequality (Jørgensen, Troels (1976), "On discrete groups of Möbius transformations", American Journal of Mathematics 98 (3): 739–749). It says that if $A$ and $B$ generate a non-elementary discrete subgroup of $SL_2(C)$, then
$$
|\text{Tr}(A)^2 ... |
1,105,971 | <p>I've got two independent bernoulli distributed random variables $X$ and $Y$ with parameter $\frac{1}{2}$. Based on those I define two new random variables </p>
<p>$X' = X + Y , E(X') = 1$</p>
<p>$Y' = |X - Y|, E(Y') = \sum_{x=0}^1\sum_{y=0}^1|x-y|*P(X=x)*P(Y=y) = \frac{1}{2}$ </p>
<p><strong>How can I calculate E... | Ian | 83,396 | <p>For general dependent variables with finite variance, $E((X'-E(X'))(Y'-E(Y')))$ could be anything between $-\sqrt{E((X'-E(X'))^2) E((Y'-E(Y'))^2)}$ and $\sqrt{E(X'^2) E(Y'^2)}$. The fact that this must hold follows from the Cauchy-Schwarz inequality, while the fact that any value can be attained can be proven by con... |
3,679,806 | <p>this is Probability Density Function(pdf)
if <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independent, how to prove <span class="math-container">$X^2$</span> and <span class="math-container">$Y^2$</span> are also independent</p>
| Jonathan Hole | 661,524 | <p>That <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independent just means that the sigma algebras they generate are independent. But the sigma algebra <span class="math-container">$X^2$</span> generates is containedin the sigma algebra generates by <span class="math-containe... |
368,461 | <p>Let <span class="math-container">$G=(V,E)$</span> be a finite simple graph. We say a map <span class="math-container">$p:V\to [n]:=\{1,\ldots,n\}$</span> is a <em>pseudo-coloring</em> if for all <span class="math-container">$a\neq b\in[n]$</span> there is <span class="math-container">$v\in\psi^{-1}(\{a\})$</span> an... | bof | 43,266 | <p>Here is a very simple example for <span class="math-container">$\psi(G\times H)\gt\max\{\psi(G),\psi(H)\}$</span>.</p>
<p>The graph <span class="math-container">$G=H=K_3$</span> has <a href="https://en.wikipedia.org/wiki/Complete_coloring" rel="nofollow noreferrer">achromatic number</a> and pseudo-chromatic number e... |
4,545,364 | <blockquote>
<p>Solve the quartic polynomial :
<span class="math-container">$$x^4+x^3-2x+1=0$$</span>
where <span class="math-container">$x\in\Bbb C$</span>.</p>
<p>Algebraic, trigonometric and all possible methods are allowed.</p>
</blockquote>
<hr />
<p>I am aware that, there exist a general quartic formula. (Ferrari... | Yuri Negometyanov | 297,350 | <p>HINT.</p>
<p><span class="math-container">$$4(x^4+x^3-2x+1)=(2x^2+x-1)^2+3(x-1)^2$$</span>
<span class="math-container">$$=\big(2x^2+(1+i\sqrt3\,)(x-1)\big)\big(2x^2+(1-i\sqrt3)(x-1)\big),$$</span></p>
<p>DETAILS.</p>
<ol>
<li><p>The first quadratic term can be chosen in the form of <span class="math-container">$(2x... |
2,179,253 | <p>$$n{n-1 \choose 2}={n \choose 2}{(n-2)}$$
Give a conceptual
explanation of why this formula is true.</p>
| PSPACEhard | 140,280 | <p>Suppose from $n$ students, you want to select $1$ student to clean the blackboard and $2$ students to clean the ground. There are $n$ ways to choose a student to clean the blackboard, and from the remaining students there are totally $\binom{n-1}{2}$ ways to select students to clean the ground. Therefore, the total ... |
1,537,676 | <p>$f(x,y)=\begin{cases}
|\frac{y}{x^2}|exp(-|\frac{y}{x^2}|) & , x\ne0\\
0 & , x=0
\end{cases}$</p>
<p>I need to show that $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ the limit does not exist in $(0,0)^T$. I tried to prove it with the sequence criteria but I could not find a good sequence. Polar form doesn't get ... | Michael Medvinsky | 269,041 | <p>On $y=x^2$ you have
$$\lim\limits_{(x,y)\to(0,0)}\left|\frac{y}{x^2}\right|\exp\left(-\left|\frac{y}{x^2}\right|\right)
=\lim\limits_{(x,y)\to(0,0)}\exp(-1)=\frac{1}{e}$$</p>
<p>Denote
$$f(x,y)=\left|\frac{y}{x^2}\right|\exp(-\left|\frac{y}{x^2}\right|)
$$
on $y=0$
we have $f(x,0)=0$
and therefore
$$\lim\limits_{... |
1,112,081 | <p>Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? Thanks in advance.</p>
| idm | 167,226 | <p>set $u=\sqrt x$ then
$$...=\int_0^\infty 2u^2e^{-u^2}du=-\int_0^\infty u(-2ue^{-u^2})du.$$</p>
<p>Set $f'(u)=-2ue^{-u^2}$ and $g(u)=u$, and by part you'll get
$$-\int_0^\infty u(-2ue^{-u^2})du=\underbrace{-[ue^{-u^2}]_0^\infty}_{=0} +\int_0^{\infty }e^{-u^2}du$$</p>
<p>Now use the fact that $\int_{-\infty }^\infty... |
1,890,040 | <p>This is related to <a href="https://math.stackexchange.com/questions/1888881/expanding-a-potential-function-via-the-generating-function-for-legendre-polynomi">this previous question of mine</a> where (with lots of help) I show that $$\sum_{l=0}^\infty \frac{R^l}{a^{l+1}}P_l(\cos\theta)=\frac{1}{\sqrt{a^2-2aR\cos\the... | hmakholm left over Monica | 14,366 | <p>Here's a graphical count, based on intersecting the configuration with spheres of different sizes -- though in order to make things visible actually I'm normalizing the size of the spheres, such that the images here actually shows a sphere intersected by the planes of a dodecahedron of increasing size.</p>
<p>The 1... |
1,890,040 | <p>This is related to <a href="https://math.stackexchange.com/questions/1888881/expanding-a-potential-function-via-the-generating-function-for-legendre-polynomi">this previous question of mine</a> where (with lots of help) I show that $$\sum_{l=0}^\infty \frac{R^l}{a^{l+1}}P_l(\cos\theta)=\frac{1}{\sqrt{a^2-2aR\cos\the... | Nate | 91,364 | <p>There is a nice combinatorial formula for the number of regions in a hyperplane arrangement in $\mathbb{R}^d$.</p>
<p>$$r(\mathcal{A}) = (-1)^d\chi_\mathcal{A}(-1)$$</p>
<p>Where $r(\mathcal{A})$ denotes the number of regions and $\chi_\mathcal{A}(t)$ is the characteristic polynomial of the underlying poset of fla... |
3,294,369 | <p>I am reading a paper which asserts that the value of the function <span class="math-container">$f(x) = x^{\frac{1}{x}}$</span> at <span class="math-container">$x=0$</span> is equal to <span class="math-container">$0$</span>. I can believe that this is so if I write <span class="math-container">$f(x) = e^{{\frac{1}{x... | Robert Israel | 8,508 | <p>The limit of <span class="math-container">$x^{1/x}$</span> as <span class="math-container">$x \to 0+$</span> is indeed <span class="math-container">$0$</span>, as you see by writing <span class="math-container">$f(x) = \exp(\ln(x)/x)$</span>. The limit as <span class="math-container">$x \to 0-$</span>, on the other... |
510,130 | <p>Let $(r_i)_{i=1}^m$ be a sequence
of positive reals such that
$\sum_i r_i < 1$
and let $t$ be a positive real.
Consider the sequence $T(n)$
defined by $T(0) = t$,
$T(n) = \sum_i T(\lfloor r_i n \rfloor) $
for $n \ge 1$.</p>
<p>Show that
$T(n) = o(n)$,
that is,
$\lim_{n \to \infty} \dfrac{T(n)}{n}
= 0
$.</p>
<p>... | marty cohen | 13,079 | <p>Here is what I have
been able to come up with.</p>
<p>I will show that $T(n)/n$ can be made arbitrarily small,
although the number of steps
seems to be exponential
in the reciprocal of the bound.
More specifically,
for a real $0 < c < 1$,
there are
explicitly computable constants
$A$ and $B$
which both ... |
2,636,712 | <p>I have question about my proof. I could not tell whether it is sufficient enough since my professor approached it differently. </p>
<p><strong>The problem:</strong></p>
<blockquote>
<p>Let $z \in \mathbb{C}^{*}$. If $|z| \neq 1$, prove that the order of $z$ is infinite. </p>
</blockquote>
<p><strong>My proof:</... | Patrick Stevens | 259,262 | <p>The proof looks fine, but I always prefer to do things without contradiction where possible.</p>
<p>Let $z \in \mathbb{C}^*$, and consider the sequence of reals whose $n$th term is $|z^n| = |z|^n$.</p>
<p>If $|z| > 1$, then this sequence is strictly increasing, and hence in particular the sequence $z^n$ cannot ... |
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | NiloCK | 308 | <p>Most experienced mathematicians will immediately identify $x = 0$ as a solution to this equation, and then do the same division (factoring) as your student in order to obtain the other solution at $x=2$. The only difference is that the experienced mathematician has already done due diligence with respect to the poss... |
7,237 | <p>this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and one of the students did $$\frac{6x^2}{6x}=\frac{12x}{6x}$$ $$x = 2$$ which is a valid solution but this method elimin... | user21820 | 1,550 | <p>Firstly, division must obey rules regardless of what you are dividing. If the division is performed over real numbers, then you have to respect the rules for division of real numbers, that is no division by <span class="math-container">$0$</span>. In other structures <span class="math-container">$0$</span> is not th... |
1,818,764 | <p>I think everything I have done is kosher, but unless I am missing an identity it is a different answer than the online quiz and wolfram alpha give.</p>
<p>I tried to use the trig substitution
$$ x=2\sin(\theta)\Rightarrow dx=2\cos(\theta)$$</p>
<p>Which yields
$$\int\frac{x^2}{\sqrt{4-x^2}}dx=\int\frac{4\sin^2(\t... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
\int \frac{x^{2}}{\sqrt{4-x^{2}}} d x &=-\int x d \sqrt{4-x^{2}} \\
&=-x \sqrt{4-x^{2}}+\int \sqrt{4-x^{2}} d x \\
&=-x \sqrt{4-x^{2}}+\frac{x \sqrt{4-x^{2}}}{2}+2 \sin ^{-1}\left(\frac{x}{2}\right)+C\\&= -\frac{x \sqrt{4-x^{2}}}{2}+2 \sin ^{-1}\left(\f... |
2,638,679 | <p><a href="https://i.stack.imgur.com/S4p0Y.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/S4p0Y.jpg" alt="enter image description here"></a></p>
<p>Due apologies for this rustic image. But while drawing this lattice arrangement about the "square numbers" , I discovered a pattern here wherein if I ... | Joca Ramiro | 264,635 | <p>The eigenvalues $\lambda_1$ and $\lambda_2$ are the roots of the characteristic polynomial $\det(A-\lambda 1)=a\lambda^2+b\lambda+c$, where $a,b,c$ are real numbers. It is easy to see that $a=1$. Being the polynomial second degree, the sum of its roots is $\lambda_1+\lambda_2=-b/a=3$ and their product $\lambda_1\c... |
2,638,679 | <p><a href="https://i.stack.imgur.com/S4p0Y.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/S4p0Y.jpg" alt="enter image description here"></a></p>
<p>Due apologies for this rustic image. But while drawing this lattice arrangement about the "square numbers" , I discovered a pattern here wherein if I ... | Widawensen | 334,463 | <p>Every matrix satisfies its own characteristic equation so </p>
<p>$A$ satisfies </p>
<ul>
<li><p>$A^2-\operatorname{tr}(A)A+\det(A)I=0 $</p>
<p>and $A^{-1}$ satisfies </p></li>
<li><p>$A^{-2}-\operatorname{tr}(A^{-1})A^{-1}+\det(A^{-1})I=0$.</p></li>
</ul>
<p>With multiplying both sides of the first equati... |
1,747,696 | <p>First of all: beginner here, sorry if this is trivial.</p>
<p>We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ .</p>
<p>My question is: what if instead of moving by 1, we moved by an arbitrary number, say 3 or 11? $ 11+22+33+44+\ldots+11n = $ ?
The way I've understood the usual formula is that the first n... | gnasher729 | 137,175 | <p>There's a simple rule for linear progressions: The sum equals the number of items, times the average of the first and last item. So in this case n (11n + 11) / 2. </p>
<p>If you added 7, 17, 27, 37 and so on, the first item would be 7, the last would be 10n - 3, the average (10n - 3 + 7) / 2 = 5n + 2, so the sum wo... |
4,226,455 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a set and <span class="math-container">$Y$</span> a topological space. What is the topology on <span class="math-container">$X$</span> induced by constant maps <span class="math-container">$f:X \to Y$</span>?</p>
</blockquote>
<p>The induced topology is <sp... | mohottnad | 955,538 | <p>Regarding your claim:</p>
<blockquote>
<p>My book says that the proposition "∀x∈D,P(x)" can be vacuously true, because it can be turned into the form "P→Q"</p>
</blockquote>
<p>You seem to conflate the truth value of the material conditional P→Q with quantified formula ∀xP(x) where x ranges over ... |
837,570 | <p>Prove that $\arctan{x}=\frac{1}{x^2}$ has only one solution on the set of real numbers.</p>
<p>I need some help with it, would greatly appreciate it.</p>
| JimmyK4542 | 155,509 | <p>For $x > 0$, $\arctan x$ is strictly increasing while $\dfrac{1}{x^2}$ is strictly decreasing. </p>
<p>For $x < 0$, $\arctan x < 0 < \dfrac{1}{x^2}$. </p>
<p>What does this tell you about the number of positive and negative solutions?</p>
|
4,255,587 | <p>There was a quiz posted on F*cebook by someone. Here's the problem.</p>
<p><a href="https://i.stack.imgur.com/fGu2W.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fGu2W.jpg" alt="enter image description here" /></a></p>
<p>And here's my attempt:</p>
<p><a href="https://i.stack.imgur.com/EMP6g.jpg... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$\Delta ABC$</span> be our triangle, <span class="math-container">$AB=BC$</span>, <span class="math-container">$PQRL$</span> be a square,</p>
<p>where <span class="math-container">$P\in AB$</span>, <span class="math-container">$Q\in BC$</span>, <span class="math-container">$R\in QC$<... |
2,934,973 | <p><a href="https://i.stack.imgur.com/XQ80d.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XQ80d.png" alt="enter image description here"></a> </p>
<p>Let <span class="math-container">$\theta = \angle BAC$</span>. Then we can write <span class="math-container">$\cos \theta = \dfrac{x}{\sqrt{2}}$</sp... | clathratus | 583,016 | <p>Assuming that </p>
<p><span class="math-container">$A=(1,1), B=(2,3), C=(4,0)$</span>,</p>
<p>we know that <span class="math-container">$\vec{AB}=(2-1)\vec{x}+(3-1)\vec{y}=\vec{x}+2\vec{y}$</span>, and <span class="math-container">$\vec{AC}=(4-1)\vec{x}+(0-1)\vec{y}=3\vec{x}-\vec{y}$</span>.</p>
<p>Thus, <span cl... |
965,851 | <p>I have a computer problem that I was able to reduce to an equation in quadratic form, and thus I can solve the problem, but it's a little messy. I was just wondering if anybody sees any tricks to simplify it?</p>
<p>$$\sin^2\beta ⋅ d^4 + c^2\left(\cos^2\beta⋅\cos^2\alpha-\frac{\cos^2\beta}{2}-\frac12\right)d^2 + ... | Aaron Maroja | 143,413 | <p>Try this, first suppose $x > 0$, then you take $x<0$. </p>
<p>$$\begin{align}x^2 = 2^x &\Rightarrow (x^2)^{\frac{1}{2}} = (2^x)^\frac{1}{2} \Rightarrow x= 2^\frac{x}{2} \\ & \Rightarrow x \ e^{-x\frac{\ln\ 2}{2}} = 1 \Rightarrow -x \frac{\ln\ 2}{2}\ e^{-x\frac{\ln\ 2}{2}} = -\frac{\ln \ 2}{2} \\ &a... |
965,851 | <p>I have a computer problem that I was able to reduce to an equation in quadratic form, and thus I can solve the problem, but it's a little messy. I was just wondering if anybody sees any tricks to simplify it?</p>
<p>$$\sin^2\beta ⋅ d^4 + c^2\left(\cos^2\beta⋅\cos^2\alpha-\frac{\cos^2\beta}{2}-\frac12\right)d^2 + ... | fleablood | 280,126 | <p>I wrote about this in another post.</p>
<p>The other solution is 4.</p>
<p>For $x^a = a^x; a > 0$ in general... Well, notice the shapes of the graphs for $x \ge 0$ are such that they intersect twice if $a \ne e$ and intersect once if $a = 1$ (at $x = e$).</p>
<p>In my other post I went into great detail about ... |
75,791 | <p>When will a probabilistic process obtained by an "abstraction" from a deterministic discrete process satisfy the Markov property?</p>
<p>Example #1) Suppose we have some recurrence, e.g., $a_t=a^2_{t-1}$, $t>0$. It's a deterministic process. However, if we make an "abstraction" by just considering the one partic... | Craig | 15,279 | <p>The simple answer is: every discrete deterministic process is already a Markov process. The "probabilities" in question are just restricted to be 0 or 1.</p>
|
1,837,356 | <p>I'm reading <a href="http://www.careerbless.com/aptitude/qa/permutations_combinations_imp8.php" rel="nofollow">this passage</a> and wondering why</p>
<p>Number of ways in which
k identical balls can be distributed into
n distinct boxes =</p>
<p>$$\binom {k+n-1}{n-1}$$</p>
<p>could someone explain it to me plea... | JMoravitz | 179,297 | <p>The proof technique is what is often referred to as <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="nofollow noreferrer">stars and bars</a>.</p>
<p>Imagine if you will you have $n$ distinct boxes and $k$ identical balls.</p>
<p>We arrange a sequence of $k$ $\star$'s and $n-1$ $|$'s to d... |
825,703 | <p>I have been working with vector spaces for a while and I now take for granted what the vector space does. I feel like I dont really understand why multiplication and addition must be defined on a vector space. For example, it feels like adding two vectors and having their sum contained within the space is just a nam... | Ittay Weiss | 30,953 | <p>Imagine the plane $\mathbb R^2$ as a vector space. Now truncate it on all sides so you are just left with a little rectangle about the origin. Now, you this is your world and you take two vectors in it and add them, you may land inside the rectangle or you may not. You don't want to have to worry about such issues, ... |
825,703 | <p>I have been working with vector spaces for a while and I now take for granted what the vector space does. I feel like I dont really understand why multiplication and addition must be defined on a vector space. For example, it feels like adding two vectors and having their sum contained within the space is just a nam... | Empy2 | 81,790 | <p>A lot of the calculus problems we can solve are linear. For these problems, if we know two solutions, then the sum is another solution. So is a scalar multiple of either. So the set of solutions to a large set of problems is a vector space. It makes sense to understand vector spaces.</p>
<p>Many things aren't v... |
2,245,631 | <blockquote>
<p>$x+x\sqrt{(2x+2)}=3$</p>
</blockquote>
<p>I must solve this, but I always get to a point where I don't know what to do. The answer is 1.</p>
<p>Here is what I did: </p>
<p>$$\begin{align}
3&=x(1+\sqrt{2(x+1)}) \\
\frac{3}{x}&=1+\sqrt{2(x+1)} \\
\frac{3}{x}-1&=\sqrt{2(x+1)} \\
\frac{(3-x... | Jack | 138,160 | <p>As in the comment $x-1$ is a factor of $f(x) = -2x^3 -x^2-6x+9$. After an easy long division we get $f(x) = (x-1)(-2x^2-3x-9)$. From this we can use the quadratic equation to see the other two roots are not real.</p>
<p>We can do the long division in the following way:</p>
<p>We need to divide $-2x^3-x^2-6x+9$ by ... |
23,846 | <p>I'm stuck with this algebra question.</p>
<p>I try to prove that the exterior algebra $R$ over $k^d$, that is, the $k$-algebra that is generated by $x_1,\ldots,x_d$ and $x_ix_j=- x_jx_i$ for each $i,j$, has just one simple module which is not faithful.</p>
<p>I think the only simple module is $k$, but I am not rea... | rschwieb | 29,335 | <p>Assuming that $k$ is a field for the same reasons Mariano gave, the short answer is:</p>
<blockquote>
<p>The exterior algebra is <strong><em>local</em></strong>, and so any simple module over it looks like $R/M$ where $M$ is the unique maximal right ideal.</p>
</blockquote>
<p>This is detailed in <a href="https:... |
1,685,895 | <blockquote>
<blockquote>
<p>Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal.</p>
</blockquote>
</blockquote>
<hr>
<p>My attempt:</p>
<p>$\displaystyle \binom{n}{7}=\binom{n}{8} $</p>
<p>$$ n(n-1)(n-... | Upstart | 312,594 | <p>The coefficient of $x^7$ will be ${n\choose 7}{2^{n-7}\over 3^7}$ and the coefficient of $x^8$ will be ${n\choose 8}{2^{n-8}\over 3^8}$ and not what you have got .you just missed out the factor of 3 that plays a role as a coefficient.So equating it you get
$${n\choose 7}{2^{n-7}\over 3^7}={n\choose 8}{2^{n-8}\over 3... |
155,024 | <p>I am trying to plot a polynomial inside <code>Manipulate</code> with indexed coefficients.</p>
<p>Surprisingly, only an empty plot is generated.</p>
<p>Any help welcome.</p>
<pre><code>Block[{n = 3},
Manipulate @@ {
Column[{
Sum[a[i] x^i, {i, 0, n}],
Plot[Evaluate@Sum[a[i] x^i, {i, 0, n}], {x, -5, 5}]
}],
... | rhermans | 10,397 | <p>Best so far is <code>BoolEval[Rest[y] == Most[y]]</code></p>
<p><img src="https://i.stack.imgur.com/ensyM.png" alt="Mathematica graphics"></p>
<hr>
<pre><code>data = Union @ RandomReal[E + {0, 10^-7}, 10^6];
</code></pre>
<hr>
<pre><code>pwb1[y_] := Boole[SameQ @@@ Transpose[{Rest[y], Most[y]}]]
pwb1[data]; //... |
1,180,199 | <p>I am not quite sure how to deal with discrete IVP</p>
<p>Find self-similar solution
\begin{equation}
u_t=u u_x\qquad -\infty <x <\infty,\ t>0
\end{equation}</p>
<p>satisfying initial conditions</p>
<p>\begin{equation}
u|_{t=0}=\left \{\begin{aligned}
-1& &x\le 0,\\
1& &x> 0
\end{aligne... | EditPiAf | 418,542 | <p>This is a Riemann problem for the PDE <span class="math-container">$u_t - u u_x = 0$</span>, which cannot be solved by applying the method of characteristics alone: weak solutions must be considered. This PDE becomes equal to the inviscid Burgers' equation <span class="math-container">$v_t + vv_y = 0$</span> under t... |
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