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Power Series Solution Consider the initial value problem $$xu''+\sin(x)u=0 \ \ \ \ u(0)=0, u'(0)=2$$ Derive the first $4$ non-zero terms of a power series solution to this problem about the point $x=0$. I know the solution will have the from $$u(x)=\sum_{k=0}^{\infty}A_kx^k$$ But upon differentiating, I get that $$\sum_{k=0}^{\infty}k(k-1)A_kx^{k-2}+\sin(x)\sum_{k=0}^{\infty}A_kx^{k-1}=0$$ and do not know how to proceed. Would an initial substitution help?
|
[
{
"id": "corpus-0",
"score": 0.8827380537986755,
"text": "Convergence of the Power Series for $xu''+\\sin(x)u=0$ Consider the initial value problem $$xu''+\\sin(x)u=0 \\ \\ \\ \\ u(0)=0, u'(0)=2$$ What can be said about the radius of convergence of this series? I have determined that the first four nonzero terms (about the point $x=0$) of the power series are $$2x,-x^2,\\frac{1}{6}x^3,-\\frac{1}{72}x^4$$ How can I test for convergence? Intuitively it looks not to converge, but how can I show this? edit The solution will have the from $$u(x)=\\sum_{k=0}^{\\infty}A_kx^k.$$ Differentiating, \\begin{align} \\sum_{k=2}^{\\infty} k(k-1)A_kx^{k-1}+\\sin(x)\\sum_{k=0}^{\\infty}A_kx^k&=0 \\\\ \\sum_{k=2}^{\\infty} k(k-1)A_kx^{k-1}+\\left(\\sum_{k=0}^{\\infty} \\frac{(-1)^k x^{2k+1}}{(2k+1)!}\\right)\\sum_{k=0}^{\\infty}A_kx^k&=0 \\\\ (2A_2x+6A_3x^2+12A_4x^3+20A_5x^4+30A_6x^5+42A_7x^6+..)+\\left(2x^2+A_2x^3+\\left(A_3-\\frac{1}{3}\\right)x^4+\\left(A_4-\\frac{A_2}{6!}\\right)x^5+\\left(A_5-\\frac{A_3}{3!}+\\frac{2}{5!}\\right)x^6+..\\right)=0 \\end{align} Equating coefficients, $x^{2k}=0$ for $k\\in\\mathbb{Z^+}$. $$6A_3+2=0\\implies A_3=-\\frac{1}{3}$$ $$20A_5+A_3-\\frac{1}{3}=0\\implies A_5=\\frac{1}{30}$$ $$42A_7+A_5-\\frac{A_3}{3!}+\\frac{2}{5!}=0\\implies A_7=-\\frac{19}{7560}$$ Hence $$u=2x-\\frac{1}{3}x^3+\\frac{1}{30}x^5-\\frac{19}{7560}x^7+..$$"
}
] |
[
{
"id": "corpus-130786",
"score": 0.7994771003723145,
"text": "Solution of $y''+xy=0$ The differential equation $y''+xy=0$ is given. Find the solution of the differential equation, using the power series method. That's what I have tried: We are looking for a solution of the form $y(x)= \\sum_{n=0}^{\\infty} a_n x^n$ with radius of convergence of the power series $R>0$. Then: $$y'(x)= \\sum_{n=1}^{\\infty} n a_n x^{n-1}= \\sum_{n=0}^{\\infty} (n+1) a_{n+1} x^n$$ $$y''(x)= \\sum_{n=1}^{\\infty} (n+1) n a_{n+1} x^{n-1}= \\sum_{n=0}^{\\infty} (n+2) (n+1) a_{n+2} x^n$$ Thus: $$\\sum_{n=0}^{\\infty} (n+2) (n+1) a_{n+2} x^n+ x \\sum_{n=0}^{\\infty} a_n x^n=0 \\\\ \\Rightarrow \\sum_{n=0}^{\\infty} (n+2)(n+1) a_{n+2} x^n+ \\sum_{n=0}^{\\infty} a_n x^{n+1}=0 \\\\ \\Rightarrow \\sum_{n=0}^{\\infty} (n+2)(n+1) a_{n+2} x^n+ \\sum_{n=1}^{\\infty} a_{n-1} x^n=0 \\\\ \\Rightarrow 2a_2+\\sum_{n=1}^{\\infty} \\left[ (n+2) (n+1) a_{n+2}+ a_{n-1}\\right] x^n=0$$ So it has to hold: $$a_2=0 \\\\ (n+2) (n+1) a_{n+2}+a_{n-1}=0, \\forall n=1,2,3, \\dots$$ For $n=1$: $3 \\cdot 2 \\cdot a_3+ a_0=0 \\Rightarrow a_3=-\\frac{a_0}{6}$ For $n=2$: $4 \\cdot 3 \\cdot a_4+a_1=0 \\Rightarrow a_4=-\\frac{a_1}{12}$ For $n=3$: $5 \\cdot 4 \\cdot a_5+a_2=0 \\Rightarrow a_5=0$ For $n=4$: $6 \\cdot 5 \\cdot a_6+a_3=0 \\Rightarrow 30 a_6-\\frac{a_0}{6}=0 \\Rightarrow a_6=\\frac{a_0}{6 \\cdot 30}=\\frac{a_0}{180}$ For $n=5$: $7 \\cdot 6 \\cdot a_7+ a_4=0 \\Rightarrow 7 \\cdot 6 \\cdot a_7-\\frac{a_1}{12}=0 \\Rightarrow a_7=\\frac{a_1}{12 \\cdot 42}$ Is it right so far? If so, how could we find a general formula for the coefficients $a_n$? EDIT: Will it be as follows: $$a_{3k+2}=0$$ $$a_{3k}=(-1)^k \\frac{a_0}{(3k)!} \\prod_{i=0}^{k-1} (3i+1)$$ $$a_{3k+1}=(-1)^k \\frac{a_1}{(3k+1)!} \\prod_{i=0}^{k-1} (3i+2)$$ If so, then do we have to write seperately the formula for the coefficients of $x^0, x^1$, because otherwhise the sum would be from $0$ to $-1$ ?",
"topk_rank": 0
},
{
"id": "corpus-166542",
"score": 0.7790071964263916,
"text": "$f'' + f =0$: finding $f$ using power series How can one solve the following differential equation $$f'' + f =0$$ with the usage of power series? Writing: $$f(x) = \\displaystyle\\sum_{k=0}^{\\infty} a_k x^k$$ $$f'(x) = \\displaystyle\\sum_{k=1}^{\\infty} a_k k x^{k-1}$$ $$f'' (x) = \\displaystyle\\sum_{k=2}^{\\infty} k (k-1) a_k x^{k-2}$$ How to continue?",
"topk_rank": 1
},
{
"id": "corpus-88563",
"score": 0.7691946625709534,
"text": "The differential equation $xu_{x}+yu_{y}=2u$ satisfying the initial condition $y=xg(x),u=f(x)$ I was thinking about the following problem : Find out which of the following option(s) is/are correct? The differential equation $xu_{x}+yu_{y}=2u$ satisfying the initial condition $y=xg(x),u=f(x)$, with (a)$f(x)=2x,g(x)=1$ has no solution, (b)$f(x)=2x^2,g(x)=1$ has infinite number of solutions, (c)$f(x)=x^3,g(x)=x$ has a unique solution, (d)$f(x)=x^4,g(x)=x$ has a unique solution. My Attempt: Using lagrange's method, we see that $\\frac{dx}{x}=\\frac{dy}{y}=\\frac{du}{2u} $ gives $y/x=c_1,$ and $y=\\sqrt uc_2$ ,where $c_1,c_2$ being constants. Now we have to apply to the given initial conditions and check the given options.Here ,I observe that $y=xg(x)=x.1=x$,[as $g(x)=1$] and $u=f(x)=2x^2$But,this relation does not satisfy $xu_{x}+yu_{y}=2u$ and so the choice"(a)$f(x)=2x,g(x)=1$ has no solution," is right. Now I am stuck here and could not progress further.Am i going in the right direction? Please help.Thanks in advance for your time.",
"topk_rank": 2
},
{
"id": "corpus-34793",
"score": 0.756020724773407,
"text": "How to solve the diffencial equation for power series $(1+x²)y'' - 6xy=0$ in $x_{0}=0$ I have tried for the power series but when suppose the solution is $y(x)=\\sum _{n=0}^\\infty a_nx^n$ appeared sums dependents of $a_n$, $a_{n+2}$ and $a_{n-1}$ and this is not possible.",
"topk_rank": 3
},
{
"id": "corpus-47884",
"score": 0.7554469108581543,
"text": "$(x-x_0)^0$ in power series When I first studied power series in high school, the teacher gave the following general definition: \\begin{equation} f(x)=\\sum_{n=0}^{\\infty}a_n (x-x_0)^n \\end{equation} He then proceeded to explain that when both $x=0$ and $x_0 =0$, the first term of the power series is defined to simply be $a_0$. How does one justify $0^0 = 1$ in this case?",
"topk_rank": 4
},
{
"id": "corpus-223809",
"score": 0.7543363571166992,
"text": "Solving functional differential equation $f'(x)=2f(2x)-f(x)$ Show that there is at least a nonzero function $f$, differentiable on $[0,+\\infty)$, satisfying $$f'(x)=2f(2x)-f(x) \\qquad \\forall x>0 $$ $$M_n:=\\int_{0}^{\\infty}x^nf(x)dx<\\infty \\qquad \\forall n\\in \\mathbb{N} $$ My best idea so far is to assume that the solution is a power series, i.e. $$ f(x)=\\sum_{n=0}^{\\infty}a_nx^n\\qquad \\forall x>0$$ Then the equation becomes $$ \\sum_{n=0}^{\\infty}na_nx^{n-1}=2\\sum_{n=0}^{\\infty}a_n2^nx^n-\\sum_{n=0}^{\\infty}a_nx^n$$ equating all the coefficients of the same degree I get $$na_n=(2^{n}-1)a_{n-1}\\qquad \\forall n\\geq 1$$ So setting $a_0=1$, I get $$a_{n}=\\frac{1}{n!}\\prod_{k=1}^{n}(2^k-1) \\qquad \\forall n$$ But does the power series actually converge? Using Hadamard's formula, and that $2^{k}-1\\geq 2^{k-1}$, $$ |a_n|^{1/n}\\geq\\frac{1}{(n!)^{1/n}}\\left[2^{n(n-1)/2}\\right]^{1/n}\\sim\\frac{e}{n(2\\pi n)^{1/2n}}2^{(n-1)/2}\\to \\infty$$ so the radius of converge of the series is $0$, so it doesn't actually define a solution on $[0,+\\infty)$.",
"topk_rank": 5
},
{
"id": "corpus-34698",
"score": 0.7493466734886169,
"text": "solve initial value problem ivp I have equations \\begin{align} 7z'(x) &= x^2 + y(x)^2 + z(x)^2 \\\\ y''(x) &= -(y'(x) + 7y(x))\\sin(z(x)) \\end{align} where $$(y(0),y'(0),z(0))=(1.5,-2.6,0.5)$$ The task is to solve this problem using a computer. Can someone help with explaining what goes in $u'(x)$? Thanks. I am guessing $u(x) = (y'',z')$.",
"topk_rank": 6
},
{
"id": "corpus-220587",
"score": 0.739952027797699,
"text": "$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\\begin{cases}1 & \\text{ if } 0\\le x\\le 1\\\\0 & \\text{ if } x>1\\end{cases}$$Then, $y(3/2)$ equals to (A) $\\frac{\\sinh(1)}{e^3}$ (B) $\\frac{\\cosh(1)}{e^3}$ (C) $\\frac{\\sinh(1)}{e^2}$ (D) $\\frac{\\cosh(1)}{e^2}$ Integrating both sides we get , $$\\int_0^xd(y(x))+2\\int_0^xy(x)\\,dx=\\int_0^xf(x)\\,dx$$ $$\\implies y(x)+2\\int_0^xy(x)\\,dx=\\int_0^xf(x)\\,dx=1$$ $$\\implies y(x)=1-2C$$where , $$C=\\int_0^xy(x)\\,dx=(1-2C)x\\implies C =\\frac{x}{1+2x}$$ Then , $y(x)=\\frac{1}{1+2x}$ and so, $y(3/2)=1/4$. But no option match..Please help..",
"topk_rank": 7
},
{
"id": "corpus-100361",
"score": 0.7343854308128357,
"text": "Understanding ProofWiki's proof of differentiation of sine power series At I've read that being $\\sin(x)=\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ by using we get $\\sin'(x)=\\sum_{n=0}^{\\infty}(2n+1) \\frac{(-1)^n}{(2n+1)!}x^{2n}=\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{(2n)!}x^{2n}=\\cos(x)$ but the page says that if $f(x)=\\sum_{n=0}^{\\infty}c_n(x-a)^n$ then inside the radius of convergence of the power series we have $f'(x)=\\sum_{n=1}^{\\infty}nc_n(x-a)^{n-1}$ so shouldn't it be $\\sin'(x)=\\sum_{n=1}^{\\infty}(2n+1)\\frac{(-1)^n}{(2n+1)!}x^{2n+1-1}=\\sum_{n=1}^{\\infty}\\frac{(-1)^n}{(2n)!}x^{2n}$? Now I understand that this last result has a missing term due to the index of the series being $1$ and not $0$ but being the $1$ mandatory because of the above theorem and since trying to shift index doesn't help (we get $\\sin'(x)=\\sum_{n=0}^{\\infty}\\frac{(-1)^{n+1}}{(2n+2)!}x^{2n+2}$) how can the result that ProofWiki shows be rigorously justified? (I mean, I think we can't just say \"oh, there's a missing term!\" after having applied the theorem and magically shift the index of the series to obtain the desired result)",
"topk_rank": 8
},
{
"id": "corpus-172669",
"score": 0.732843279838562,
"text": "Find a general solution for $\\int_{0}^{\\infty} \\sin\\left(x^n\\right)\\:dx$ So, I was recently working on the Sine Fresnal integral and was curious whether we could generalise for any Real Number, i.e. $$I = \\int_{0}^{\\infty} \\sin\\left(x^n\\right)\\:dx$$ I have formed a solution that I'm uncomfortable with and was hoping for qualified eyes to have a look over. So, the approach I took was to employ Complex Numbers (I forget the name(s) of the theorem that allows this). But $$\\sin\\left(x^n\\right) = \\Im\\left[-e^{-ix^n}\\right]$$ And so, n $$ I = \\int_{0}^{\\infty} \\sin\\left(x^n\\right)\\:dx = \\Im\\left[\\int_{0}^{\\infty} -e^{-ix^n}\\:dx \\right]= -\\Im\\left[\\int_{0}^{\\infty} e^{-\\left(i^{\\frac{1}{n}}x\\right)^{n}}\\:dx \\right]$$ Applying a change of variable $u = i^{\\frac{1}{n}}x$ we arrive at: \\begin{align} I &= -\\Im\\left[i^{-\\frac{1}{n}}\\int_{0}^{\\infty} e^{-u^{n}}\\:du \\right] \\\\ &= -\\Im\\left[i^{-\\frac{1}{n}}\\frac{\\Gamma\\left(\\frac{1}{n}\\right)}{n} \\right]\\\\ &= \\sin\\left(\\frac{\\pi}{2n}\\right)\\frac{\\Gamma\\left(\\frac{1}{n}\\right)}{n} \\end{align} My area of concern is in the substitution. As $i^{-\\frac{1}{n}} \\in \\mathbb{C}$, I believe the limits of the integral should have been from $0$ to $i^{-\\frac{1}{n}}\\infty$. Is that correct or not? I'm also struggling with bounds on $n$ for convergence. Is this expression valid for all $n\\in\\mathbb{R}$ Any guidance would be greatly appreciated",
"topk_rank": 9
},
{
"id": "corpus-133892",
"score": 0.7289009094238281,
"text": "General solution to the PDE $xU_x+yU_y=0$ Solve the equation $xU_x+yU_y=0$. From Partial Differential Equations: An Introduction, 2nd Ed. (Strauss, pg. 10). There are no boundary conditions. Solution: The PDE can be written $(x,y)\\cdot\\nabla U(x,y)=0$, implying that the vector $(x,y)$ is tangent to the PDE's characteristic equation. Then I can write $dy/dx=y/x$, and to solve this ODE, $$\\frac{dy}{dx}=\\frac yx \\\\ \\ln y=\\ln x+c_1 \\\\ y=cx \\\\ y/x=c$$ Hence, the PDE is dependent on $c$ and I can write $U(x,y)=f(y/x)$ for some function $f$, solving the problem. Question: Is this solution correct? The hint for the question (supplied by the professor) said I should use a substitution of variables, but this seems reasonably valid.",
"topk_rank": 10
},
{
"id": "corpus-138131",
"score": 0.7258530259132385,
"text": "How to find sum of power series $\\sum_{n=0}^\\infty\\frac{1}{n!(n+2)}$ by differentiation and integration? Let $$ S = \\sum_{n=0}^\\infty \\frac{1}{n!(n+2)} $$ Integrate the Taylor Series of $xe^x$ to show that S = 1 Also, Differentiate the Taylor series of $\\frac{e^x - 1}{x}$ to show that S = 1 For the integration one, I got $\\int xe^x dx$ = $x^2\\sum_{n=0}^\\infty \\frac{1}{n!(n+2)} x^n$ and pretty much stuck. I can see the S term but i have no idea how to move on from there",
"topk_rank": 11
},
{
"id": "corpus-217707",
"score": 0.7192398309707642,
"text": "Can I solve this differential equation at a point? I want to solve this differential equation evaluted at $x=0$. I will start with an example with $m=2$ $$ f(0)+f'(0)+f''(0)=2\\lambda_{1}\\lambda_{2}+4 $$ where $m$,$\\lambda_{1}$ and $\\lambda_{1}$ are positive constants. The $f(x)$ that satisfies the previous equation is $$ \\prod_{k=1}^{2}\\left(\\lambda_{k}.x+2(1-\\lambda_{k})\\right) $$ I have encountered with this equation while solving an integral. The result was $$ \\left. \\sum_{n=0}^{m+1} \\left( \\frac{d^n}{d x^n}\\left( x^{m-1}\\prod_{k=1}^{2}\\left(\\lambda_{k}.x+m(1-\\lambda_{k})\\right)\\right)\\right)\\right|_{x=0}=m\\lambda_{1}\\lambda_{2}+m^2 $$ So I want to find f(x) that satisfies $$ \\left. \\sum_{n=0}^{m+1} \\left( \\frac{d^n}{d x^n}\\left( f(x) \\right)\\right)\\right|_{x=0}=m_{1}\\lambda_{1}\\lambda_{2}+m_{1}m_{2} $$ Note: $m_{1}\\neq m_{2}$.",
"topk_rank": 12
},
{
"id": "corpus-242699",
"score": 0.718234658241272,
"text": "Solving ODE with negative expansion power series I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this: $\\xi^r\\partial_r g_{rr}+2g_{tt}\\partial_t\\xi^t=\\mathcal{O}(r)$ $g_{ii}$ are given, from metric, but that's not important. I need to assume that the solution (since I'm looking for components of $\\xi^\\mu$, which is a vector with components $\\xi^t,\\xi^r,\\xi^\\phi$) is given with power series of the form: $\\xi^\\mu=\\sum\\limits_{n}\\xi^\\mu_n(t,\\phi)r^n$, and this is to be seen as expansion around $1/r$ (expansion around $r=\\infty$). Now when I plug this in the ODE I get this $\\frac{2}{l^2}\\sum_n\\xi^r_nr^{n+1}+2\\sum_n\\xi^t_{n,t} r^n+\\frac{2}{l^2}\\sum_n\\xi^t_{n,t}r^{n+2}=\\mathcal{O}(r)$, where $\\xi^\\mu_{n,i}$ is the derivative with the respect to i-th component. What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my $\\mathcal{O}(r)$ terms cancel each other out? Or? I'm kinda stuck :\\",
"topk_rank": 13
},
{
"id": "corpus-128598",
"score": 0.7174898982048035,
"text": "Find the solution of the pde $xu_y-yu_x=u$. I am trying to find the equation of characteristic curves and solution of the partial differential equation $$x\\frac{\\partial u}{\\partial y}-y\\frac{\\partial u}{\\partial x}=u$$ $u(x,0)=\\sin(\\frac{\\pi}{4}x)$ . What I did: $$\\frac{dx}{-y}=\\frac{dy}{x}=\\frac{du}{u}$$ with From first two equalities $$\\frac{dx}{-y}=\\frac{dy}{x}$$ by solving we get $x^2+y^2=c_1$ for some constant $c_1$. Now I am stuck with the third and rest of the equalities to come up with a suitable manipulation to solve the problem. How can I do this? Any help would be great. Thanks.",
"topk_rank": 14
},
{
"id": "corpus-69922",
"score": 0.7164829969406128,
"text": "For which $a$ does the equation $f(z) = f(az) $ has a non constant solution $f$ For which $a \\in \\mathbb{C} -\\ \\{0,1\\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$. My attempt: First, we can see that any such solution must satisfy: $f(z)=f(a^kz)$ for all $k \\in \\mathbb{N} $. If $|a|<1$: The series $z_{k} = a^k$ converges to 0 which is an accumulation point, and $f(z_i)=f(z_j)$ for all $i, j\\in \\mathbb{N} $. Thus $f$ must be constant. If $|a|=1$: For all $a \\neq 1$ , $f$ must be constant on any circle around $0$, so again $f$ must be constant. My quesions are: Am I correct with my consclusions? Also, I'm stuck in the case where $|a|>1$. Any ideas? Thanks",
"topk_rank": 15
},
{
"id": "corpus-207771",
"score": 0.7161478996276855,
"text": "For which $a$ does the equation $f(z) = f(az) $ has a non constant solution $f$ For which $a \\in \\mathbb{C} -\\ \\{0,1\\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$. My attempt: First, we can see that any such solution must satisfy: $f(z)=f(a^kz)$ for all $k \\in \\mathbb{N} $. If $|a|<1$: The series $z_{k} = a^k$ converges to 0 which is an accumulation point, and $f(z_i)=f(z_j)$ for all $i, j\\in \\mathbb{N} $. Thus $f$ must be constant. If $|a|=1$: For all $a \\neq 1$ , $f$ must be constant on any circle around $0$, so again $f$ must be constant. My quesions are: Am I correct with my consclusions? Also, I'm stuck in the case where $|a|>1$. Any ideas? Thanks",
"topk_rank": 16
},
{
"id": "corpus-150129",
"score": 0.7144859433174133,
"text": "Find the sum of the power series $\\sum_{n=1}^\\infty n*(n+1)*x^n$ $\\sum_{n=1}^\\infty n*(n+1)*x^n$ Hello everyone, I need help in solving the question above. I started with $\\frac{1}{1-x} = \\sum_{n=0}^\\infty x^n$, if differentiated it once so it became $\\frac{1}{(1-x)^2} =\\sum_{n=1}^\\infty nx^{n-1} $ but from here I don't know how to continue. Thanks.",
"topk_rank": 17
},
{
"id": "corpus-22380",
"score": 0.7130986452102661,
"text": "$y'=$ ${-x+\\sqrt{(x^2+4y)}}\\over 2$ , $y(2)=-1$ Prove that the solutions $y_1$ and $y_2$ for the initial value problem $y'=$${-x+\\sqrt{(x^2+4y)}}\\over 2$ , $y(2)=-1$ are: $y_1=1-x$ $y_2={{-x^2}\\over 4}$ And explain why the two solutions not contradiction with the theory of existence and uniqueness. ( sorry I don't speak English well ) My answer: For the initial condition : $y_1=1-x$ Satisfy the initial condition $y(2)=-1$ $y_1=1-2=-1$ And Satisfy the differential equation ${y_1}^{\\prime}=-1$ ${{-x+\\sqrt{x^2+4y}}\\over 2}={{-x+\\sqrt{x^2+4(1-x)}}\\over 2}$ $={{-x+\\sqrt{(x-2)^2}}\\over 2}$ $={{-x+x-2}\\over 2}=-1$ L.h.s.=R.h.s. $y^{\\prime}={{-x+\\sqrt{x^2+4y}}\\over 2}$ Also $y_2$ Satisfy the initial condition $y_2={{-x^2}\\over4}={-4\\over4}=-1$ And satisfy the differential equation. $y_2^{\\prime}={-x\\over 2}$ ${{-x+\\sqrt{x^2+4y}}\\over 2}={{-x+\\sqrt{x^2+4({-x^2\\over 4})}}\\over 2}={-x\\over 2}$ And for second part of exercise: ${\\partial f\\over \\partial y}={1\\over \\sqrt{x^2+4y}}$ and this not defined when $x=2, \\ y=-1$ So $f(x,y)$ no satisfy lipshtize condition in any rectangular has (2,-1), Lipshtize condition including uniqueness of solution, so no contradict with the existence and uniqueness. True ? If this also wrong I well be delet it , I answered by the same way of my teacher :(",
"topk_rank": 18
},
{
"id": "corpus-48361",
"score": 0.7130986452102661,
"text": "$y'=$ ${-x+\\sqrt{(x^2+4y)}}\\over 2$ , $y(2)=-1$ Prove that the solutions $y_1$ and $y_2$ for the initial value problem $y'=$${-x+\\sqrt{(x^2+4y)}}\\over 2$ , $y(2)=-1$ are: $y_1=1-x$ $y_2={{-x^2}\\over 4}$ And explain why the two solutions not contradiction with the theory of existence and uniqueness. ( sorry I don't speak English well ) My answer: For the initial condition : $y_1=1-x$ Satisfy the initial condition $y(2)=-1$ $y_1=1-2=-1$ And Satisfy the differential equation ${y_1}^{\\prime}=-1$ ${{-x+\\sqrt{x^2+4y}}\\over 2}={{-x+\\sqrt{x^2+4(1-x)}}\\over 2}$ $={{-x+\\sqrt{(x-2)^2}}\\over 2}$ $={{-x+x-2}\\over 2}=-1$ L.h.s.=R.h.s. $y^{\\prime}={{-x+\\sqrt{x^2+4y}}\\over 2}$ Also $y_2$ Satisfy the initial condition $y_2={{-x^2}\\over4}={-4\\over4}=-1$ And satisfy the differential equation. $y_2^{\\prime}={-x\\over 2}$ ${{-x+\\sqrt{x^2+4y}}\\over 2}={{-x+\\sqrt{x^2+4({-x^2\\over 4})}}\\over 2}={-x\\over 2}$ And for second part of exercise: ${\\partial f\\over \\partial y}={1\\over \\sqrt{x^2+4y}}$ and this not defined when $x=2, \\ y=-1$ So $f(x,y)$ no satisfy lipshtize condition in any rectangular has (2,-1), Lipshtize condition including uniqueness of solution, so no contradict with the existence and uniqueness. True ? If this also wrong I well be delet it , I answered by the same way of my teacher :(",
"topk_rank": 19
}
] |
query-1
|
Question on the existence of finite open subsets in $\mathbb{R}^{k}$ Dear reader of this post, can any subset $S \subset \mathbb{R}^{k}$, with the euclidean distance for $x,y \in \mathbb{R}^{k}$, be open and contain only finitely many elements? I am looking forward for your reply!
|
[
{
"id": "corpus-1",
"score": 0.7808202505111694,
"text": "Is every open subset of $ \\mathbb{R} $ uncountable? Is every open subset of $\\mathbb{R}$ uncountable? I was crafting a proof for the theorem that states every open subset of $\\mathbb{R}$ can be written as the union of a countable number of disjoint intervals when this question came up. I feel like the answer is yes, but I'm not sure how to go about proving it or whether there is a crazy construction (like the Cantor Set) that creates a countable, open subset of $\\mathbb{R}$. Any ideas?"
}
] |
[
{
"id": "corpus-8796",
"score": 0.741574764251709,
"text": "$\\mathbb{R}^n$ not homeomorphic to open subset of $\\mathbb{R}^m$ I am a little confused on this topic, I have managed to prove that if $n < m$ then this is true, but the proof does not seem to work if $m < n$. Indeed, if $n < m$ and $f :\\mathbb{R}^n \\to U$ is a homeomorphism with $U$ being an open subset of $\\mathbb{R}^m$, then $\\mathbb{R}^n$ will also be homeomorphic to the subset of $\\mathbb{R}^m$ consisting of tuples where the last $m-n$ entries are zero, this set $A$ is closed, and so we have an homeomorphism $F: U \\to A$, but since $U$ is open so must $A$ be by the invariance of domain theorem, but $A$ can't be both close and open since the only such sets in $\\mathbb{R}^m$ is the empty set and $\\mathbb{R}^m$ itself, contradiction. But what if $m < n$? This proof does not seem to work then?",
"topk_rank": 0
},
{
"id": "corpus-33926",
"score": 0.7413583397865295,
"text": "How is every subset of the set of reals with the finite complement topology compact? While we know that every finite set is compact, how to prove that every infinite subset of $\\mathbf{R}$, the set of all real numbers, is also compact in the finite-complement topology (also called the co-finite topology)? What is the situation in the case of an arbitrary infinite set $X$ with the finite-complement topology?",
"topk_rank": 1
},
{
"id": "corpus-150464",
"score": 0.7413583397865295,
"text": "How is every subset of the set of reals with the finite complement topology compact? While we know that every finite set is compact, how to prove that every infinite subset of $\\mathbf{R}$, the set of all real numbers, is also compact in the finite-complement topology (also called the co-finite topology)? What is the situation in the case of an arbitrary infinite set $X$ with the finite-complement topology?",
"topk_rank": 2
},
{
"id": "corpus-15362",
"score": 0.7408134937286377,
"text": "in a topological space only finite subsets are compact sets If in a topological space only finite subsets are compact sets, is it then the discrete topological space? Thank you.",
"topk_rank": 3
},
{
"id": "corpus-31340",
"score": 0.7408134937286377,
"text": "in a topological space only finite subsets are compact sets If in a topological space only finite subsets are compact sets, is it then the discrete topological space? Thank you.",
"topk_rank": 4
},
{
"id": "corpus-67624",
"score": 0.7408134937286377,
"text": "in a topological space only finite subsets are compact sets If in a topological space only finite subsets are compact sets, is it then the discrete topological space? Thank you.",
"topk_rank": 5
},
{
"id": "corpus-142770",
"score": 0.7405524849891663,
"text": "Prove that the inverse image of an open set is open Let $ X \\subset \\mathbb{R}$ be a non-empty, open set and let $f: X \\rightarrow \\mathbb{R}$ be a continuous function. Show that the inverse image of an open set is open under f, i.e. show: If $M \\subset \\mathbb{R}$ is open, then $f^{-1}(M)$ is open as well. I think that this should basically follow from the definitions, but I'm still having some troubles with the proof. Suppose $M \\subset \\mathbb{R}$ is open, then it follows that $\\forall z \\in M, \\exists r > 0: K_r(z) \\subseteq M$ Since f is continuous it follows for each $x_0 \\in X$ that for each $\\epsilon > 0 ,\\exists \\delta > 0 : |f(x)-f(x_0)|< \\epsilon, \\forall x \\in X: |x-x_0|<\\delta$ Now I need to show that $f^{-1}(M)$ is open, so that for each $z \\in M, \\exists r>0: K_r(f^{-1}(z_0)) \\subseteq f^{-1}(M)$ I'm not sure how to prove this. I guess since $z = f(x)$ and since M is open we have that $\\exists r>0:\\{z:|z-z_0|<r\\}\\subseteq M$. And since f is continuous: $|f(x)-f(x_0)| = |z-z_0|< \\epsilon$ for all $|x-x_0|<\\delta$, but since $|x-x_0|=|f^{-1}(z)-f^{-1}(z_0)|<\\delta$ it somehow follows from that that $f^{-1}(M)$ is open?! Which is just horrible and probably completely wrong. But I'm really confused from all the definitions right now and need some help please.",
"topk_rank": 6
},
{
"id": "corpus-189690",
"score": 0.7400239109992981,
"text": "$(0,1)$ is an open subset of $\\mathbb{R}$ but not of $\\mathbb{R}^2$, when we think of $\\mathbb{R}$ as the x-axis in $\\mathbb{R}^2$. Prove this. It's problem #1 of Chapter 2 in Pugh's book. I think I got the first part by taking $r=1$ and pointing out that $d(p,q) < 1 \\implies q \\in (0,1)$ for all $p \\in (0,1)$ but I have no idea how to show it isn't a subset of $\\mathbb{R}^2$. I tried proof by contradiction and got no where. I've been working at this problem for hours and not getting anywhere. How should I do it?",
"topk_rank": 7
},
{
"id": "corpus-154011",
"score": 0.73934006690979,
"text": "Topology: Example of a compact set but its closure not compact Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.",
"topk_rank": 8
},
{
"id": "corpus-182904",
"score": 0.73934006690979,
"text": "Topology: Example of a compact set but its closure not compact Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.",
"topk_rank": 9
},
{
"id": "corpus-107732",
"score": 0.7389827370643616,
"text": "Prove that Open Sets in $\\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice There are several proofs I have seen of this, but they all seem to use choice subtely at some point. Is there any way to prove this without choice, or is it possibly unproveable?",
"topk_rank": 10
},
{
"id": "corpus-117410",
"score": 0.7389827370643616,
"text": "Prove that Open Sets in $\\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice There are several proofs I have seen of this, but they all seem to use choice subtely at some point. Is there any way to prove this without choice, or is it possibly unproveable?",
"topk_rank": 11
},
{
"id": "corpus-204370",
"score": 0.7387124300003052,
"text": "what are all the open subgroups of $(\\mathbb{R},+)$ I am not able to find out what are all the open subgroups of $(\\mathbb{R},+)$, open as a set in usual topology and also subgroup.",
"topk_rank": 12
},
{
"id": "corpus-198616",
"score": 0.7385989427566528,
"text": "Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace? Moderator's note: I've copied the title of the post from a more recently posted version of this question by the OP. The text below is from the original question: I am not sure whether the motivations still applied, hence I left it untouched. It is well known that any connected metrizable space with more than two points is uncountable. And any connected metrizable space has infinitely many open spheres; as each such sphere is connected, there are infinitely many connected subspaces. But must it be the case that any connected metrizable space has uncountably many connected subspaces?",
"topk_rank": 13
},
{
"id": "corpus-203167",
"score": 0.7383633255958557,
"text": "Is the null set a subset of every set? Ever since day one of of my Mathematical Logic course, this fact has really bothered me. I cannot wrap my head around how an empty set is a subset of every possible set. Could someone kindly explain how this is true? Any help is appreciated!",
"topk_rank": 14
},
{
"id": "corpus-192561",
"score": 0.7379785180091858,
"text": "Is the null set a subset of every set? Ever since day one of of my Mathematical Logic course, this fact has really bothered me. I cannot wrap my head around how an empty set is a subset of every possible set. Could someone kindly explain how this is true? Any help is appreciated!",
"topk_rank": 15
},
{
"id": "corpus-55470",
"score": 0.7370325922966003,
"text": "Cardinality and infinite sets: naturals, integers, rationals, bijections I have alot of questions. Do Infinite sets have the same cardinality when there is a bijection between them? Are $\\mathbb{N}$ and $\\mathbb{Z}$ infinite sets? I assume they are, but why? Why does there not exist a bijection between them? Why do $\\mathbb{N}, \\mathbb{Z}, \\mathbb{Q}$ have the same cardinality? I assume there exists a bijection between them, but how can I find this function?",
"topk_rank": 16
},
{
"id": "corpus-14281",
"score": 0.7367130517959595,
"text": "Every finite set contains its supremum: proof improvement. Every finite subset of $\\mathbb R$ contains its supremum (and its infimum) Proof Let $A=\\{a_1,...,a_n\\}$ be a finite subset of $\\mathbb{R}$. Since it is non-empty and it is bounded ($\\max A$ is an upper bound), it has supremum, that is $\\exists \\sup A$ and by definition $\\forall a \\in A \\;\\, a \\leq \\sup A$. Let's suppose that $\\sup A \\not\\in A$ then, since $\\max A \\in A$ we have that $\\max A < \\sup A$. But considering that $\\mathbb Q$ is dense in $\\mathbb R$ we can conclude that $\\exists r \\in \\mathbb Q$ s.t. $\\max A < r < \\sup A$, but this is absurd since $r$ is an upper bound of $A$ and it is lower than the supremum. Necessarily, $\\sup A\\in A$. Is there anything wrong? Is there any way to prove this without using density of $\\mathbb Q$ or another property? Thanks in advance.",
"topk_rank": 17
},
{
"id": "corpus-90715",
"score": 0.7365139722824097,
"text": "Is convex hull of a finite set of points in $\\mathbb R^2$ closed? Is the convex hull of a finite set of points in $\\mathbb R^2$ closed? Intuitively, yes. But not sure how to show that. Thanks!",
"topk_rank": 18
},
{
"id": "corpus-51045",
"score": 0.735917329788208,
"text": "Maps from subsets of $\\mathbb{R}^2$ that are either open/closed/continuous I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one. For each of the following properties, give an example consisting of two subsets $X, Y \\subset \\mathbb{R}^2 $, both considered as topological spaces with their Euclidean topologies, together with a map $f: X \\rightarrow Y$ that has the indicated property. Open but not closed/continuous Closed but not open/continuous Continuous but not open/closed etc... (all cases) I think the reason for my difficulty is a lack of geometric intuition of what open, closed and continuous maps look like as deformations of shapes in $\\mathbb{R}^2$ (I can construct counterexamples from artificial made up spaces). I tried squashing balls and 'pumping space' into lines, as well as joining strips in annulus shapes, but all the examples I seem to come up with are either too well behaved (i.e. all 3) or too misbehaved. Instead of a case by case answered I would prefer a couple worked out cases and some intuition on how to visualise open / closed maps in this context.",
"topk_rank": 19
}
] |
query-2
|
Hyperdrive vs Warp drive. Which is fastest? In Star Trek Warp Factor 36 is the highest FTL speed while in star wars I think the fastest canon was 0.4. So which is faster?
|
[
{
"id": "corpus-2",
"score": 0.9113600850105286,
"text": "Is a warp drive faster than a hyperdrive? What's faster: a warp drive from Star Trek, or a hyperdrive from Star Wars?"
}
] |
[
{
"id": "corpus-50925",
"score": 0.7908008694648743,
"text": "Which ship can go faster, the Millennium Falcon or the USS Enterprise-D? As far as I can tell, both universes have equally objective ideas of speed. Do we know how fast the two ships, Millennium Falcon and USS Enterprise-D, can go? Which is faster?",
"topk_rank": 0
},
{
"id": "corpus-59289",
"score": 0.7809740304946899,
"text": "What happened to warp drives destroying the universe? I remember a TNG episode where some scientists said that warp drives were causing a lot of damage to subspace. At the end of the episode Star Fleet decided that warp speeds would be restricted to warp 5 except for emergency situations. Of course the next episode they were cruising along at warp 7. Is this ever mentioned again in Star Trek?",
"topk_rank": 1
},
{
"id": "corpus-128838",
"score": 0.7678200006484985,
"text": "What factors influence the speed of the rebel fleet? I'm just playing my first round of FTL and I'm wondering what can affect the speed of the rebel fleet that is pursuing me. Does the time I spend in realtime in each system matter, or is only the number of jumps important? What ways are there to slow down the fleet?",
"topk_rank": 2
},
{
"id": "corpus-102940",
"score": 0.7594102621078491,
"text": "How fast can the Galaxy class Starships go in reverse? I've seen them maneuver. They definitely can go backwards. How fast can they go in reverse? Is warp speed possible, or is reverse limited to impulse power? If it is limited to impulse power, can they travel backwards under impulse power as fast as they travel forwards under impulse power?",
"topk_rank": 3
},
{
"id": "corpus-61072",
"score": 0.7486103773117065,
"text": "Speed in Star Trek How fast is Impulse, Warp and Warp X speed? As I have understood it, Warp (1) speed = 1x Speed of Light (SoL), i.e. ~300.000 km/s. However, in ST:VOY, they are hurled 70.000 LY from Earth, which will take 70 years to travel at Warp 9, i.e. Warp 9 = 1000x SoL. That makes no sense to me. If Warp doubles for each full number (binary numbers), then Warp 10 would be 1024x SoL ~ 68 years journey. That works fine, in approximation, given they need to make various stops on the way and/or take small detours to fly through star systems on the way. But no... Not only is Warp 10 impossible (why is that, anyway?), but anything over Warp 9,0 is reserved for burst only. Thus sustained travel is limited to Warp 9, which would (by my calculations) be 512x SoL ~ 135-140 years journey. But this doesn't match any known numbers.. Bonus question: How fast is Impulse drive? You can travel in a shuttle from Earth to Jupiter in a few hours. We only ever hear about quarter, half and full Impulse speed. But at the same scale, you can travel to Jupiter in a few hours, but you can also navigate a safe landing on a planet/moon, through the atmosphere.. Again, I see no consistancy in this..",
"topk_rank": 4
},
{
"id": "corpus-90181",
"score": 0.7475086450576782,
"text": "How does Picard choose a warp speed? How does Picard, or anyone for that matter, decide at what warp speed to travel? Obviously there are times when they need to get somewhere in a hurry and he says, warp 9 or maximum warp, but the rest of the time he just seems to pick a number at random. He rolls a warp 6.5 Is there some guide as to recommended speeds that he's following? Is there a speed that is the most efficient?",
"topk_rank": 5
},
{
"id": "corpus-136144",
"score": 0.7475086450576782,
"text": "How does Picard choose a warp speed? How does Picard, or anyone for that matter, decide at what warp speed to travel? Obviously there are times when they need to get somewhere in a hurry and he says, warp 9 or maximum warp, but the rest of the time he just seems to pick a number at random. He rolls a warp 6.5 Is there some guide as to recommended speeds that he's following? Is there a speed that is the most efficient?",
"topk_rank": 6
},
{
"id": "corpus-64265",
"score": 0.7381971478462219,
"text": "Star Wars canon, small ships and hyperdrive I distinctly remember in Star Wars, A New Hope, in the scene when the Millennium Falcon arrives at the remains of what was once Alderaan, a TIE fighter passes close by and Han Solo gives chase. Obi-Wan says \"A fighter that size couldn't get this deep into space on its own.\" Is he implying that TIE fighters don't have Hyperdrives or that they are too far from Imperial space? I ask this because I distinctly remember that, at least by \"ROTJ\" X-Wings are very capable of hyperdrive as in the scene at the beginning of the Battle of Endor when the entire rebel fleet makes the jump into hyperspace including a squadron of X-Wings. X-Wings are not much bigger than TIE fighters. Thank you",
"topk_rank": 7
},
{
"id": "corpus-200818",
"score": 0.723966658115387,
"text": "Does warp drive permit safe, near- or above- lightspeed travel through atmospheres? As XKCD recently educated some of us in a \"What If\" blog post, playing (near) light-speed baseball on a planet with an atmosphere . The TL;DR version translates to: A baseball moving at 0.9c is traveling too fast for particles in the atmosphere to get out of the way. The resulting atomic collisions set off a chain reaction that feeds off of the rest of the surrounding atmosphere, turning into a massive nuclear explosion. This does not bode well for the batter, the pitcher, the catcher, or anyone else within a few miles of the reaction's epicenter. Bringing this issue into context with the Star Trek universe, there was a point in the 2009 movie where the Enterprise came out of warp amidst the clouds of Saturn's sixth moon, Titan. Presumably, this means that the Enterprise at some point was traveling through Titan's atmosphere near or above the speed of light. Is there some principle of warp drive technology that allows the Enterprise to avoid the fate of the baseball in this situation, or should that interaction have actually destroyed (at least) a good part of the Enterprise and Titan?",
"topk_rank": 8
},
{
"id": "corpus-38162",
"score": 0.7121832370758057,
"text": "Are There Levels of Canon in Star Trek, Like in Star Wars? Star Wars has different levels of canon and explanations for each level (movies, books, games and so on). From what I've seen in Star Trek, there seems to be one level of canon: If it was on screen (other than the animated series), it's canon (although some people claim the 2009 movie and Star Trek: The Final Frontier don't really count). If it's not on screen, it's not canon. Is that it? Or are there some games that are a second level of canon, or some books or other material considered to be a higher level than other media? Or is there just one level of canon in Star Trek?",
"topk_rank": 9
},
{
"id": "corpus-117525",
"score": 0.7119284868240356,
"text": "Have the rules on hyperspace changed in canon now? In Star Wars - hyperspace travel has some limitations. For one, you can't get to close to a gravitational body. We see this in Ep IV - the Falcon must lift off on repulsorlifts and travel at sublight away from Tatooine before entering hyperspace. I seem to recall something in the novelization about \"six planetary radii\" when it explained how the Death Star moved using hyperspace and sublight engines. In the now Legends canon - this gravity limitation was the explanation for the Kessel Run being measure in parsecs (basically, how much of a corner a ship can cut past a cluster of black holes), and even used tactically by the Empire with their Interdictor cruisers. So, in the new film, how could the Falcon fly through hyperspace to get under the Starkiller Base shields?",
"topk_rank": 10
},
{
"id": "corpus-99688",
"score": 0.7110877633094788,
"text": "What is the speed of the Force? For instance, if a Jedi Force-pushes something 1000 meters away, is the effect instantaneous or is there a propagation delay, something like the speed of light?",
"topk_rank": 11
},
{
"id": "corpus-191233",
"score": 0.7094601392745972,
"text": "Are those stars moving past the window when the ship is at warp? When the enterprise (ENT) is moving at warp we see numerous stars flying past in the windows: However, according to , at warp 5 (the maximum speed of the 1st enterprise) it would take 7.4 days to get from the Sun to Alpha Centari, our nearest star(s) approximately 4 light years away. So to have stars whizzing past every few seconds would imply that the stars are very close together. Is this an inaccurate depiction of star movement at warp, or is there some explanation like the warp bubble creates a lensing effect? Now I realise the classic answer is \"suspend your disbelief, it's a tv show\", but the animators have at least thought about the star movement as they have added in a bit of red shift and blue shift which you can see in the image.",
"topk_rank": 12
},
{
"id": "corpus-104149",
"score": 0.7091847062110901,
"text": "What defines the upper limit for speed of Subspace Communications? mentions there to be a definite upper limit as to the speed of Subspace Communications. Is the reason for this ever touched on in any of the Star Trek media (Canon or not) or is it \"speed of plot\"? This would suggest speed of plot but I'm curious what would provide an upper limit if light doesn't?",
"topk_rank": 13
},
{
"id": "corpus-17394",
"score": 0.7082675099372864,
"text": "What difference will the more skillful (1 star or 2 star) crew members make compared to a non-skilled one in FTL? I checked out all tooltips in the game and their official website. Since FTL did not come with a manual I have no clue here. I know that they have something to do with recharge rates and percentages but I can't find anything specific about it.",
"topk_rank": 14
},
{
"id": "corpus-88590",
"score": 0.7082450985908508,
"text": "How does \"warp drive\" not violate Special Relativity causality constraints? I'm talking about this nonsense: Now, I'm aware that there are problems with the practicalities (or possibilities) of methods involving exotic matter with negative mass, and that kind of thing has been addressed here: But in this case my question is different and more specific. Sweeping all the other problems of constructing such a beast under the rug, wouldn't this violate causality? SR says that the simultaneity and relative ordering of events is dependent on the relative motion of the observer, but that no matter how it works out, if event A causes event B, no observer will see B first. One of the many problems with FTL travel (or signals) is that if any information is sent faster than light, there will exist an inertial frame in which the signal arrives before it is sent, a blatant violation of causality. (I think we can all agree, can we not, that the threshold should be extremely high before breaking assumptions that we're in a causal universe?) So how is any scheme to travel faster than light, even if it may locally satisfy the equations of GR, not a blatant violation causality according to SR? Is my understanding of GR incomplete and there's something in it that obviates the causality implications of SR? Or am I correct that FTL claims should be dismissed with extreme prejudice on this basis alone? (Pending a serious replicable experiment showing causality to be breakable.)",
"topk_rank": 15
},
{
"id": "corpus-107568",
"score": 0.7082450985908508,
"text": "How does \"warp drive\" not violate Special Relativity causality constraints? I'm talking about this nonsense: Now, I'm aware that there are problems with the practicalities (or possibilities) of methods involving exotic matter with negative mass, and that kind of thing has been addressed here: But in this case my question is different and more specific. Sweeping all the other problems of constructing such a beast under the rug, wouldn't this violate causality? SR says that the simultaneity and relative ordering of events is dependent on the relative motion of the observer, but that no matter how it works out, if event A causes event B, no observer will see B first. One of the many problems with FTL travel (or signals) is that if any information is sent faster than light, there will exist an inertial frame in which the signal arrives before it is sent, a blatant violation of causality. (I think we can all agree, can we not, that the threshold should be extremely high before breaking assumptions that we're in a causal universe?) So how is any scheme to travel faster than light, even if it may locally satisfy the equations of GR, not a blatant violation causality according to SR? Is my understanding of GR incomplete and there's something in it that obviates the causality implications of SR? Or am I correct that FTL claims should be dismissed with extreme prejudice on this basis alone? (Pending a serious replicable experiment showing causality to be breakable.)",
"topk_rank": 16
},
{
"id": "corpus-132756",
"score": 0.7044802308082581,
"text": "Why did the Star Trek writers decide Warp 10 would be infinite? From onward, has a basically cubic scale from warps 1 - 9. But then, close to warp 10, it suddenly develops its own puzzling scale, as can be seen in . In this scale, , occupying all points in the universe simultaneously and therefore effectively teleporting instantaneously. My question is why have this confusing scale? Why not have a uniform cubic scale? Was there ever any \"official\" explanation for this? Clarification: I'm looking for why the writers decided on this, not the \"in-universe\" explanation.",
"topk_rank": 17
},
{
"id": "corpus-82171",
"score": 0.7042145729064941,
"text": "Why did the Star Trek writers decide Warp 10 would be infinite? From onward, has a basically cubic scale from warps 1 - 9. But then, close to warp 10, it suddenly develops its own puzzling scale, as can be seen in . In this scale, , occupying all points in the universe simultaneously and therefore effectively teleporting instantaneously. My question is why have this confusing scale? Why not have a uniform cubic scale? Was there ever any \"official\" explanation for this? Clarification: I'm looking for why the writers decided on this, not the \"in-universe\" explanation.",
"topk_rank": 18
},
{
"id": "corpus-17705",
"score": 0.6982517242431641,
"text": "What are some scenarios where FTL information transfer would violate causality? I've always heard people saying, \"Faster than light information transfer can't happen because it would violate causality! Effects can precede their causes!\" However, I'm trying to think of a situation where this would happen. I'm sure it has something to do with relativity or something like that. I sort of understand that people going faster perceive time slower. Can someone help illuminate this for me by showing a scenario where causality is clearly violated due to FTL information transfer?",
"topk_rank": 19
}
] |
query-3
|
sed to redirect data to a file I am trying to replace >>> with "" in file cat table_files.md | egrep ">>>$" \>>> \>>> \>>> \>>> \>>> \>>> \>>> \>>> \>>> replace it with sed cat table_files.md | sed "s\>>>$\\g" > table_files.md Unfortunately, all my data gone, it feedback to me a blank file. I check this should be done using -i option in place. How could that happen, is it possible to recover the data?
|
[
{
"id": "corpus-3",
"score": 0.7184659838676453,
"text": "redirect to the same file as the source file processed by the command Redirection is not allowed here. Is cat not allowing? But isn't redirection independent of any command running with it? $ cat bk > bk cat: bk: input file is output file Why does the following redirect empty the file: $ less bk > bk awk also works in the similar way as less in regard to redirection. awk is the one with which I actually found the problem, while the above examples are meaningless In general, what are the right way to use redirect?"
}
] |
[
{
"id": "corpus-153784",
"score": 0.6825014352798462,
"text": "Replace line M in file1 with line N in file2 Let M=3 and N=4. I want to replace line M in file1 with line N in file2. I can replace with a string using sed: sed -i '3s/.*/stringToReplace/' file1 And I can use awk to get line N from file2 awk 'NR==4' \"file2\" How can I combine these two? If I try sed -i '3s/.*/{awk 'NR==4' \"file2\"}/' file1 then I replace line M with literally {awk 'NR==4' \"file2\"}.",
"topk_rank": 0
},
{
"id": "corpus-246116",
"score": 0.6822612881660461,
"text": "Using sed to delete a whole line if it contains only specific number I have a txt file 123 456 789 456 123456 I want to delete specific characters from the file i.e (123). I tried with $ sed -i '/123/d' dummy.sh $ vi dumm.txt 456 789 456 In the below command when I run both the words (123 and 123456) are getting deleted but I need to delete only 123 from the file $ sed -i 's/123//g' dummy.sh $ vi dumm.txt 456 789 456 456 when I run the below command the 123 is getting replaced with null. Can anyone say how to delete a whole line if it contains only specific number?",
"topk_rank": 1
},
{
"id": "corpus-104984",
"score": 0.6822378039360046,
"text": "Redirect stderr and stdout in Bash I want to redirect both stdout and stderr of a process to a single file. How do I do that in Bash?",
"topk_rank": 2
},
{
"id": "corpus-116842",
"score": 0.6822378039360046,
"text": "Redirect stderr and stdout in Bash I want to redirect both stdout and stderr of a process to a single file. How do I do that in Bash?",
"topk_rank": 3
},
{
"id": "corpus-202506",
"score": 0.682058572769165,
"text": "How to detect end of line with sed I'm looking for a way to only execute replacement when the last character is a newline, using sed. For instance: lettersAtEndOfLine is replaced, but this is not: lettersWithCharacterAfter& Since sed does not work well with newlines, it is not as simple as $ sed -E \"s/[a-zA-Z]*\\n/replace/\" file.txt How can this be accomplished?",
"topk_rank": 4
},
{
"id": "corpus-183991",
"score": 0.6820352077484131,
"text": "What is the purpose of \"~\" in the command \"sed 's~ ~~g'\"? I found line sed 's~ ~~g' in a shell script on a Linux system. What is this ~?",
"topk_rank": 5
},
{
"id": "corpus-135728",
"score": 0.6813018321990967,
"text": "Delete specific line number(s) from a text file using sed? I want to delete one or more specific line numbers from a file. How would I do this using sed?",
"topk_rank": 6
},
{
"id": "corpus-215873",
"score": 0.6812456250190735,
"text": "What does & mean exactly in output redirection? I see stuff like command 1> out or with 2>&1 to redirect stderr, but sometimes I also see &> by itself, etc. What is the best way to understand & and what it means exactly?",
"topk_rank": 7
},
{
"id": "corpus-134156",
"score": 0.6809055805206299,
"text": "How could I remember how to use redirection? I know what program > /dev/null 2>&1 does. It redirects the output to /dev/null and 2>&1 means to redirect the error output in the same place where the output is sent. My problem is I always have to google it because I never remember it. So, I try &2>1, 1>2&, 1>&2... I try every combination until I google it... What's the trick to remember it easily?",
"topk_rank": 8
},
{
"id": "corpus-173868",
"score": 0.680867612361908,
"text": "Bash string range and replace To print part of the string, replace , with . I use command: echo \"${q:16:6}\" | sed 's/,/./' Is it possible to use something like: echo \"${q:16:6/,/.}\" because it does not work?",
"topk_rank": 9
},
{
"id": "corpus-374",
"score": 0.6807162165641785,
"text": "Are there zero-width assertions in sed? Is it possible to make lookahead or lookbehind zero-width assertions in sed? I want to emulate Perl's (?=) and family. My sed is not GNU sed version 4.0.",
"topk_rank": 10
},
{
"id": "corpus-15390",
"score": 0.6805747747421265,
"text": "How to delete whole text from the beginning of a line with a certain string I use sed -n '/string/,$p' file >> otherfile to copy text from a line with certain string to another file. Now I want the copied text to be deleted in the original file. I tried different methods with sed and awk but nothing worked. How to use sed to delete the text of a file from the beginning line with a certain string in it? texta texta1 textb <- string, delete rest of the text from here textb1 textc textc1 or would it be easier to cut instead of copying with sed and what would be the command instead?",
"topk_rank": 11
},
{
"id": "corpus-88517",
"score": 0.6804854273796082,
"text": "sed command not giving proper output for regular expression used I am trying to fetch data between a range of time intervals from the log file. However, the complete data is not fetched for the end date. Below is the command used: sed -n '/26/Nov/2018/,/30/Nov/2018/ p' /path/file.log The data I am having in the above log file is somewhat like: 192.168.0.123 - - [25/Nov/2018:13:24:59 +0530] \"Data 1\" 192.168.0.123 - - [26/Nov/2018:13:24:59 +0530] \"Data 2\" 192.168.0.123 - - [26/Nov/2018:13:25:09 +0530] \"Data 3\" 192.168.0.123 - - [27/Nov/2018:13:25:09 +0530] \"Data 4\" 192.168.0.123 - - [27/Nov/2018:13:25:11 +0530] \"Data 5\" 192.168.0.123 - - [27/Nov/2018:13:25:53 +0530] \"Data 6\" 192.168.0.123 - - [28/Nov/2018:13:25:53 +0530] \"Data 7\" 192.168.0.123 - - [29/Nov/2018:13:25:53 +0530] \"Data 8\" 192.168.0.123 - - [30/Nov/2018:13:25:53 +0530] \"Data 9\" 192.168.0.123 - - [30/Nov/2018:13:25:53 +0530] \"Data 10\" 192.168.0.123 - - [01/Dec/2018:13:25:53 +0530] \"Data 11\" 192.168.0.123 - - [01/Dec/2018:13:25:53 +0530] \"Data 12\"",
"topk_rank": 12
},
{
"id": "corpus-45147",
"score": 0.6803460121154785,
"text": "Using a reference to a bash string variable in sed I'm a relative Linux novice. Suppose that I have a text file a.txt that contains the following text: A B C Now, if I want to change the text on line 2 (which contains B), I can use the following sed command: sed -i '2s/^.*$/X/' a.txt to change the B to X, for example. But now suppose that I want to write an sed command that changes the text on line 2 to something that includes a bash string variable. I want to change the text on line 2 to fname=\"myaddress\", where the quotation marks are included explicitly, and where myaddress is a bash string variable (defined somewhere else). To try to do this, I have created a bash shell script called mytest.sh and containing the following: #!/bin/bash myaddress=\"/home/\" echo fname=\\\"$myaddress\\\" head a.txt sed -i '2s/^.*$/fname=\\\"$myaddress\\\"/' a.txt head a.txt Now, echo fname=\\\"$myaddress\\\" prints fname=\"/home/\", which is the text that I want to be on line 2 of a.txt. However, sed instead prints fname=\"$myaddress\" to line 2 of a.txt, probably because it is inside the surrounding apostrophes. In other words, the output of the above bash shell script is: fname=\"/home/\" A B C A fname=\"$myaddress\" C My question is, how can I get sed to actually print the value stored in myaddress and referenced by $myaddress?",
"topk_rank": 13
},
{
"id": "corpus-170508",
"score": 0.6803460121154785,
"text": "Using a reference to a bash string variable in sed I'm a relative Linux novice. Suppose that I have a text file a.txt that contains the following text: A B C Now, if I want to change the text on line 2 (which contains B), I can use the following sed command: sed -i '2s/^.*$/X/' a.txt to change the B to X, for example. But now suppose that I want to write an sed command that changes the text on line 2 to something that includes a bash string variable. I want to change the text on line 2 to fname=\"myaddress\", where the quotation marks are included explicitly, and where myaddress is a bash string variable (defined somewhere else). To try to do this, I have created a bash shell script called mytest.sh and containing the following: #!/bin/bash myaddress=\"/home/\" echo fname=\\\"$myaddress\\\" head a.txt sed -i '2s/^.*$/fname=\\\"$myaddress\\\"/' a.txt head a.txt Now, echo fname=\\\"$myaddress\\\" prints fname=\"/home/\", which is the text that I want to be on line 2 of a.txt. However, sed instead prints fname=\"$myaddress\" to line 2 of a.txt, probably because it is inside the surrounding apostrophes. In other words, the output of the above bash shell script is: fname=\"/home/\" A B C A fname=\"$myaddress\" C My question is, how can I get sed to actually print the value stored in myaddress and referenced by $myaddress?",
"topk_rank": 14
},
{
"id": "corpus-166282",
"score": 0.679914653301239,
"text": "Redirect output from all commands to file, not just last I want to add a header to a file with echo, then use a command to create the rest of the file. This means I'll be using two separate commands. How do I write the output from both commands to a file with redirection? I've tried echo \"header line\" | cut -c 1-5 input_file > output_file echo \"header line\"; cut -c 1-5 input_file > output_file This only redirects the output from the cut command. The following command works, but feels clumsy: echo \"header line\" > output_file; cut -c 1-5 input_file >> output_file What is the clever way of solving my problem?",
"topk_rank": 15
},
{
"id": "corpus-202215",
"score": 0.6798051595687866,
"text": "Decoding URL encoding (percent encoding) I want to decode URL encoding, is there any built-in tool for doing this or could anyone provide me with a sed code that will do this? I did search a bit through and on the internet but I couldn't find any command line tool for decoding url encoding. What I want to do is simply in place edit a txt file so that: %21 becomes ! %23 becomes # %24 becomes $ %26 becomes & %27 becomes ' %28 becomes ( %29 becomes ) And so on.",
"topk_rank": 16
},
{
"id": "corpus-34339",
"score": 0.6791940927505493,
"text": "How to split long sed expression into multiple lines? sed 's/[long1][long2]/[long3][long4]/' file.txt I would like to split this command onto multiple lines - f.x. something like this: sed 's/ [long1] [long2] / [long3] [long4] /' file.txt Using \\ or separating strings didn't work.",
"topk_rank": 17
},
{
"id": "corpus-231262",
"score": 0.6781678795814514,
"text": "Replace comma with newline following with just a command and space I have a list like so. blue-image1.jpg, blue-image2.jpg, blue-image3.jpg red-image1.jpg, red-image2.jpg, red-image3.jpg It's was originally a csv format but I added newlines because they needed a prefix at first. Now I need to put them together again so it becomes like this. blue-image1.jpg,blue-image2.jpg,blue-image3.jpg red-image1.jpg,red-image2.jpg,red-image3.jpg I tried cat list.txt | tr ',\\n' ',' and cat list.txt | sed 's/,\\n/,/g' but neither seem to work properly.",
"topk_rank": 18
},
{
"id": "corpus-103392",
"score": 0.6779318451881409,
"text": "What is the differences between &> and 2>&1 There are two forms of redirection the standard output and standard error into standard output. But which one is better? and why the &> is considered the perfect? I can't find what is the differences so that many tutorials and even bash manual state that &> is better! So Why shall I use &> and not 2>&1 Mainly using bash shell EDIT: Thanks for commentors Only >& works in csh or tcsh In ksh only 2>&1 works. dash use >file 2>&1 redirection only Then which one to use to ensure my script is compatible with other systems whatever the used shells are!",
"topk_rank": 19
}
] |
query-4
|
Trying to install Ubuntu 13.04 alongside Windows 7 I installed Ubuntu 13.04 alongside the existing Windows 7 OS. The Ubuntu installation completed successfully, and it asked me to restart my computer. When I restarted, it was just Windows 7 that loaded as usual, but Ubuntu wasn't there. Can anyone tell me what's wrong?
|
[
{
"id": "corpus-4",
"score": 0.7665112018585205,
"text": "How can I repair grub? (How to get Ubuntu back after installing Windows?) I installed Windows 7, which ate Ubuntu's boot file. When starting up the computer, it now goes straight to Windows, without giving me the option of booting Ubuntu. How can I get Ubuntu back?"
}
] |
[
{
"id": "corpus-45783",
"score": 0.7279748916625977,
"text": "How can I install Ubuntu on a second hard drive with Windows being on the first? I have recently got an Alienware M17x R4 and since it has two hard drive bays I was thinking of putting my hard drive from my old laptop and installing Ubuntu on it without making any changes to the drive that has Windows on it. Also if possible to add a entry to the Windows boot loader to boot Ubuntu instead of using GRUB.",
"topk_rank": 0
},
{
"id": "corpus-109348",
"score": 0.7275742292404175,
"text": "Can I install Ubuntu inside Windows? Ubuntu 12.04 won't come with the Windows Installer (Wubi) pre-added. But is there any way to install 12.04 using Wubi?",
"topk_rank": 1
},
{
"id": "corpus-150822",
"score": 0.7275742292404175,
"text": "Can I install Ubuntu inside Windows? Ubuntu 12.04 won't come with the Windows Installer (Wubi) pre-added. But is there any way to install 12.04 using Wubi?",
"topk_rank": 2
},
{
"id": "corpus-8048",
"score": 0.7239404916763306,
"text": "What's the Best Practice reinstalling windows 7 on a dual boot with ubuntu 11.04 I've posted a question that I have a 32bits windows 7 on a dual boot machine with ubuntu 10.04 upgraded to 11.04. From the responses I had from the questions, I decided to reinstall the windows 7 but this time a 64bits version. I've install the grub on the first section of the HDD on which ubuntu is install, meaning the 2nd Drive. I still want guidance before I proceed. What would you suggest me as best way to achieve that without disturbing the boot loading. thank you for reading this.",
"topk_rank": 3
},
{
"id": "corpus-90223",
"score": 0.7239404916763306,
"text": "What's the Best Practice reinstalling windows 7 on a dual boot with ubuntu 11.04 I've posted a question that I have a 32bits windows 7 on a dual boot machine with ubuntu 10.04 upgraded to 11.04. From the responses I had from the questions, I decided to reinstall the windows 7 but this time a 64bits version. I've install the grub on the first section of the HDD on which ubuntu is install, meaning the 2nd Drive. I still want guidance before I proceed. What would you suggest me as best way to achieve that without disturbing the boot loading. thank you for reading this.",
"topk_rank": 4
},
{
"id": "corpus-209351",
"score": 0.7231636643409729,
"text": "How to load Windows 7 from GRUB rescue I had Windows 7 and Ubuntu installed on the same partition Some time later, decided to uninstall Ubuntu.So I booted Ubuntu,Changed the hd related to it to NTFS,I did NOT extended my main drive restarted-> GRUB rescue>. No surprise since I had just wiped out the whole GRUB directory staff Now I just want to boot into my windows.Move my necessary files to an external memory and reinstall the windows seven What I have in my hands: Ubuntu live CD, A copy of win7 DVD I tried to make use of the win7 DVD(I had lost my DVD) to fix the mbr but the problem is when I choose repair, It says that this version of Windows is incompatible with yours What should I do?",
"topk_rank": 5
},
{
"id": "corpus-142216",
"score": 0.7226884961128235,
"text": "Do I keep my Windows installation if I install Ubuntu with Wubi, the Windows installer? I want to know if I will keep and still be able to use my Windows 7 installation after I use the Wubi Windows Installer (from ) to install Ubuntu? Thanks in advance, Bas.",
"topk_rank": 6
},
{
"id": "corpus-229899",
"score": 0.7224135994911194,
"text": "Live USB for Ubuntu 13.10 does't recognize Windows 8 OS I have created a Live USB for Ubuntu 13.10 64 bit using Universal-USB-Installer-1.9.4.4 I have restarted and booted from the USB; tried Ubuntu without installing. Everything works fine there including web browsing, videos, installing new software etc. So I decide to install Ubuntu as dual boot option to Windows 8. When I tried installing, the options do not show the existing Windows 8 OS. It shows only the option to install Ubuntu as standalone OS. The step by step installation instructions available online mention that it should show option like \"Install Ubuntu alongside Windows 8\" I have a Dell Inspiron 3521 with Windows 8; UEFI boot. I have 3 options for the boot menu: \"UEFI Security boot On\", \"UEFI Security boot Off\" and \"Legacy boot\". Please advise how I can get to dual boot Windows 8 and Ubuntu. I know that USB setup is good as I am able to boot and try from it without installing. May be boot settings? Thanks in Advance! ~KK",
"topk_rank": 7
},
{
"id": "corpus-210842",
"score": 0.7223040461540222,
"text": "How to remove Windows after installing Ubuntu? I installed Ubuntu 13.04 on my Vista 64 bit machine and I thought it was going to uninstall Vista but it didn't, now I have both systems. How do I get rid of Vista?",
"topk_rank": 8
},
{
"id": "corpus-4234",
"score": 0.7213079333305359,
"text": "How do I set Ubuntu to my default boot? I have Windows 7 and 8 installed on my laptop, I just installed Ubuntu. It normally boots up in Windows 8, how do I change that?",
"topk_rank": 9
},
{
"id": "corpus-22337",
"score": 0.7201576828956604,
"text": "Ubuntu installation (multiboot) I have a windows 7 Computer. I want to install ubuntu. I know if I install to the unallocated it will set up the multiboot like it should. However, I want to set the partitions up manually, I want different stuff in different partitions, e.g. logs. If I do this, will the installer still set up the multiboot like it should?",
"topk_rank": 10
},
{
"id": "corpus-51211",
"score": 0.7200013399124146,
"text": "Grub and Ubuntu disappeared after Windows update I had Windows 10 and Ubuntu 14.04 installed in dual boot. Roughly 125GB each. After the Windows update the PC only showed me the grub rescue terminal. I could neither use Ubuntu nor use Windows. I inserted the liveCD and ran boot-repair. After that Windows appeared normally, but no GRUB and no Ubuntu. This is what I get when I run sudo fdisk -l: Ubuntu must be either /dev/sda2 or /dev/sda4, since they are the biggest partitions, but neither of them is recognized as a Linux system. What should I do?",
"topk_rank": 11
},
{
"id": "corpus-195239",
"score": 0.7189235091209412,
"text": "Can not boot Windows on Legacy BIOS after installing Ubuntu on UEFI Long story short, I installed ubuntu while I was watching netflix and feeling very lazy, and may have lost all my windows 7 files. Quick summary: -Windows 7 was installed on Legacy bios and I installed ubuntu on UEFI bios -ubuntu installer didn't detect any other OS on my computer and wanted to write over the disk, so I had to manually edit the partition -Despite repeated warnings about partitioning windows in the ubuntu installer, I did this anyway, assigning 140 gigs to ubuntu and 500 to windows Now, my computer won't boot into anything but Ubuntu -in Gparted, it still shows that I have two partitions, and windows 7 boot loader -However, I can't boot into the non-linux partition, and boot loader says I need a recovery disk. My question is does anyone know if what I've done is fixable? I'm hoping that I can fix it using the way outlined in this thread: Thanks for the help everyone! Note: gparted shows: Foudn linux image: /boot/vmlinuz - 3.13.0.30 - generic Found linitrd image: /boot/initrd.img - 3.13.0 - 30 - generic Found linux image: /boot/vmlinuz - 3.13.0 - 24 generic Found initrd image: /boot/initrd.img - 3.13.0 - 24- generic memtest memtest Found windows 7 (loader) on /dev/sda1",
"topk_rank": 12
},
{
"id": "corpus-115403",
"score": 0.7187484502792358,
"text": "Is it possible to install Ubuntu without a CD or USB drive? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive. The laptop has windows 7 installed with ubuntu 11.04 installed through wubi. What I want to do is remove windows completely, and make ubuntu the only OS installed on the system. Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?) or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in).",
"topk_rank": 13
},
{
"id": "corpus-126176",
"score": 0.7187484502792358,
"text": "Is it possible to install Ubuntu without a CD or USB drive? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive. The laptop has windows 7 installed with ubuntu 11.04 installed through wubi. What I want to do is remove windows completely, and make ubuntu the only OS installed on the system. Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?) or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in).",
"topk_rank": 14
},
{
"id": "corpus-127452",
"score": 0.7187484502792358,
"text": "Is it possible to install Ubuntu without a CD or USB drive? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive. The laptop has windows 7 installed with ubuntu 11.04 installed through wubi. What I want to do is remove windows completely, and make ubuntu the only OS installed on the system. Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?) or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in).",
"topk_rank": 15
},
{
"id": "corpus-128757",
"score": 0.7187484502792358,
"text": "Is it possible to install Ubuntu without a CD or USB drive? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive. The laptop has windows 7 installed with ubuntu 11.04 installed through wubi. What I want to do is remove windows completely, and make ubuntu the only OS installed on the system. Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?) or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in).",
"topk_rank": 16
},
{
"id": "corpus-163691",
"score": 0.7187484502792358,
"text": "Is it possible to install Ubuntu without a CD or USB drive? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive. The laptop has windows 7 installed with ubuntu 11.04 installed through wubi. What I want to do is remove windows completely, and make ubuntu the only OS installed on the system. Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?) or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in).",
"topk_rank": 17
},
{
"id": "corpus-193690",
"score": 0.7187484502792358,
"text": "Is it possible to install Ubuntu without a CD or USB drive? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive. The laptop has windows 7 installed with ubuntu 11.04 installed through wubi. What I want to do is remove windows completely, and make ubuntu the only OS installed on the system. Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?) or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in).",
"topk_rank": 18
},
{
"id": "corpus-43691",
"score": 0.7185115814208984,
"text": "How to restore Windows boot loader after removing Ubuntu I am just about to install Ubuntu alongside Windows, just need to make sure I know how to remove it later. I know from past the major problem was when I removed Ubuntu, I either still got into the GRUB booting menu, or I couldn't boot at all, and had to format HDD and start over. Now I found an article practically saying the Windows loader should be back upon deleting Linux partition, so I want to make sure: If I have Windows and Ubuntu dualboot, and I want to get back to Windows booting quickly itself without choosing from any booting menu, is it enough to just delete the Ubuntu partition? Will the Windows loader override GRUB? Thanks!",
"topk_rank": 19
}
] |
query-5
|
Removing malware injecting into Google Chrome I had seen this popup across different sites, but I realized my computer was infected once I got hit with it on stackexchange! I'm using Google Chrome on Mac computer, and I've checked my extensions. Nothing is suspicious about the extensions I have. How do I find the process running this code injection and remove it from my computer? And what tools are best for the job? The linked question is how to remove viruses from my PC. The question states that I have a Mac.
|
[
{
"id": "corpus-5",
"score": 0.736211359500885,
"text": "How can I remove malicious spyware, malware, adware, viruses, trojans or rootkits from my PC? What should I do if my Windows computer seems to be infected with a virus or malware? What are the symptoms of an infection? What should I do after noticing an infection? What can I do to get rid of it? how to prevent from infection by malware? This question comes up frequently, and the suggested solutions are usually the same. This community wiki is an attempt to serve as the definitive, most comprehensive answer possible. Feel free to add your contributions via edits."
}
] |
[
{
"id": "corpus-139120",
"score": 0.6970067024230957,
"text": "How do I completely uninstall Google Chrome and Chromium? I'm having a problem with Chrome and one of its extensions, so I want to completely uninstall everything. I've uninstalled Chrome through the Software Center, but when I reinstall it, all of my extensions and preferences are still there. If I want to start over from scratch, what do I need to do?",
"topk_rank": 0
},
{
"id": "corpus-54233",
"score": 0.6949208378791809,
"text": "Ubuntu vulnerable to malware sites? I just browsed a site a while ago that was flagged as an \"attack site\" or has malware, but Firefox didn't warn before I went into it. Only when I opened a new tab with another page from the site did it block the page. How do I check for vulnerabilities on my system since I think there are no anti-malware available for Ubuntu? If there are, please do tell. Thanks!",
"topk_rank": 1
},
{
"id": "corpus-60779",
"score": 0.69001305103302,
"text": "Deleting a suspicious malware (imklaunchagent) in Mac I have a MacBook Pro (macOS Mojave) laptop and have recently noticed a suspicious process in Activity Monitor with this name : imklaunchagent I checked this folder: /System/Library/Frameworks/InputMethodKit.framework/Versions/A/Resources/imklaunchagent and it seems imklaunchagent and its related files cannot be removed. I tried to remove it via Terminal and this message showed up: \"Operation not permitted\". However, in \"Security and Privacy\" of \"System Preferences\", I chose Terminal to have full disk access, still can't delete the file and the same message appears. I installed the free version of Combo Cleaner and it detected other viruses, none of which are imklaunchagent. Is imklaunchagent a malware? How can it be deleted?",
"topk_rank": 2
},
{
"id": "corpus-105195",
"score": 0.6882840991020203,
"text": "Goodle Ads and malicious links Had my website hacked through a plugin 3 weeks ago. rebuilt the site with a new wordpress site (hosted on Godaddy wordpress pro). this was a fresh build. All the built in malware checks in wordpress plugins and also google's console checks are good and clean. Google ads wont run as it reports I have three malicious links in my website. They gave me the link addresses but they are nothing I can locate. Anyone have ideas on how to find what google found?",
"topk_rank": 3
},
{
"id": "corpus-239661",
"score": 0.6818935871124268,
"text": "Re-enable extensions not coming from Chrome Web Store on Chrome v35+ (with enhanced security) Does anybody know how to re-enable your own extensions after they were disabled by the Chrome v35 update? (Mostly greasemonkey scripts in my case, thus simple .js files previously drag-n-dropped in to the extension windows. ) When I started up Chrome today I got the warning that some non-ChromeWebStore extensions were disabled. More info was giving on this : Extensions disabled by Chrome You're seeing this notification because one or more of your Chrome extensions has been turned off to make Chrome safer. The extensions didn't come from the Chrome Web Store or were installed without your permission. For your protection, you can only use Chrome extensions that you get from the Chrome Web Store. To see a list of your extensions: Click the Chrome menu Chrome menu on the browser toolbar. Select Tools. Select Extensions. Extensions that have been disabled are grayed out and you won't be able to re-enable them. I was hoping I could still enable them by activating developer mode for my extensions but still no luck.. Any tips anyone? P.s. this is not a duplicate from as this is related to the chrome v35 update",
"topk_rank": 4
},
{
"id": "corpus-61094",
"score": 0.6814680695533752,
"text": "How do you tell what Bing is detecting as malware on a page? Bing Webmaster Tools started flagging some of my pages as infected with malware. Despite my best efforts, I have been unable to figure out what the problem might be. The \"help\" message about the malware is: Malware Network Reference Reference to known malware distribution network detected. This could be via inclusion of an iframe or frame tag, an embedded object, or a script tag. So far I have: Verified that the source code for the website is unchanged (I verified that it matches what is checked into git, and that there is nothing weird in the commit history) I fetched the page as Bingbot and compared it to a version that I downloaded using another user agent to make sure that there isn't something serving Bing an altered page I grepped the page source for case insensitive script, frame, object, and embed to make sure that I recognize all the code. I grepped the page source for case insensitive https?:// to make sure I recognize all the links What else should I be looking for? The only scripts that are in the page are either inline scripts that I wrote myself, or tags that include AdSense ads via DFP. Could the ads be inserting content that Bingbot is complaining about? Is there any way to get further information from Bing? I was unable to find anything further that might be helpful in their webmaster tools. I couldn't find that they said anything about which specific malware, or what code snippets from my site might be problematic.",
"topk_rank": 5
},
{
"id": "corpus-66507",
"score": 0.6787136197090149,
"text": "How to list Explorer extensions and disable them? I have reason to believe that I may have malware on my system in the form of an Explorer extension. I suspect this because Procmon shows that Explorer.exe keeps re-writing a certain Registry key that is intended to run a certain EXE at startup. How do I find out what Explorer extensions are installed, and how do I remove them?",
"topk_rank": 6
},
{
"id": "corpus-22103",
"score": 0.6750680804252625,
"text": "Anti-virus comparison for Mac OS X This is not a \"I use OS X, do I need Anti-Virus?\" question. I am looking at comparison and user opinion of existing AV products. I know I can avoid most (if not all) threats by being careful on the sites I visit, and the emails I download. But I need an anti-virus, for a variety of reasons - mostly to prevent my Mac as a carrier for viruses. (Many friends' USB drives turn out to be infected). I was looking for the \"Best\" of the anti virus software out there. I am looking prominently at anti-virus. \"Internet Security\" and the like are not important, and would try to avoid them if it would save money or not make my mac slower. By \"Best\" I mean the best combination of detect-rate (or whatever that is called) and performance. I don't intend to run it always in the background. Maybe I would do a onetime scan of my system and hard drives, and then it would be mostly on a need basis, especially on USB drives/portable hard disks of friends. So primary objective: High Detect rate I have used and (got this with one of those software bundles sometime before, not the latest version) and had problems with both. Both sometimes hog my CPU, and somehow \"lock\" my USB drives preventing me from Ejecting it (I had to force eject them or just pull them off from the port) (thought off late Norton seemed a little better) Ok, thats a long one. But my question is simple: What anti-virus do you recommend for the Mac? Thanks.",
"topk_rank": 7
},
{
"id": "corpus-93746",
"score": 0.6750680804252625,
"text": "Anti-virus comparison for Mac OS X This is not a \"I use OS X, do I need Anti-Virus?\" question. I am looking at comparison and user opinion of existing AV products. I know I can avoid most (if not all) threats by being careful on the sites I visit, and the emails I download. But I need an anti-virus, for a variety of reasons - mostly to prevent my Mac as a carrier for viruses. (Many friends' USB drives turn out to be infected). I was looking for the \"Best\" of the anti virus software out there. I am looking prominently at anti-virus. \"Internet Security\" and the like are not important, and would try to avoid them if it would save money or not make my mac slower. By \"Best\" I mean the best combination of detect-rate (or whatever that is called) and performance. I don't intend to run it always in the background. Maybe I would do a onetime scan of my system and hard drives, and then it would be mostly on a need basis, especially on USB drives/portable hard disks of friends. So primary objective: High Detect rate I have used and (got this with one of those software bundles sometime before, not the latest version) and had problems with both. Both sometimes hog my CPU, and somehow \"lock\" my USB drives preventing me from Ejecting it (I had to force eject them or just pull them off from the port) (thought off late Norton seemed a little better) Ok, thats a long one. But my question is simple: What anti-virus do you recommend for the Mac? Thanks.",
"topk_rank": 8
},
{
"id": "corpus-183930",
"score": 0.6741743087768555,
"text": "How to prevent malicious code execution loading blend files? Since I download from time to time example files from blogs, wikis etc. I would like to know how I can avoid execution of malicious code. Is it enough to disable the and see what is in the text editor?",
"topk_rank": 9
},
{
"id": "corpus-201045",
"score": 0.6738973259925842,
"text": "Viruses in wine? The Story I installed wine some time ago, and after trying out some versions of Cinema 4D that were \"free\", I looked in the comments of the videos, and saw some people complaining about viruses. I got worried, downloaded and installed MalwareBytes, and scanned my \"system\". Of course, since MalwareBytes was runnning through Wine, it only scanned ~/.wine/C:. It found many viruses/trojans and I opted to delete them. I did so. Questions: Assuming there is any malware left, can this malware spread into my OS? Tell me the entire truth on what can happen. I deleted my entire .wine directory and am planning to uninstall wine later. I am the only user out of the three on this system that runs wine programs, but wine is installed for all. Is that enough, or must I do more? Please be specific on what I should do because I want to be 100% sure that there are no viruses. I am sure that many others have these questions too, so please answer this question to the point. Additional information: Wine was installed with sudo-apt get. The exact commands can be found here: Commands: sudo add-apt-repository ppa:ubuntu-wine/ppa -y sudo apt-get update sudo apt-get install wine1.6 -y I did try to install a font that was a ttf font. I tried to convert it to otf with no success. Then, I placed it in the fonts directory directly. I then saw that some people said it was malicious, and I removed it. This question was flagged as a duplicate of this question: It is not the same. This question is more precise, as it specifies two scenarios involving viruses, not risks of viruses. The other question talked about Internet Explorer; this one does not mention the installation of any programs. Note: wine was installed with sudo privileges, so does that mean it runs with sudo privileges?",
"topk_rank": 10
},
{
"id": "corpus-105846",
"score": 0.6729212403297424,
"text": "Why can I get a virus or trojan from visiting a website? I have seen these a lot lately. You click the link, and instant trojan. No need to download or anything. How is this possible? Is there a way I can scan the links before visiting to make sure I won't be infected?",
"topk_rank": 11
},
{
"id": "corpus-60829",
"score": 0.6720933318138123,
"text": "Where should I start in tracking down Firefox malware Switched my sister to Ubuntu because I got tired of re-installing the other OS every 6 months. Now she managed to get some malware in her Firefox on Ubuntu. Without access to the computer (or when I get it next week), where should I look, what questions can I ask, what could I tell a computer novice to try over the phone? Symptoms: While surfing some recipe site she had an ad-window pop up with no window controls. She rebooted the computer and when she re-started Firefox it came back, full screen, no controls, on top. I told her to just use -F2 and xkill to get rid of it, which it did but stopped Firefox completely. On re-start it was back, I told her to hit F11 which did take it back to a large screen, but not full-screen, so she could see that there was a normal browser window running underneath. She topped the normal window but the only other thing I had time to check was plug-ins, which there wasn't anything that sounded suspicious. I'm looking for ideas of what to try over the phone, or where to start next week. I'm comfortable with the command line, and using about:config if that makes any difference in the answer.",
"topk_rank": 12
},
{
"id": "corpus-20401",
"score": 0.6719089150428772,
"text": "Is it generally advisable to get anti-virus software for Mac OS X? Is it worth to buy an anti-virus software for Mac OS X? I heard both opinions like \"it's not possible to get a virus on Mac OS X\" and \"Mac OS X is also vulnerable against virus... \" Which is true? What is the accepted practice, if any? Under what circumstances would anti-virus be advisable?",
"topk_rank": 13
},
{
"id": "corpus-83223",
"score": 0.6719089150428772,
"text": "Is it generally advisable to get anti-virus software for Mac OS X? Is it worth to buy an anti-virus software for Mac OS X? I heard both opinions like \"it's not possible to get a virus on Mac OS X\" and \"Mac OS X is also vulnerable against virus... \" Which is true? What is the accepted practice, if any? Under what circumstances would anti-virus be advisable?",
"topk_rank": 14
},
{
"id": "corpus-203742",
"score": 0.6719089150428772,
"text": "Is it generally advisable to get anti-virus software for Mac OS X? Is it worth to buy an anti-virus software for Mac OS X? I heard both opinions like \"it's not possible to get a virus on Mac OS X\" and \"Mac OS X is also vulnerable against virus... \" Which is true? What is the accepted practice, if any? Under what circumstances would anti-virus be advisable?",
"topk_rank": 15
},
{
"id": "corpus-23129",
"score": 0.6715167760848999,
"text": "There is by now a captcha-problem Tried several times with several captchas. is the data explorer page I tried to access. with space between : To the non-believers :) : System produced on: Windows 7 Chrome Version 33.0.1750.146",
"topk_rank": 16
},
{
"id": "corpus-57085",
"score": 0.6710217595100403,
"text": "Check if process is running Mac OS X then execute code I am creating a script which clears the cache for Google Chrome. However, I would like to check if Chrome is open and if so not run the code but if it isn't then it will execute the code. I can see that the Process Name is Google Chrome but the code doesn't work. This is what I have done so far. What am I doing wrong? SERVICE='Google Chrome' if ps ax | grep -v grep | grep $SERVICE then RUNS THE CODE else echo \"PLEASE CLOSE GOOGLE CHROME\" fi Any help would be appreciated :)",
"topk_rank": 17
},
{
"id": "corpus-194168",
"score": 0.6702133417129517,
"text": "How do you launch the JavaScript debugger in Google Chrome? When using Google Chrome, I want to debug some JavaScript code. How can I do that?",
"topk_rank": 18
},
{
"id": "corpus-52938",
"score": 0.6699856519699097,
"text": "My iPad has a popup saying I have a virus When I went to a website in Safari, it opened up another website with popups claiming my iPad has a virus. I deleted some of my apps to try and get rid of the virus, but the popup keeps showing up every time I open Safari. Can someone help me?",
"topk_rank": 19
}
] |
query-6
|
Completely brand new to GIS - Help picking technology from the pros I'm new to GIS and need to learn for an internship. I've been reading a lot and now need help picking technologies to use... particularly the map serving technologies. Project requirements - Free Web Based Service Able to display layers/features over Google Maps/Bing Maps Able to select individual features/layers (Seems like open layers does this?) e.g. select individual/multiple points/polygons Works on 64 bit Red Hat Compatible with PHP, PostGIS (seems like they all do) What I've narrowed it down to so far... GeoServer - Seems to be the easiest to learn and install. Not sure if I can use Google Maps with it though. Maybe with a WFS? Most questions tagged on this site. MapServer - Seems tricky to install on 64 bit. (No MapTools FGS yet for 64 bit) 2nd most questions tagged on this site. MapGuide - Looks cool and has cool features but it looks "youngest" and the documentation is behind so it might be hard to get things going fast. (by the time the internships over.) Least most tagged.
|
[
{
"id": "corpus-6",
"score": 0.648548424243927,
"text": "How to Start Web Mapping? I want to start working on a web map at work to map some of our statistics by city. I have an idea, but I'm not exactly sure where to start. Are there any definitive resources that provide step by step guidance on how to do this?"
}
] |
[
{
"id": "corpus-78635",
"score": 0.6160330772399902,
"text": "Can I use Openlayers Plugin with a different Datum/Projection? There is probably already a post for my problem, but after long searching, I could not find a solution for my issue. I am working with QGIS und I am using the following projection/datum: MGI / Austria GK East (EPSG:31256) (+proj=tmerc +lat_0=0 +lon_0=16.33333333333333 +k=1 +x_0=0 +y_0=-5000000 +ellps=bessel +towgs84=577.326,90.129,463.919,5.137,1.474,5.297,2.4232 +units=m +no_defs) I just want to add a layer form the openlayers plugin. For example Google satelite. the Google satelite uses the following projection/Datum: WGS 84 / Pseudo Mercator (EPSG:3857) (+proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +wktext +no_defs). The Problem that I am facing with, is that I can not get the Google layer in the same projection with my other layers (EPSG:31256)! They are not overlapping correct! Is there a way to do this?",
"topk_rank": 0
},
{
"id": "corpus-61911",
"score": 0.6159593462944031,
"text": "Defining coordinate reference system with rotation in GeoServer? I am using GeoServer and have a layer in EPSG:900913 (\"Google Mercator\"). I need to \"rotate\" the map around certain point (say, 1500000, 7000000) by certain degree (say, 30 degrees clockwise). How could I define such a coordinate system based on EPSG:900913? GeoServer's does not work for my purposes as I need to tile the map later on. As far as I understand this, my only option is to define an own coordinate system. For GeoServer I'd need to . The configuration seems to be straightforward, but I have a difficulty defining my rotated CRS in . I am wondering how to apply a rotation around certain point onto a CRS like Google Mercator: PROJCS[\"WGS84 / Google Mercator\", GEOGCS[\"WGS 84\", DATUM[\"World Geodetic System 1984\", SPHEROID[\"WGS 84\", 6378137.0, 298.257223563, AUTHORITY[\"EPSG\",\"7030\"]], AUTHORITY[\"EPSG\",\"6326\"]], PRIMEM[\"Greenwich\", 0.0, AUTHORITY[\"EPSG\",\"8901\"]], UNIT[\"degree\", 0.017453292519943295], AXIS[\"Longitude\", EAST], AXIS[\"Latitude\", NORTH], AUTHORITY[\"EPSG\",\"4326\"]], PROJECTION[\"Mercator_1SP\"], PARAMETER[\"semi_minor\", 6378137.0], PARAMETER[\"latitude_of_origin\", 0.0], PARAMETER[\"central_meridian\", 0.0], PARAMETER[\"scale_factor\", 1.0], PARAMETER[\"false_easting\", 0.0], PARAMETER[\"false_northing\", 0.0], UNIT[\"m\", 1.0], AXIS[\"x\", EAST], AXIS[\"y\", NORTH], AUTHORITY[\"EPSG\",\"900913\"]] My questions, specifically: How to write a WKT which transform an existing CRS? My guess would be that I need a new PROJCS wrapping an existing one and adding a PROJECTION clause. How would I found out the projection id (like Mercator_1SP above) and the required parameters (the PARAMETER clauses)? Can I \"reference\" EPSG:900913 in CRS WKT instead of copy-pasting the whole PROJCS clause?",
"topk_rank": 1
},
{
"id": "corpus-11899",
"score": 0.6156058311462402,
"text": "What are useful websites for scouting GIS related jobs? What are useful websites for scouting GIS related jobs?",
"topk_rank": 2
},
{
"id": "corpus-169906",
"score": 0.6156058311462402,
"text": "What are useful websites for scouting GIS related jobs? What are useful websites for scouting GIS related jobs?",
"topk_rank": 3
},
{
"id": "corpus-118172",
"score": 0.6155855655670166,
"text": "How can I learn the basics of SharePoint 2013 in a few days? I have very little knowledge of SharePoint, and want to learn about SharePoint 2013. Most of the resources that I've found are either about what's new in 2013, or they have a bunch of links to SharePoint 2010. I also don't have the time to read a thousand page book. A hundred or so would be fine. Can anyone recommend a resource (site / book / video / tutorial) for a beginner to learn the basics of SharePoint 2013 in a few days? I've downloaded a Kindle book, \"Getting Started With SharePoint 2013.\" It's perfect for absolute beginners, who just want to know how to upload & download documents, but I'm looking for more than that.",
"topk_rank": 4
},
{
"id": "corpus-164273",
"score": 0.6155855655670166,
"text": "How can I learn the basics of SharePoint 2013 in a few days? I have very little knowledge of SharePoint, and want to learn about SharePoint 2013. Most of the resources that I've found are either about what's new in 2013, or they have a bunch of links to SharePoint 2010. I also don't have the time to read a thousand page book. A hundred or so would be fine. Can anyone recommend a resource (site / book / video / tutorial) for a beginner to learn the basics of SharePoint 2013 in a few days? I've downloaded a Kindle book, \"Getting Started With SharePoint 2013.\" It's perfect for absolute beginners, who just want to know how to upload & download documents, but I'm looking for more than that.",
"topk_rank": 5
},
{
"id": "corpus-227041",
"score": 0.615424394607544,
"text": "Converting MapInfo TAB and/or MIF to SHP using free tools or ArcGIS for Desktop core product? How do I convert MapInfo TAB and/or MIF files to SHP using only free tools or ArcGIS itself? I've not had much luck trying to do this in the past and I don't want to have to shell out for the ArcGIS Data Interoperability extension which seems to be the 'official' way to go about this.",
"topk_rank": 6
},
{
"id": "corpus-122808",
"score": 0.6152889728546143,
"text": "Can Geoserver return the raster value of a lat/lon point I have a GeoTIFF published by Geoserver 2.2 and being used by Leaflet via WMS. How can I get Geoserver to return the raster value at a lat/long point? I'm guessing it has to do with using WCS, but no examples seem to exist for doing this! This will be similar to querying PostGIS with ST_Value(raster, ST_SetSRID(ST_Point(lon,lat),4326)). I dont really want to import the same raster into PostGIS just to find the raster value at a point. Or is this the recommended solution?",
"topk_rank": 7
},
{
"id": "corpus-160354",
"score": 0.6144874691963196,
"text": "Misplaced Google Maps with OpenLayers plugin in QGIS? I was using Google Maps with OpenLayers plugin in QGIS with my vector layer with projection EPSG:4326 with google map projection EPSG:3857 and \"on the fly\" transformation enabled. It was matching all the vector layer with Google Maps but now it's not working. How can I reproject Google Maps so that it matches with my vector layer as before?",
"topk_rank": 8
},
{
"id": "corpus-156288",
"score": 0.6143741011619568,
"text": "What can I use to detect pinpoint location (inches) outdoors? I am getting back in to EE after some time so please excuse my ignorance. I am looking for a way to detect pinpoint location outdoors to navigate a robot for a project with my sons. Is there a low cost way to triangulate or use GPS? I am looking for down to the inch precision. Also I do not care if I need to place some transmitters in different locations to give the device a reference. This is for a robotic lawnmower. I have a 2 acre yard and my house is near the middle with several trees as obstacles. Two of my three boys (14yrs, 11yrs, 5rys old) brought up the idea so the real goal of this project is to spend time with them and pique their interest in EE & CE. With that said cost is a factor but I do not care if we work on it for the next 2 years and spend a little along and along. Here are my current plans Include a Windows PC on board so I can code against the sensors. Microsoft Connect on board to help with obstacle detection (reason for Windows PC) Include a USB GPS for general location Include camera just for the fun of it In 2 years if I have some money in it that is okay but I do not want to start with a crazy expensive GPS. Thanks to everyone who has help me!!!!",
"topk_rank": 9
},
{
"id": "corpus-23985",
"score": 0.6141574382781982,
"text": "A beginner's guide to SQL database design Do you know a good source to learn how to design SQL solutions? Beyond the basic language syntax, I'm looking for something to help me understand: What tables to build and how to link them How to design for different scales (small client APP to a huge distributed website) How to write effective / efficient / elegant SQL queries",
"topk_rank": 10
},
{
"id": "corpus-29280",
"score": 0.6141574382781982,
"text": "A beginner's guide to SQL database design Do you know a good source to learn how to design SQL solutions? Beyond the basic language syntax, I'm looking for something to help me understand: What tables to build and how to link them How to design for different scales (small client APP to a huge distributed website) How to write effective / efficient / elegant SQL queries",
"topk_rank": 11
},
{
"id": "corpus-139654",
"score": 0.614113450050354,
"text": "What's in \"standard system utilities\" in the server install? My googlefu has failed me. What packages are part of the \"standard system utilities\" that is preselected during install? TIA! (-:",
"topk_rank": 12
},
{
"id": "corpus-90045",
"score": 0.614089846611023,
"text": "What are the requirements for MAAS and OpenStack? What is the minimum number of computers needed to install MAAS and OpenStack? Also, what are the hardware requirements for all nodes with OpenStack?",
"topk_rank": 13
},
{
"id": "corpus-196377",
"score": 0.6139374375343323,
"text": "Does Ubuntu Touch have WebGL support? I would like to create a WebGL game for Ubuntu Touch, building on the HTML5 app template. Is this a possible option?",
"topk_rank": 14
},
{
"id": "corpus-124879",
"score": 0.6139339208602905,
"text": "What's in \"standard system utilities\" in the server install? My googlefu has failed me. What packages are part of the \"standard system utilities\" that is preselected during install? TIA! (-:",
"topk_rank": 15
},
{
"id": "corpus-99659",
"score": 0.6138371229171753,
"text": "Mixed geometry types styling with GeoServer 2.2 I have a PostGIS table with geometry column. In this column I have geometries of several types (polygons, points, lines). I publish this this table on GeoServer. Now I want to create OpenLayers layer with this data. But how to make a style for this data? I've already read and don't want to create gtype in PostGIS if this possible. I try to create new style within GeoServer but I think that do something wrong: <?xml version=\"1.0\" encoding=\"ISO-8859-1\"?> <StyledLayerDescriptor version=\"1.0.0\" xsi:schemaLocation=\"http://www.opengis.net/sld StyledLayerDescriptor.xsd\" xmlns=\"http://www.opengis.net/sld\" xmlns:ogc=\"http://www.opengis.net/ogc\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:xsi=\"http://www.w3.org/2001/XMLSchema-instance\"> <!-- a Named Layer is the basic building block of an SLD document --> <Rule> <ogc:PropertyIsEqualTo> <ogc:Function name=\"geometryType\"> <ogc:PropertyName>the_geom</ogc:PropertyName> </ogc:Function> <ogc:Literal>Polygon</ogc:Literal> </ogc:PropertyIsEqualTo> <PolygonSymbolizer> <Stroke> <CssParameter name=\"stroke\">#0000ff</CssParameter> <CssParameter name=\"stroke-width\">1</CssParameter> </Stroke> </PolygonSymbolizer> </Rule> </StyledLayerDescriptor> And get a error: line 10: cvc-complex-type.2.4.a: Invalid content was found starting with element 'Rule'. One of '{\"http://www.opengis.net/sld\":Name, \"http://www.opengis.net/sld\":Title, \"http://www.opengis.net/sld\":Abstract, \"http://www.opengis.net/sld\":NamedLayer, \"http://www.opengis.net/sld\":UserLayer}' is expected. line 11: cvc-complex-type.2.4.a: Invalid content was found starting with element 'ogc:PropertyIsEqualTo'. One of '{\"http://www.opengis.net/sld\":Name, \"http://www.opengis.net/sld\":Title, \"http://www.opengis.net/sld\":Abstract, \"http://www.opengis.net/sld\":LegendGraphic, \"http://www.opengis.net/ogc\":Filter, \"http://www.opengis.net/sld\":ElseFilter, \"http://www.opengis.net/sld\":MinScaleDenominator, \"http://www.opengis.net/sld\":MaxScaleDenominator, \"http://www.opengis.net/sld\":Symbolizer}' is expected. What am I doing wrong? UPDATE I try to show a points and make this rule for this geom type: <Rule> <ogc:Filter> <ogc:PropertyIsEqualTo> <ogc:Function name=\"geometryType\"> <ogc:PropertyName>the_geom</ogc:PropertyName> </ogc:Function> <ogc:Literal>Point</ogc:Literal> </ogc:PropertyIsEqualTo> </ogc:Filter> <PolygonSymbolizer> <Stroke> <CssParameter name=\"stroke\">#0000ff</CssParameter> <CssParameter name=\"stroke-width\">2</CssParameter> </Stroke> </PolygonSymbolizer> </Rule> But not see then on my map. What's wrong?",
"topk_rank": 16
},
{
"id": "corpus-209357",
"score": 0.6136809587478638,
"text": "Creating \"average\" polygon? I found in the old ESRI forums with the same type of problem I am facing today. The solution to the problem is explained in little detail and since I am not an expert, I was wondering if somebody could describe it to me in little more detail. We are doing a mental map analysis. We have let's say 50 polygons of a certain city square which are all different in shape, size and location. Our goal is to create a single polygon which would represent the answers from our whole input data. So basically, we need a method for generating a single polygon out of these 50 which would be an \"average\" polygon in size, shape and location. Can somebody explain the process?",
"topk_rank": 17
},
{
"id": "corpus-66423",
"score": 0.6136506795883179,
"text": "How can Google be so fast? What are the technologies and programming decisions that make Google able to serve a query so fast? Every time I search something (one of the several times per day) it always amazes me how they serve the results in near or less than 1 second time. What sort of configuration and algorithms could they have in place that accomplishes this? Side note: It is kind of overwhelming thinking that even if I was to put a desktop application and use it on my machine probably would not be half as fast as Google. Keep on learning I say. Here are some of the great answers and pointers provided: Hardware - cluster farms and massive number of cheap computers Caching and Load Balancing",
"topk_rank": 18
},
{
"id": "corpus-164629",
"score": 0.6135431528091431,
"text": "OL3 How to select features of a WMS layer I want to show on the map the features of the saved layers from a GeoServer server, but I don't know how to do this... Like this, but with the dragbox interaction... I have this code: var select = new ol.interaction.Select(); mapa.addInteraction(select); var selectedFeatures = select.getFeatures(); var dragBox = new ol.interaction.DragBox({ condition: ol.events.condition.always, style: new ol.style.Style({ stroke: new ol.style.Stroke({ color: [0, 0, 255, 1] }) }) }); mapa.addInteraction(dragBox); dragBox.on('boxend', function(e) { if (capaActiva != null) { var info = []; var extent = dragBox.getGeometry().getExtent(); var source = capaActiva.getSource(); console.log(source); var url = source.getGetFeatureInfoUrl(extent, viewResolution, 'EPSG:3857',{'INFO_FORMAT': 'text/plain'}); console.log(url); } else { Ext.MessageBox.show({ title: 'Error', msg: 'Check a layer', buttonText: { ok: 'Aceptar', } }); } });",
"topk_rank": 19
}
] |
query-7
|
Labelling the x-axis \begin{tikzpicture} \begin{axis}[ symbolic x coords={-2,-1,0,1,2,3,4,5,6}, xtick=data ] \addplot[ybar,fill=blue] coordinates { (-2,0) (-1,730974) (0,20059) (1,5492) (2,729) (3,151) (4,30) (5,5) (6,1) }; \end{axis} \end{tikzpicture} \end{figure} When the graph is plotted, I get an axis from 0*10^5 to 8*10^5. How may I change the axis so that I get numbers 0 to 800000 instead?
|
[
{
"id": "corpus-7",
"score": 0.7357663512229919,
"text": "How to prevent pgfplots from using the 10^n notation for axis ticks When creating a plot like this one \\documentclass{article} \\usepackage{pgfplots} \\begin{document} \\begin{tikzpicture} \\begin{axis} \\addplot coordinates { (0.01,0.01) (0.09,0.09) }; \\end{axis} \\end{tikzpicture} \\end{document} pgfplot uses the following xticklabels: Is there any way to make pgfplots create labels as 0.01,0.02,... without specifying them all manually using xticklabels and yticklabels?"
}
] |
[
{
"id": "corpus-175449",
"score": 0.6982250213623047,
"text": "How to read the x-axis of this qqplot? I am new to qqplot and trying to figure out the labels on the x-axis of this plot: The left plot is the Verbal SAT score of 105 college students majoring in CS. The right plot is the GPA score. I understand that the qqplot means "quantile vs quantile", the y-axis is each student's test score, but what is the x-axis and why it is from -2 to 2? Update: For anybody whose looking for a more visual explanation of QQ-plot, check out this video on youtube: After reading and watching dozens of articles and videos, let me explain what is the x-axis and the y-axis in layman's term. The y-axis is the values you recorded. Like the verbal SAT score in my example. Because there are 105 of verbal SAT sores, there should be 105 values on the x-axis as well. There are three steps involved in order to get these 105 values: Step 1 Sort the 105 values in ascending order and give each one of them an index number starting from 1, 2, 3 .... all the way up to 105. Let's use i to denote each index number. Step 2 Use this formula to find a list of percentages (I will explain what are they later, just keep on reading): $\\left(\\dfrac{i-0.5}{n}\\right)$ So you will get 105 values that looks like this: $\\left(\\dfrac{1-0.5}{105}\\right)$, $\\left(\\dfrac{2-0.5}{105}\\right)$, $\\left(\\dfrac{3-0.5}{105}\\right)$ ... $\\left(\\dfrac{105-0.5}{105}\\right)$ Step 3 Find the z-score of the values we just calculated in step 2. These z-scores are the x-axis of the qq-plot Now, plot the verbal SAT scores as the y-axis and the z-scores as the x-axis, you will get a qq-plot. Ok, now it is the time to explain the meaning of step 2 & 3. Let's start with the definition of cumulative distribution function. Let's say we have a test and the maximum score a participant can get is 100 points. After taking the test, 50% of the participants scored 60 points or less, 40% of the participants scored 50 points or less, 30% of the participants scored 40 points or less, and so on. The cumulative distribution function (aka. CDF) takes one input, a test score, and it outputs a percentage which tells you the percentage of people who scored less than or equal to that test score. With the example case above, If I put 60 points into the CDF function, I should get 50% back. If I put 50 points I get 40%. The inverse of the cumulative distribution function takes one input, a percentage and it outputs the corresponding test score. If I put in 50%, it should give me 60 points. It does nothing fancy other than reverse the function of CDF. Step 2 basically calculated a list of percentages and when we feed that list of percentages into Step 3 and get back a list of z-scores. So Step 2 + Step 3 = calculating the inverse of the cumulative distribution function. The list of percentages calculated in Step 2 looks like this: 0.005, 0.014, ... 0.995. The list of z-scores in Step 3 looks like this: -2.575829304, -2.197286377, ... 2.575829304. It means for participants who scored less than or equal to 0.5% (0.005 x 100) of the total participants, the corresponding z-score is -2.57. So what does the qq-plot tell us in our example: If the verbal SAT score indeed follows a standard normal distribution （mean=0, variance=1）, then the outline of the histogram of the test scores should resemble the bell curve of the standard normal distribution. But it is really hard to judge if a histogram matches the shape of a bell curve, a straight line is probably a better choice for the naked eyes. Let's match the observed values against the z-scores.",
"topk_rank": 0
},
{
"id": "corpus-20674",
"score": 0.694805920124054,
"text": "Pgfplots: tick labels with zerofill except for 0 tick The standard pgfplots notation for tick labels is without final zeros when no needed but the standard one for scientific papers is to have all ticks with the same number of digits after the dot a part for the 0 that is often print simply as 0. I am able to set the same precision for all ticks but I would like to print the 0 simply as 0 and not for example 0.0, how can I do that? The first image is the pgfplots default, the second one is what I am able to do and the third is what I desire to do. Minimal code for the second image \\documentclass{article} \\usepackage{pgfplots} \\begin{document} \\begin{tikzpicture} \\begin{axis} [ymin=0, ymax=2, xmin=0.2, xmax=2, y tick label style={ /pgf/number format/fixed, /pgf/number format/fixed zerofill, /pgf/number format/precision=1}, ] \\addplot[only marks] {x}; \\end{axis} \\end{tikzpicture} \\end{document}",
"topk_rank": 1
},
{
"id": "corpus-145236",
"score": 0.6941174864768982,
"text": "Labelling Angle Issues I am trying to label the angles of a triangle. I can't figure out why my code is kicking the name to the outside. My apologies for any formatting issues; this is my first post here. \\documentclass{article} \\usepackage{tkz-euclide} \\usetkzobj{all} \\begin{document} \\begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt}] \\path (80:5) node [dot,label=above left:A]{} coordinate (A) (20:9) node[dot, label=right:B]{} coordinate (B) (0:0) node[dot,label=below left:C]{} coordinate(C); \\path coordinate (Q)at((C)!(A)!(B)) node at (Q)[]{} ; \\draw[-] (A)--(C); \\draw[-] (C)--(B); \\draw[-] (A)--(B); \\draw[purple!70!black] (A)--(Q); \\tkzMarkRightAngle(A,Q,C); \\draw[|<->|] ((P)!−7mm!90:(B))--node[fill=white,sloped] {x} ((B)!−7mm!−90:(P)); \\draw[|<->|] ((Q)!−4.9mm!90:(A))--node[fill=white] {10} ((A)!−3mm!−90:(Q)); \\tkzMarkAngle[size=0.75cm,color=black,label=α]; \\tkzMarkAngle[size=1cm,color=black,label=β]; \\tkzMarkRightAngle(B,A,C); \\end{tikzpicture} \\end{document} I am trying to fit $\\beta$ inside the triangle instead of its current location.",
"topk_rank": 2
},
{
"id": "corpus-193645",
"score": 0.6937701106071472,
"text": "Are there any guidelines for labeling axes in plots/graphs? What are best practices (if any exist) for labeling axes in plots/graphs for academic documents, such as theses, publications, or presentations? Possible options are to use the variable, spell out the variable, or a combination of both. For example, to label a time axis, we could write: t (s) or Time (s) or Time, t (s). Are there any objective reasons to prefer one style over another, assuming the publisher does not specify which to use? Or sources suggesting a particular format? I have browsed many journal requirements, but haven't found any that specifically address this.",
"topk_rank": 3
},
{
"id": "corpus-129411",
"score": 0.6933799386024475,
"text": "Adding Labels and arrows in Tikz graph I'm trying to do something similar to this epsilon-delta graph I've found. Specifically, I'd like to copy the placements and arrows, like the L+e and L-e coming out of L. Same thing with the x0 + d and x0-d coming out of the x0. Can someone help me with this? This is what I have so far. \\begin{tikzpicture} \\begin{axis}[ , legend pos=north east, axis x line=middle, axis y line=middle, x tick label style={/pgf/number format/fixed, /pgf/number format/fixed zerofill, /pgf/number format/precision=1}, y tick label style={/pgf/number format/fixed, /pgf/number format/fixed zerofill, /pgf/number format/precision=1}, grid = major, width=8cm, height=6cm, grid style={dashed, gray!30}, xmin=0 , % start the diagram at this x-coordinate xmax= 4, % end the diagram at this x-coordinate ymin= 0, % start the diagram at this y-coordinate ymax= 11, % end the diagram at this y-coordinate %axis background/.style={fill=white}, xlabel=$x$, ylabel=$f(x)$, tick align=outside, enlargelimits=false] % plot the stirling-formulae %\\addplot[domain=0:200, red, ultra thick,samples=100] {K/(1 + ((K-X0)/X0) * exp(r*x)) }; \\addplot[domain=0:200, red, ultra thick,samples=100] {3*x-1 }; \\addplot[line width=1pt, dashed, mark=none, black] coordinates {(0,8) (3,8)}; \\addplot[line width=1pt, dashed, mark=none, black] coordinates {(3,0) (3,8)}; \\addplot[line width=1pt, dashed, mark=none, blue] coordinates {(3.33,0) (3.333,9)}; \\addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (2.666,0)}; \\addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (0,7)}; \\addplot[line width=1pt, solid, mark=none, blue] coordinates {(3.333,9) (0,9)}; \\addplot[holdot] coordinates{(60,8.66)}; % 1\\addlegendentry{$f(x) = 10 \\sin (x)$} \\end{axis} \\end{tikzpicture} which gives",
"topk_rank": 4
},
{
"id": "corpus-99488",
"score": 0.693278968334198,
"text": "How to insert degree Celsius symbol in node text in pgfplots/tikz? Here is my code. \\resizebox{\\textwidth}{!}{ \\begin{tikzpicture} \\selectcolormodel{gray} \\begin{axis}[xlabel=Voltage,ylabel=Current,title=IV Characteristics of PV Module] \\node[label={Set 1}] at (axis cs:2,60){}; \\node[label=Set 2] at (axis cs:2,150){}; \\node[label=Set 3] at (axis cs:2,240){}; \\node[label=Set 4] at (axis cs:2,305){}; \\addplot table [x=v1,y=i1, col sep=comma,mark=none] {SolarCellIV.csv}; \\addplot table [x=v2,y=i2, col sep=comma,mark=none] {SolarCellIV.csv}; \\addplot table [x=v3,y=i3, col sep=comma,mark=none] {SolarCellIV.csv}; \\addplot table [x=v4,y=i4, col sep=comma,mark=none] {SolarCellIV.csv}; \\node at (axis cs:12,360) {Set 1: 44.9 degreeCelsius}; \\end{axis} \\end{tikzpicture}} I want the symbol for Degree Celsius in place of the text. How do I do so?",
"topk_rank": 5
},
{
"id": "corpus-179715",
"score": 0.6928008198738098,
"text": "Displaying two graphs with the x-axes aligned and with titles centered I have two plots of rational functions displayed separately on the Cartesian plane. Two modifications are needed; I think one modification will give both modifications. I want the x-axes aligned. I think the reason that they are not aligned is that the title under one plot occupies four lines and the title under the other plot occupies two lines. How do I keep the displays of the titles as I have it and get the x-axes aligned? \\documentclass[10pt]{amsart} \\usepackage{tikz} \\usetikzlibrary{calc,intersections,} \\usepackage{pgfplots} \\pgfplotsset{compat=1.11} \\usepackage{mathtools,array} \\begin{document} \\begin{tikzpicture} \\begin{axis}[width=3in, height=3in, axis equal image, axis lines=middle, xmin=-10,xmax=10, ymin=-10,ymax=10, restrict y to domain=-10:10, xlabel=$x$,ylabel=$y$, axis line style={latex-latex}, enlargelimits={abs=0.25cm}, xtick={\\empty},ytick={\\empty}, xlabel style={at={(ticklabel* cs:1)},anchor=north west}, ylabel style={at={(ticklabel* cs:1)},anchor=south west} ] \\addplot[blue, domain=-10:-0.1] {1/x}; \\addplot[blue, domain=0.1:10] {1/x}; \\end{axis} %Title for the plot of the first rational function. \\node[font=\\bfseries, anchor=north, inner sep=0, align=center] at ($(current bounding box.south) +(0,-0.3)$) {\\mbox{An illustration of the continuity of} \\\\ \\mbox{a rational function defined on} \\\\ \\mbox{the complement of a singleton} \\\\ \\mbox{set in \\boldmath$\\mathbb{R}$\\unboldmath}}; \\end{tikzpicture} % \\qquad \\quad % \\begin{tikzpicture} \\begin{axis}[width=3in, height=3in, axis equal image, clip=false, axis lines=middle, xmin=-5,xmax=5, ymin=-5,ymax=5, restrict y to domain=-5:5, xtick={\\empty},ytick={\\empty}, xlabel=$x$,ylabel=$y$, enlargelimits={abs=0.25cm}, axis line style={latex-latex}, xlabel style={at={(ticklabel* cs:1)},anchor=north west}, ylabel style={at={(ticklabel* cs:1)},anchor=south west} ] \\addplot[blue, domain=-5:5] {x^3/(x^2 + 1)}; \\end{axis} %Title for the plot of the second rational function. \\node[font=\\bfseries, anchor=north, inner sep=0, align=center] at ($(current bounding box.south) +(0,-0.3)$) {\\mbox{An illustration of the continuity of} \\\\ \\mbox{a rational function defined on \\boldmath$\\mathbb{R}$\\unboldmath}}; \\end{tikzpicture} \\end{document}",
"topk_rank": 6
},
{
"id": "corpus-116021",
"score": 0.6869876980781555,
"text": "Locating tick marks at integral multiples of pi/2 Originally in my code, I had xtick={-2*pi, -(3/2)*pi, -(1/2)*pi, (1/2)*pi,(3/2)*pi, 2*pi}, to put tick marks at integral multiples of $pi/2$. It would not compile. I had to replace it with cumbersome decimal approximations. Why won't the code compile with expressions like pi? To specify width of the x-axis in the axis environment, I used xmin=-2*pi, and xmax=2*pi,, and to specify the domain of the cosine function, I used domain=-2*pi:2*pi. \\documentclass{amsart} \\usepackage{mathtools} \\usepackage{tikz} \\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings,backgrounds,patterns} \\usepackage{pgfplots} \\pgfplotsset{compat=1.11} \\begin{document} \\begin{tikzpicture} \\begin{axis}[width=4.5in,clip=false, xmin=-2*pi,xmax=2*pi, ymin=-1.25,ymax=1.25, axis lines=middle, %xtick={-2*pi, -(3/2)*pi, -pi, -(1/2)*pi, (1/2)*pi, pi, (3/2)*pi, 2*pi}, xtick={-6.28318, -4.7123889, -3.14159, -1.5708, 1.5708, 3.14159, 4.7123889, 6.28318}, xticklabel style={font=\\footnotesize,fill=white}, xticklabels={$-2\\pi$,,$-\\pi$,,,$\\pi$,,$2\\pi$}, ytick={-1}, yticklabel style={font=\\footnotesize,fill=white}, yticklabels={$-1$}, extra y ticks={1}, extra y tick labels={$1$}, extra y tick style={yticklabel style={font=\\footnotesize,fill=white,anchor=south east}}, ticklabel style={font=\\footnotesize,fill=white}, axis line style={latex-latex}, axis line style={shorten >=-7.5pt, shorten <=-7.5pt}, xlabel style={at={(ticklabel* cs:1)},anchor=north west}, ylabel style={at={(ticklabel* cs:1)},anchor=south west} ] \\addplot [samples=501,mark=none, thick, red,domain=-2*pi:2*pi] {cos(deg(x))}; \\end{axis} \\end{tikzpicture} \\end{document}",
"topk_rank": 7
},
{
"id": "corpus-28721",
"score": 0.6867856979370117,
"text": "How can I write the name show in axis of 3D array using TiKz I am drawing a 3D array like the attached picture, which actually describes a multidimensional array. However, I can not put the name of the axis and take the value from the matrix like the attached picture. This is the code that I have: \\documentclass[margin=5mm]{standalone} \\usepackage{tikz} \\usetikzlibrary{matrix,calc} \\begin{document} \\begin{tikzpicture}[every node/.style={anchor=north east,fill=white,minimum width=1.4cm,minimum height=7mm}] \\matrix (mA) [draw,matrix of math nodes] { (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ }; \\matrix (mB) [draw,matrix of math nodes] at ($(mA.south west)+(1.5,0.7)$) { (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ }; \\matrix (mC) [draw,matrix of math nodes] at ($(mB.south west)+(1.5,0.7)$) { (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ (1,1,3) & (1,1,3) & (1,1,3) & (1,1,3) \\\\ }; \\draw[dashed](mA.north east)--(mC.north east); \\draw[dashed](mA.north west)--(mC.north west); \\draw[dashed](mA.south east)--(mC.south east); \\end{tikzpicture} \\end{document}",
"topk_rank": 8
},
{
"id": "corpus-214532",
"score": 0.6863741278648376,
"text": "Pgfplots with custom axis markers Assume you have a trigonometric function, say for simplicity sin(x). When plotting the function, how can one place the x-axis markers on 0, pi/2, pi, 3pi/2, 2pi, instead of the default numeric values that pgfplots uses?",
"topk_rank": 9
},
{
"id": "corpus-249844",
"score": 0.6862233281135559,
"text": "Pgfplots with custom axis markers Assume you have a trigonometric function, say for simplicity sin(x). When plotting the function, how can one place the x-axis markers on 0, pi/2, pi, 3pi/2, 2pi, instead of the default numeric values that pgfplots uses?",
"topk_rank": 10
},
{
"id": "corpus-97538",
"score": 0.6862059235572815,
"text": "Label PGFPlot table points with coordinates I would like to label each plotted point with its x and y coordinate. I am able to label with a single one using the MWE below. I must avoid manually labeling each point since my actual table size is larger and will be changed out regularly. \\documentclass{standalone} \\usepackage{tikz} \\usepackage{pgfplots} \\pgfplotsset{compat=1.16} \\begin{document} \\pgfplotstableread[col sep=&,row sep=\\\\]{ x & y \\\\ 0.940 & 0.992 \\\\ 0.992 & -0.940 \\\\ -0.992 & 0.940 \\\\ -0.940 & -0.992 \\\\ }\\mytable \\begin{tikzpicture} \\begin{axis} \\addplot+[only marks, nodes near coords]table[x=x,y=y,meta=x]{\\mytable}; \\end{axis} \\end{tikzpicture} \\end{document} which creates the following figure The desired output is for each point to be labelled with both coordinates. For specific example the top left point would be labelled with "(0.94, 0.992)".",
"topk_rank": 11
},
{
"id": "corpus-107484",
"score": 0.6860113143920898,
"text": "Flip x and y axes in Excel graph This should not be very difficult, but I cannot figure out how to do it. I have a table similar to this %low %high 0 0 12 1 13 26 ... 19 90 94 20 95 100 When I graph it, excel defaults to having the first column on the x axis and plotting the second and third column as y values. I want the first column to be on the y axis instead. I assume there is an easy way to do this, but I cannot figure it out. Most of the things I have found from searching have suggested the \"Switch Row/Column\" button, but that does something else. Thanks for the help.",
"topk_rank": 12
},
{
"id": "corpus-84342",
"score": 0.6855476498603821,
"text": "x tick label style issue, anchor=east and text width command not working together I have a figure where one of my labels is very long. I would like this label to be split to two rows. Currently my code is like this \\documentclass{article} \\usepackage{pgfplots} \\pgfplotsset{compat=1.10} \\begin{document} \\begin{tikzpicture} \\pgfplotsset{width=10 cm} \\begin{axis} [ symbolic x coords={0, Label 1, Label 2, This is a very very very long label, Label 4, Label 5, Label 6, Label 7,Label 8}, xtick={Label 1, Label 2, This is a very very very long label, Label 4, Label 5, Label 6, Label 7}, x tick label style={rotate=45, anchor=east, align=center}, axis lines*=left, ymajorgrids = true, legend style={at={(0.5,-0.30)},anchor=north}, ymin=0, ymax=175, bar width=5mm, ybar=-0.5cm, enlarge x limits={abs=0.6cm}, nodes near coords, every node near coord/.append style={color=black}, ] \\addplot [red,fill=red] coordinates{ (Label 1,100) (Label 2,90) (This is a very very very long label,80) (Label 4,140) } ; \\addplot [blue,fill=blue] coordinates{ (Label 5,80) (Label 6,25) (Label 7,66) } ; \\addlegendentry{Legend 1} \\addlegendentry{Legend 2} \\end{axis} \\end{tikzpicture} \\end{document} Which produces this: Now if I try to change the x tick label style to be x tick label style={rotate=45, anchor=east, align=center,text width=3.5cm} This happens: So the anchor=east command doesn't seem to go along with the text width command. Other anchor commands like anchor=west, anchor=base seem to be working perfectly normal. Does anyone know why the anchor east command doesn't seem to work with the text width command and is there a way to go around this?",
"topk_rank": 13
},
{
"id": "corpus-139837",
"score": 0.6852933168411255,
"text": "Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \\pi in PGFPlots?",
"topk_rank": 14
},
{
"id": "corpus-52551",
"score": 0.6851189136505127,
"text": "Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \\pi in PGFPlots?",
"topk_rank": 15
},
{
"id": "corpus-55843",
"score": 0.6851189136505127,
"text": "Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \\pi in PGFPlots?",
"topk_rank": 16
},
{
"id": "corpus-57149",
"score": 0.6851189136505127,
"text": "Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \\pi in PGFPlots?",
"topk_rank": 17
},
{
"id": "corpus-212044",
"score": 0.6851189136505127,
"text": "Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \\pi in PGFPlots?",
"topk_rank": 18
},
{
"id": "corpus-231750",
"score": 0.6851189136505127,
"text": "Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \\pi in PGFPlots?",
"topk_rank": 19
}
] |
query-8
|
Asymptotically unbiased estimator vs consistent estimator I'm wondering if there is a difference between an asymptotically unbiased estimator and a consistent estimator. For asymptotically unbiased estimators, the expected value of the estimator converges to the parameter, while for a consistent estimator, the estimator converges in probability to the parameter. These sound nearly identical, but I believe that being consistent might actually be stronger than being asymptotically unbiased, since being consistent, i.e. converging in probability, implies that not only will the estimates be roughly centered around the true value, but will also be getting closer to it as n increases. Can someone confirm or refute this claim?
|
[
{
"id": "corpus-8",
"score": 0.8343149423599243,
"text": "What's the difference between asymptotic unbiasedness and consistency? Does each imply the other? If not, does one imply the other? Why/why not? This issue came up in response to a comment on an answer I posted . Although google searching the relevant terms didn't produce anything that seemed particularly useful, I did notice on the math stackexchange. However, I thought that this question was appropriate for this site too. EDIT after reading the comments Relative to the math.stackexchange answer I was after something more in depth, covering some of the issues dealt with in . Also, as I see it the math.stackexchange question shows that consistency doesn't imply asymptotically unbiasedness but doesn't explain much if anything about why. The OP there also takes for granted that asymptotic unbiasedness doesn't imply consistency, and thus the sole answerer so far doesn't address why this is."
}
] |
[
{
"id": "corpus-33102",
"score": 0.7579700946807861,
"text": "Large sample asymptotic/theory - Why to care about? I hope that this question does not get marked \"as too general\" and hope a discussion gets started that benefits all. In statistics, we spend a lot of time learning large sample theories. We are deeply interested in assessing asymptotic properties of our estimators including whether they are asymptotically unbiased, asymptotically efficient, their asymptotic distribution and so on. The word asymptotic is strongly tied with the assumption that $n \\rightarrow \\infty$. In reality, however, we always deal with finite $n$. My questions are: 1) what do we mean by large sample? How can we distinguish between small and large samples? 2) When we say $n \\rightarrow \\infty$, do we literally mean that $n$ should go to $\\infty$? e.x. for binomial distribution, $\\bar{X}$ needs about n = 30 to converge to normal distribution under CLT. Should we have $n \\rightarrow \\infty$ or in this case by $\\infty$ we mean 30 or more?! 3) Suppose we have a finite sample and suppose that We know everything about asymptotic behavior of our estimators. So what? suppose that our estimators are asymptotically unbiased, then do we have an unbiased estimate for our parameter of interest in our finite sample or it means that if we had $n \\rightarrow \\infty$, then we would have an unbiased one? As you can see from the questions above, I'm trying to understand the philosophy behind \"Large Sample Asymptotics\" and to learn why we care? I need to get some intuitions for the theorems I'm learning.",
"topk_rank": 0
},
{
"id": "corpus-82571",
"score": 0.7559425830841064,
"text": "How to show that an estimator is consistent? Is it enough to show that MSE = 0 as $n\\rightarrow\\infty$? I also read in my notes something about plim. How do I find plim and use it to show that the estimator is consistent?",
"topk_rank": 1
},
{
"id": "corpus-182081",
"score": 0.75311279296875,
"text": "Are Maximum Likelihood Estimators asymptotically unbiased? I can follow the proofs in which the asymptotic normal-distribution of a maximum likelihood estimator $\\tilde{\\theta}_n$ is derived. however, does this already imply that the maximum likelihood estimates are asymptotically unbiased, i.e. do we have $$E(\\tilde{\\theta}_n) \\to \\theta \\text{ as } n \\to \\infty?$$ Since I am aware that in general it is not true that convergence in distribution implies convergence in moments, an explanation would be nice. The result is often stated e.g. as \"...it means that the bias of the maximum likelihood estimator is equal to zero up to the order $n^{-1/2}$\") 1.) Are some kind of regularity conditions from the mle theory used to establish this result? 2.) Or is a $\\sqrt{n}$-convergence of an estimator (to a normal distribution) in general already enough to establish convergence of its moments? Note: the Wikipedia article mentions Cox, David R.; Snell, E. Joyce (1968). , as a source where the order of the bias is derived (formula (12) or (20)). However in this paper I can't follow the arguments 100%, since their Taylor approximation of $L'(\\widehat{\\beta})$ is lacking the remainder term. What is the argument used here to discard it completely?",
"topk_rank": 2
},
{
"id": "corpus-209981",
"score": 0.7402792572975159,
"text": "How can I formally show that a root n consistent estimator is weakly consistent? How can I formally show that a root n consistent estimator is weakly consistent? Heuristically its clear that if $\\sqrt{n}(\\theta_n-\\theta_0)$ is bounded in probability then since $\\sqrt{n} \\to \\infty$ as $n \\to \\infty$ clearly $\\theta_n-\\theta_0 \\in o_p(1)$ that is it goes to zero in probability but every time I try to write it formally I get stuck. Could someone help me out?",
"topk_rank": 3
},
{
"id": "corpus-137818",
"score": 0.734531044960022,
"text": "Showing that asymptotic normality implies consistency. In a statistics book i'm reading, it is postulated that asymptotic normality of an estimator implies consistency. That is $$ \\hat{\\theta}_n \\stackrel{as}{\\sim} \\mathcal{N}(\\theta_0, \\frac{1}{n}\\sigma(\\theta_0)) \\Rightarrow P_{\\theta_0}(|\\hat{\\theta}_n - \\theta_0|>\\epsilon ) \\to 0 $$ when $n\\to\\infty$, for all $\\theta_0 \\in \\Theta$ and $\\epsilon>0$. I am trying to prove this, but i can't seem to get a breakthrough. If anyone could shed some light on how this is proven, i would be very grateful.",
"topk_rank": 4
},
{
"id": "corpus-185546",
"score": 0.7334494590759277,
"text": "When is a biased estimator preferable to unbiased one? It's obvious many times why one prefers an unbiased estimator. But, are there any circumstances under which we might actually prefer a biased estimator over an unbiased one?",
"topk_rank": 5
},
{
"id": "corpus-217508",
"score": 0.7332415580749512,
"text": "When is a biased estimator preferable to unbiased one? It's obvious many times why one prefers an unbiased estimator. But, are there any circumstances under which we might actually prefer a biased estimator over an unbiased one?",
"topk_rank": 6
},
{
"id": "corpus-98033",
"score": 0.7238930463790894,
"text": "Checking the consistency and Bias of $\\frac{\\sum X_i +\\sqrt{n}/2}{n+\\sqrt{n}}$ Let $X_1,\\ldots,X_n$ be i.i.d. $B(1,\\theta)$ random variables, $0<\\theta<1$. Then, as an estimator $\\theta$, check if $T(X_1,\\ldots,X_n)= \\dfrac{\\sum_{i=1}^n X_i +\\sqrt{n}/2}{n+\\sqrt{n}}$ is consistent and/or unbiased. $$T=\\frac{\\frac{1}{n}\\sum_{i=1}^n X_i +\\frac{1}{2\\sqrt{n}}}{1+\\frac{1}{\\sqrt{n}}}$$ $$T=\\frac{\\bar{X} +\\frac{1}{2\\sqrt{n}}}{1+\\frac{1}{\\sqrt{n}}}$$ $$E(T)=\\frac{E(\\bar{X} +\\frac{1}{2\\sqrt{n}})}{1+\\frac{1}{\\sqrt{n}}}=\\dfrac{\\theta +\\frac{1}{2\\sqrt{n}}}{1+\\frac{1}{\\sqrt{n}}}.$$ So, $T$ is biased. Consistency: $\\lim\\limits_{n \\to\\infty}E(T)=\\lim\\limits_{n \\to\\infty}\\dfrac{E(\\theta +\\frac{1}{2\\sqrt{n}})}{1+\\frac{1}{\\sqrt{n}}}$ So $T$ is consistent. But The given answer is \"neither unbiased nor consistent\" Where did I go wrong ? Please advise.",
"topk_rank": 7
},
{
"id": "corpus-165707",
"score": 0.7227878570556641,
"text": "Biased and Efficient estimators Is unbiasedness a necessary condition for an estimator to be efficient? For example, if $\\hat {\\theta}= \\frac{\\sum_i^n X_i}{3}$, I assume $\\hat {\\theta}$ can't be efficient in a Cramer-Rao lower bound context because $E[\\hat {\\theta}]= \\frac {\\theta}{3}$.",
"topk_rank": 8
},
{
"id": "corpus-60355",
"score": 0.716412365436554,
"text": "Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \\left[ 0, \\theta \\right] $. Now I need to estimate $ \\theta $ based on $N$ observations and I want the estimator to be unbiased. I thought about simple estimator $ \\hat{\\theta} = \\max \\left( x_i \\right) $. Based on simulation it is not biased, yet I couldn't show it analytically. Could anyone, please, show it is unbiased? BTW, I could easily find another, easy to prove, unbiased estimator, $ \\hat{\\theta} = 2 \\mathrm{mean} \\left( {x}_{i} \\right) $",
"topk_rank": 9
},
{
"id": "corpus-145787",
"score": 0.716412365436554,
"text": "Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \\left[ 0, \\theta \\right] $. Now I need to estimate $ \\theta $ based on $N$ observations and I want the estimator to be unbiased. I thought about simple estimator $ \\hat{\\theta} = \\max \\left( x_i \\right) $. Based on simulation it is not biased, yet I couldn't show it analytically. Could anyone, please, show it is unbiased? BTW, I could easily find another, easy to prove, unbiased estimator, $ \\hat{\\theta} = 2 \\mathrm{mean} \\left( {x}_{i} \\right) $",
"topk_rank": 10
},
{
"id": "corpus-192304",
"score": 0.716412365436554,
"text": "Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \\left[ 0, \\theta \\right] $. Now I need to estimate $ \\theta $ based on $N$ observations and I want the estimator to be unbiased. I thought about simple estimator $ \\hat{\\theta} = \\max \\left( x_i \\right) $. Based on simulation it is not biased, yet I couldn't show it analytically. Could anyone, please, show it is unbiased? BTW, I could easily find another, easy to prove, unbiased estimator, $ \\hat{\\theta} = 2 \\mathrm{mean} \\left( {x}_{i} \\right) $",
"topk_rank": 11
},
{
"id": "corpus-18543",
"score": 0.7159305214881897,
"text": "Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \\left[ 0, \\theta \\right] $. Now I need to estimate $ \\theta $ based on $N$ observations and I want the estimator to be unbiased. I thought about simple estimator $ \\hat{\\theta} = \\max \\left( x_i \\right) $. Based on simulation it is not biased, yet I couldn't show it analytically. Could anyone, please, show it is unbiased? BTW, I could easily find another, easy to prove, unbiased estimator, $ \\hat{\\theta} = 2 \\mathrm{mean} \\left( {x}_{i} \\right) $",
"topk_rank": 12
},
{
"id": "corpus-58158",
"score": 0.7159305214881897,
"text": "Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \\left[ 0, \\theta \\right] $. Now I need to estimate $ \\theta $ based on $N$ observations and I want the estimator to be unbiased. I thought about simple estimator $ \\hat{\\theta} = \\max \\left( x_i \\right) $. Based on simulation it is not biased, yet I couldn't show it analytically. Could anyone, please, show it is unbiased? BTW, I could easily find another, easy to prove, unbiased estimator, $ \\hat{\\theta} = 2 \\mathrm{mean} \\left( {x}_{i} \\right) $",
"topk_rank": 13
},
{
"id": "corpus-180646",
"score": 0.7159305214881897,
"text": "Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \\left[ 0, \\theta \\right] $. Now I need to estimate $ \\theta $ based on $N$ observations and I want the estimator to be unbiased. I thought about simple estimator $ \\hat{\\theta} = \\max \\left( x_i \\right) $. Based on simulation it is not biased, yet I couldn't show it analytically. Could anyone, please, show it is unbiased? BTW, I could easily find another, easy to prove, unbiased estimator, $ \\hat{\\theta} = 2 \\mathrm{mean} \\left( {x}_{i} \\right) $",
"topk_rank": 14
},
{
"id": "corpus-246583",
"score": 0.7159305214881897,
"text": "Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \\left[ 0, \\theta \\right] $. Now I need to estimate $ \\theta $ based on $N$ observations and I want the estimator to be unbiased. I thought about simple estimator $ \\hat{\\theta} = \\max \\left( x_i \\right) $. Based on simulation it is not biased, yet I couldn't show it analytically. Could anyone, please, show it is unbiased? BTW, I could easily find another, easy to prove, unbiased estimator, $ \\hat{\\theta} = 2 \\mathrm{mean} \\left( {x}_{i} \\right) $",
"topk_rank": 15
},
{
"id": "corpus-118891",
"score": 0.7151011228561401,
"text": "Showing that estimator is consistent Let $\\hat{\\theta}_n= -\\frac{n}{\\sum_{i=1}^n \\log(X_i)}$, where $X_i$ are i.i.d. samples from distribution with pdf $\\theta x^{\\theta-1}$ for $x \\in (0,1)$. How to prove that $\\hat{\\theta}_n$ is consistent estimator of $\\theta$.",
"topk_rank": 16
},
{
"id": "corpus-147107",
"score": 0.7045711278915405,
"text": "For the binomial distribution, why does no unbiased estimator exist for $1/p$? Suppose that $X$ ~ $Binomial(n,p)$ for $0 < p < 1$ Why does no unbiased estimator exist for $1/p$? My approach: We try to find the structure of $E_p(U(x))$, where $U(x)$ is any estimator of $1/p$. Now, we will have: $\\sum{U(x)\\binom{n}{x}p^x(1-p)^{n-x}}<\\sum{U(x)\\binom{n}{x}}=M(n)<\\infty$ so that the expectation is bounded above. So this is supposed to mean that if $p < 1/M(n)$, then the expectation cannot attain $p$ but I am not sure why the above argument even makes sense and what being bounded means for the expectation.",
"topk_rank": 17
},
{
"id": "corpus-67605",
"score": 0.7028408646583557,
"text": "Find best unbiased estimator for $\\theta$ when $X_i\\sim U(-\\theta,\\theta)$ I am having an issue finding a best unbiased estimator for $\\theta$. Any help is appreciated. Let $X_1, ..., X_n$ be a random sample from a population with pdf: $f(x\\mid\\theta)=\\frac{1}{2\\theta}$ $-\\theta<x<\\theta,\\, \\theta>0$. I understand that $T(x) := |X_n|$ is the sufficient statistic since by the factorization theorem we have: $$f(x\\mid \\theta) = \\left(\\frac{1}{2\\theta}\\right)^n \\prod_{i=1}^n I[|x_i|<\\theta]$$ I think my main issue is showing that this sufficient statistic is also a complete sufficient statistic. Can somebody please aid me in this?",
"topk_rank": 18
},
{
"id": "corpus-170659",
"score": 0.699989378452301,
"text": "Consistency of an order statistic in exponential distribution I have two questions. 1) If $X_1,X_2,X_3,...,X_n$ constitute a random sample of size $n$ from an exponential distribution, show that $\\bar X$ is a consistent estimator of the parameter $\\lambda$. This is my attempt: Mean of exponential distribution is $\\lambda^{-1}$. Thus by central Limit theorem $E(\\bar X)=\\lambda^{-1}$. Using mean square consistency, if $\\lim_{n\\to \\infty} E[(\\bar X -\\lambda )^2]=0$ then $\\bar X $ is consistent for $\\lambda$. Since $$ E[(\\bar X -\\lambda )^2]= E[(\\bar X)^2]+E(\\lambda)^2-2\\lambda E(\\bar X) , $$ I get $$ \\lim_{n\\to \\infty} E[(\\bar X -\\lambda )^2] = \\lim_{n\\to \\infty} {2 \\over \\lambda ^2 }+\\lambda^2-2 . $$ Clearly this is not equal to $0$. How can I show this is consistent? I tried using Chebechev inequality as well. But even using that I couldn't. 2) Is $X_{(n)}$ a consistent estimator for $\\lambda$? If $P[|X_{(n)}-\\lambda|<\\epsilon]$ approaches $1$ as $n$ goes to infinity for all $\\epsilon > 0$ then $X_n$ is a consistent estimator for $\\lambda$. The probability density function of $X_{(n)}$ is $f_n(x) = n \\lambda e^{-\\lambda x}(1-e^{-\\lambda x})^{n-1}$, for $X>0$. Thus \\begin{align*} P[|X_{(n)}-\\lambda|<\\epsilon] &= P[\\lambda -\\epsilon<X_{n}<\\lambda+\\epsilon] \\\\ &=P[0<X_{n}<\\lambda+\\epsilon] \\\\ &=\\int_0^{\\lambda+\\epsilon} n \\lambda e^{-\\lambda x}(1-e^{-\\lambda x})^{n-1}\\\\ &= (1-e^{- \\lambda X_{(n)} }), \\end{align*} where $X{(n)}$ takes values in $[0 , \\lambda+ \\epsilon]$. Thus, can I say $\\lim_{n\\to \\infty}[1-e^{-\\lambda(\\lambda+\\epsilon)}]^n=1$?",
"topk_rank": 19
}
] |
query-9
|
Proving summation identities How would one go about proving the following identities? $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i}{z_i-z_j} = \frac{n(n-1)}{2}$$ $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i^2}{z_i-z_j} = (n-1)\sum_{i=1}^n z_i$$ $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i^3}{z_i-z_j} = (n-1)\sum_{i=j}^n z_i^2+\sum_{i<j}^n z_i z_j$$ $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i^4}{z_i-z_j} = (n-1)\sum_{i=j}^n z_i^3+\sum_{i<j}^n z_i z_j(z_i+z_j)$$ $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i^5}{z_i-z_j} = (n-1)\sum_{i=j}^n z_i^4+\sum_{i<j}^n z_i z_j(z_i^2+z_i z_j +z_j^2)$$ I see the obvious pattern here. The problem is that the algebra involving the summation is giving me some difficulty.
|
[
{
"id": "corpus-9",
"score": 0.8581988215446472,
"text": "General identity for a double summation theorem I've been reading a research paper, and I'm interested in generalizing a certain theorem but I can't seem to understand how the following results are derived: $$\\sum_{i=1}^{n}\\sum_{j\\neq i}^n \\frac{1}{z_i-z_j}=0$$ $$\\sum_{i=1}^{n}\\sum_{j\\neq i}^n \\frac{z_i}{z_i-z_j}=\\frac{1}{2}n(n-1)$$ $$\\sum_{i=1}^{n}\\sum_{j\\neq i}^n \\frac{z_i^2}{z_i-z_j}=(n-1)\\sum_{i=1}^n z_i$$ $$\\sum_{i=1}^{n}\\sum_{j\\neq i}^n \\frac{z_i^3}{z_i-z_j}=(n-1)\\sum_{i=1}^n z_i^2 + \\sum_{i<j}^n z_i z_j$$"
}
] |
[
{
"id": "corpus-59083",
"score": 0.7868978381156921,
"text": "How to prove $\\sum^n_{i=1} \\frac{1}{i(i+1)} = \\frac{n}{n+1}$? How can I prove that $\\sum^n_{i=1} \\frac{1}{i(i+1)} = \\frac{n}{n+1}$? I noticed that in the sum, the denominator has terms that cancel out, but I'm not sure how to take advantage of that.",
"topk_rank": 0
},
{
"id": "corpus-1819",
"score": 0.7859275937080383,
"text": "Let $n \\in \\mathbb{Z}^+$, prove the identity $ \\sum_{k=1}^{n-1} \\binom {n} {k} \\frac{kn^{n-k}}{k+1}=\\frac{n(n^{n}-1)}{n+1}$ Let $n \\in \\mathbb{Z}^+$, prove the identity $$ \\sum_{k=1}^{n-1} \\binom {n} {k} \\frac{kn^{n-k}}{k+1}=\\frac{n(n^{n}-1)}{n+1}$$ First of all $$ \\sum_{k=1}^{n-1} \\binom {n} {k} \\frac{kn^{n-k}}{k+1}=n^{n}\\Bigg(\\sum_{k=1}^{n-1} \\binom {n} {k} \\frac{kn^{-k}}{k+1} \\Bigg)$$ $$=n^n\\Bigg(\\sum_{k=1}^{n-1} \\binom {n}{k} \\bigg(1-\\frac{1}{k+1}\\bigg)\\bigg(\\frac{1}{n^k}\\bigg)\\Bigg)$$ $$=n^n \\sum_{k=1}^{n-1} \\binom {n} {k} \\frac{1}{n^k}-n^n \\sum_{k=1}^{n-1} \\binom {n}{k} \\bigg( \\frac{1}{k+1}\\bigg)\\bigg(\\frac{1}{n^k} \\bigg)$$ We have for the first sum $$(1+\\frac{1}{x})^n = \\sum_{k=0}^n \\binom{n}{k}\\frac{1}{x^k}.$$ For the second sum $$(1+x)^n = \\sum_{k=0}^n \\binom{n}{k}x^k.$$ Integrating both sides from $0$ to $x$, we see that $$\\frac{(1+x)^{n+1}-1}{n+1} = \\sum_{k=0}^n \\binom{n}{k}\\frac{x^{k+1}}{k+1}.$$ Putting $x=1$ yields $$\\frac{2^{n+1}-1}{n+1}=\\sum_{k=0}^n \\binom{n}{k}\\frac{1}{k+1}$$ Here where I have stopped. I could not get them similar for what I have. Would someone help me out !",
"topk_rank": 1
},
{
"id": "corpus-59319",
"score": 0.7820505499839783,
"text": "Prove the following summation inequality: $\\sum_{i=1}^n(a_i)\\sum_{i=1}^n(\\frac{1}{a_i})\\ge n^2$ $\\sum_{i=1}^n(a_i)\\sum_{i=1}^n(\\frac{1}{a_i})\\ge n^2$ is what I want to prove, where $a_i>0$ and I need to showwhen equality holds. I have tried using Cauchy-Schwarz, where inside the sums I multiply each value by 1 to do it but it doesn't work out. I have also done $\\sum_{i=1}^n(a_i)\\sum_{i=1}^n(\\frac{1}{a_i})=\\sum_{i=1}^n \\sum_{i=1}^n(\\frac{a_i}{a_i})=\\sum_{i=1}^n\\sum_{i=1}^n(1)=\\sum_{i=1}^n(n)=n^2$, but this proves an equality. Am I not allowed to do the first step I did in my work?",
"topk_rank": 2
},
{
"id": "corpus-61162",
"score": 0.7703253626823425,
"text": "Prove that $\\sum_{x=1}^{n} \\frac{1}{x (x+1)(x+2)} = \\frac{1}{4} - \\frac{1}{2 (n+1) (n+2)}$ Prove that $\\displaystyle \\sum_{x=1}^{n} \\frac{1}{x (x+1)(x+2)} = \\frac{1}{4} - \\frac{1}{2 (n+1) (n+2)}$. I tried using the partial fraction decomposition $a_j = \\frac{1}{2j} - \\frac{1}{j+1} + \\frac{1}{2(j+2)}$, but I don't see how that helps.",
"topk_rank": 3
},
{
"id": "corpus-88429",
"score": 0.7703253626823425,
"text": "Prove that $\\sum_{x=1}^{n} \\frac{1}{x (x+1)(x+2)} = \\frac{1}{4} - \\frac{1}{2 (n+1) (n+2)}$ Prove that $\\displaystyle \\sum_{x=1}^{n} \\frac{1}{x (x+1)(x+2)} = \\frac{1}{4} - \\frac{1}{2 (n+1) (n+2)}$. I tried using the partial fraction decomposition $a_j = \\frac{1}{2j} - \\frac{1}{j+1} + \\frac{1}{2(j+2)}$, but I don't see how that helps.",
"topk_rank": 4
},
{
"id": "corpus-226385",
"score": 0.767997682094574,
"text": "Prove $\\sum_{i=1}^n i! \\cdot i = (n+1)! - 1$? Prove the summation: $$\\sum_{i=1}^n i! \\cdot i = (n+1)! - 1$$ using induction. base case: $n=1$: \\begin{align*} \\sum_{i=1}^1 i! \\cdot i &= (1+1)! - 1 \\\\ 1 &= 2 - 1 \\\\ 1 &= 1 \\end{align*} This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated.",
"topk_rank": 5
},
{
"id": "corpus-159357",
"score": 0.7676194310188293,
"text": "Prove $\\sum_{i=1}^n i! \\cdot i = (n+1)! - 1$? Prove the summation: $$\\sum_{i=1}^n i! \\cdot i = (n+1)! - 1$$ using induction. base case: $n=1$: \\begin{align*} \\sum_{i=1}^1 i! \\cdot i &= (1+1)! - 1 \\\\ 1 &= 2 - 1 \\\\ 1 &= 1 \\end{align*} This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated.",
"topk_rank": 6
},
{
"id": "corpus-83780",
"score": 0.7664600014686584,
"text": "Proving the summation formula using induction: $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ I am trying to prove the summation formula using induction: $$\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$$ So far I have... Base case: Let n=1 and test $\\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ $\\frac{1}{1(1+1)} = 1-\\frac{1}{1+1}$ $\\frac{1}{2} = \\frac{1}{2}$ True for n=1 Induction Hypothesis: Assume the statement is true for the n-th case $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ Inductive Step: Prove, using the Inductive Hypothesis as a premise, that $$\\sum_{k=1}^{n+1} \\frac{1}{k(k+1)} = \\sum_{k=1}^{n} \\frac{1}{k(k+1)} + \\frac{1}{(n+1)(n+2)} = 1-\\frac1{n+1} + \\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)}{(n+1)(n+2)}+\\frac{-2-n}{(n+1)(n+2)}+\\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-2-n+1)}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-n-1}{(n+1)(n+2)} = \\frac{n^2+2n+1}{(n+1)(n+2)} = \\frac{(n+1)(n+1)}{(n+1)(n+2)} = \\frac{n+1}{n+2}$$ To prove $$ 1-\\frac{1}{n+2} = \\frac{n+1}{n+2} $$ Multiply both sides by $n+2$ to get an equivalent expression. $$ (1-\\frac{1}{n+2}) * (n+2) = (\\frac{n+1}{n+2}) * (n+2) $$ $$ n+1=n+2−1 $$ Does this all make sense? How can this be improved upon?",
"topk_rank": 7
},
{
"id": "corpus-137257",
"score": 0.7664600014686584,
"text": "Proving the summation formula using induction: $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ I am trying to prove the summation formula using induction: $$\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$$ So far I have... Base case: Let n=1 and test $\\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ $\\frac{1}{1(1+1)} = 1-\\frac{1}{1+1}$ $\\frac{1}{2} = \\frac{1}{2}$ True for n=1 Induction Hypothesis: Assume the statement is true for the n-th case $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ Inductive Step: Prove, using the Inductive Hypothesis as a premise, that $$\\sum_{k=1}^{n+1} \\frac{1}{k(k+1)} = \\sum_{k=1}^{n} \\frac{1}{k(k+1)} + \\frac{1}{(n+1)(n+2)} = 1-\\frac1{n+1} + \\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)}{(n+1)(n+2)}+\\frac{-2-n}{(n+1)(n+2)}+\\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-2-n+1)}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-n-1}{(n+1)(n+2)} = \\frac{n^2+2n+1}{(n+1)(n+2)} = \\frac{(n+1)(n+1)}{(n+1)(n+2)} = \\frac{n+1}{n+2}$$ To prove $$ 1-\\frac{1}{n+2} = \\frac{n+1}{n+2} $$ Multiply both sides by $n+2$ to get an equivalent expression. $$ (1-\\frac{1}{n+2}) * (n+2) = (\\frac{n+1}{n+2}) * (n+2) $$ $$ n+1=n+2−1 $$ Does this all make sense? How can this be improved upon?",
"topk_rank": 8
},
{
"id": "corpus-140273",
"score": 0.7664600014686584,
"text": "Proving the summation formula using induction: $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ I am trying to prove the summation formula using induction: $$\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$$ So far I have... Base case: Let n=1 and test $\\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ $\\frac{1}{1(1+1)} = 1-\\frac{1}{1+1}$ $\\frac{1}{2} = \\frac{1}{2}$ True for n=1 Induction Hypothesis: Assume the statement is true for the n-th case $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ Inductive Step: Prove, using the Inductive Hypothesis as a premise, that $$\\sum_{k=1}^{n+1} \\frac{1}{k(k+1)} = \\sum_{k=1}^{n} \\frac{1}{k(k+1)} + \\frac{1}{(n+1)(n+2)} = 1-\\frac1{n+1} + \\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)}{(n+1)(n+2)}+\\frac{-2-n}{(n+1)(n+2)}+\\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-2-n+1)}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-n-1}{(n+1)(n+2)} = \\frac{n^2+2n+1}{(n+1)(n+2)} = \\frac{(n+1)(n+1)}{(n+1)(n+2)} = \\frac{n+1}{n+2}$$ To prove $$ 1-\\frac{1}{n+2} = \\frac{n+1}{n+2} $$ Multiply both sides by $n+2$ to get an equivalent expression. $$ (1-\\frac{1}{n+2}) * (n+2) = (\\frac{n+1}{n+2}) * (n+2) $$ $$ n+1=n+2−1 $$ Does this all make sense? How can this be improved upon?",
"topk_rank": 9
},
{
"id": "corpus-242669",
"score": 0.7664600014686584,
"text": "Proving the summation formula using induction: $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ I am trying to prove the summation formula using induction: $$\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$$ So far I have... Base case: Let n=1 and test $\\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ $\\frac{1}{1(1+1)} = 1-\\frac{1}{1+1}$ $\\frac{1}{2} = \\frac{1}{2}$ True for n=1 Induction Hypothesis: Assume the statement is true for the n-th case $\\sum_{k=1}^n \\frac{1}{k(k+1)} = 1-\\frac{1}{n+1}$ Inductive Step: Prove, using the Inductive Hypothesis as a premise, that $$\\sum_{k=1}^{n+1} \\frac{1}{k(k+1)} = \\sum_{k=1}^{n} \\frac{1}{k(k+1)} + \\frac{1}{(n+1)(n+2)} = 1-\\frac1{n+1} + \\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)}{(n+1)(n+2)}+\\frac{-2-n}{(n+1)(n+2)}+\\frac{1}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-2-n+1)}{(n+1)(n+2)} = \\frac{(n+1)(n+2)-n-1}{(n+1)(n+2)} = \\frac{n^2+2n+1}{(n+1)(n+2)} = \\frac{(n+1)(n+1)}{(n+1)(n+2)} = \\frac{n+1}{n+2}$$ To prove $$ 1-\\frac{1}{n+2} = \\frac{n+1}{n+2} $$ Multiply both sides by $n+2$ to get an equivalent expression. $$ (1-\\frac{1}{n+2}) * (n+2) = (\\frac{n+1}{n+2}) * (n+2) $$ $$ n+1=n+2−1 $$ Does this all make sense? How can this be improved upon?",
"topk_rank": 10
},
{
"id": "corpus-215076",
"score": 0.766067385673523,
"text": "How can I prove that $\\sum_{n=1}^\\infty \\frac{1}{n(n+1)} = 1$? How can I prove that $$\\sum_{n=1}^\\infty \\frac{1}{n(n+1)} = 1 \\,\\,\\, ?$$ I do know a way to prove this (see my answer) but I'm curious to know what other approaches could be taken in dealing with it.",
"topk_rank": 11
},
{
"id": "corpus-146337",
"score": 0.7616211771965027,
"text": "Proof by Induction - Math Prove that for every integer n>=2: We have the summation of $$ \\frac{1}{i(i-1)}=1-\\frac{1}{n} $$ I tried the algebra with this proof, but couldn't get it. I know that you split the i-1 and the i up but I am stuck from there.",
"topk_rank": 12
},
{
"id": "corpus-152414",
"score": 0.7613592743873596,
"text": "Induction proof that $\\sum_{j=n}^{2n-1} (2j + 1) = 3n^2$ - what happened? So I have the general summation formula that I was to prove using mathemathical induction on a Calculus level. For all $n = {1,2,3,...}$ we have: $$\\sum_{j=n}^{2n-1} (2j + 1) = 3n^2$$ In the answer, the one completing the proof gets this, on the $n=k+1$: $$\\sum_{j=(k+1)}^{2(k+1)-1} (2j + 1) = \\sum_{j=k}^{2k-1} (2j + 1) - (2k+1) + (4k+1) + (4k+3)$$ $$= 3k^2 + 6k + 3 = 3(k+1)^2$$ Can anyone explain how on earth this was done? When I tried, I was unable to factorize - I tried by adding the $(2(k+1)+1)$ to the $S_n$ formula and thereby confirm it, but I couldn't do it...",
"topk_rank": 13
},
{
"id": "corpus-55759",
"score": 0.761026918888092,
"text": "Summation involving a factorial: $1 + \\sum_{j=1}^{n} j!j$ $$1 + \\sum_{j=1}^{n} j!j$$ I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ideas on how I should tackle this one?",
"topk_rank": 14
},
{
"id": "corpus-207695",
"score": 0.7608306407928467,
"text": "A finite summation of double binomial coefficients I find the following identity and have checked on Mathematica, while I have no idea how to prove it: $$\\sum_{j=0}^n(-1)^{n-j}\\binom{p+j}{j}\\binom{n+\\beta}{n-j}=\\binom{p-\\beta}{n}, \\quad \\beta>-1, \\quad p>\\beta-1.$$ It seems that proof by induction does not work and because of the gamma function, the transform technique does not work either.",
"topk_rank": 15
},
{
"id": "corpus-45480",
"score": 0.7602052688598633,
"text": "How to show that $\\sum_{k=1}^n k(n+1-k)=\\binom{n+2}3$? While thinking about I found out that this equality might be useful there: $$n\\cdot 1 + (n-1)\\cdot 2 + \\dots + 2\\cdot (n-1) + 1\\cdot n = \\frac{n(n+1)(n+2)}6$$ To rewrite it in a more compact way: $$\\sum_{k=1}^n k(n+1-k)=\\frac{n(n+1)(n+2)}6.$$ This equality is relatively easy to prove: $$\\sum_{k=1}^n k(n+1-k)= (n+1)\\sum_{k=1}^n k - \\sum_{k=1}^n k^2 = (n+1) \\frac{n(n+1)}2 - \\frac{n(n+1)(2n+1)}6 = n(n+1) \\left(\\frac{n+1}2-\\frac{2n+1}6\\right) = n(n+1)\\frac{3(n+1)-(2n+1)}6 = \\frac{n(n+1)(n+2)}6.$$ (We only used the known formulas for the and the .) Are there some other nice proofs of this equality? (Induction, combinatorial arguments, visual proofs, ...) EDIT: Now I found another question which asks about the same identity: (I have tried to search before posting. But the answers posted here so far gave me some new ideas for good keywords to search which lead me to finding that question.) The questions are, in my opinion, not exact duplicates since the other question asks specifically about combinatorial proofs and my question does not have that restriction. But I agree that this is a very minor distinction. In any case, if you think that one of them should be closed as a duplicate, then you can vote to close. I will refrain from voting to close/reopen on this question. (If one of the two questions is voted to be a duplicate of the other one, they probably cannot be merged, since the summation variables are off by one.)",
"topk_rank": 16
},
{
"id": "corpus-122136",
"score": 0.7602052688598633,
"text": "How to show that $\\sum_{k=1}^n k(n+1-k)=\\binom{n+2}3$? While thinking about I found out that this equality might be useful there: $$n\\cdot 1 + (n-1)\\cdot 2 + \\dots + 2\\cdot (n-1) + 1\\cdot n = \\frac{n(n+1)(n+2)}6$$ To rewrite it in a more compact way: $$\\sum_{k=1}^n k(n+1-k)=\\frac{n(n+1)(n+2)}6.$$ This equality is relatively easy to prove: $$\\sum_{k=1}^n k(n+1-k)= (n+1)\\sum_{k=1}^n k - \\sum_{k=1}^n k^2 = (n+1) \\frac{n(n+1)}2 - \\frac{n(n+1)(2n+1)}6 = n(n+1) \\left(\\frac{n+1}2-\\frac{2n+1}6\\right) = n(n+1)\\frac{3(n+1)-(2n+1)}6 = \\frac{n(n+1)(n+2)}6.$$ (We only used the known formulas for the and the .) Are there some other nice proofs of this equality? (Induction, combinatorial arguments, visual proofs, ...) EDIT: Now I found another question which asks about the same identity: (I have tried to search before posting. But the answers posted here so far gave me some new ideas for good keywords to search which lead me to finding that question.) The questions are, in my opinion, not exact duplicates since the other question asks specifically about combinatorial proofs and my question does not have that restriction. But I agree that this is a very minor distinction. In any case, if you think that one of them should be closed as a duplicate, then you can vote to close. I will refrain from voting to close/reopen on this question. (If one of the two questions is voted to be a duplicate of the other one, they probably cannot be merged, since the summation variables are off by one.)",
"topk_rank": 17
},
{
"id": "corpus-168594",
"score": 0.7602052688598633,
"text": "How to show that $\\sum_{k=1}^n k(n+1-k)=\\binom{n+2}3$? While thinking about I found out that this equality might be useful there: $$n\\cdot 1 + (n-1)\\cdot 2 + \\dots + 2\\cdot (n-1) + 1\\cdot n = \\frac{n(n+1)(n+2)}6$$ To rewrite it in a more compact way: $$\\sum_{k=1}^n k(n+1-k)=\\frac{n(n+1)(n+2)}6.$$ This equality is relatively easy to prove: $$\\sum_{k=1}^n k(n+1-k)= (n+1)\\sum_{k=1}^n k - \\sum_{k=1}^n k^2 = (n+1) \\frac{n(n+1)}2 - \\frac{n(n+1)(2n+1)}6 = n(n+1) \\left(\\frac{n+1}2-\\frac{2n+1}6\\right) = n(n+1)\\frac{3(n+1)-(2n+1)}6 = \\frac{n(n+1)(n+2)}6.$$ (We only used the known formulas for the and the .) Are there some other nice proofs of this equality? (Induction, combinatorial arguments, visual proofs, ...) EDIT: Now I found another question which asks about the same identity: (I have tried to search before posting. But the answers posted here so far gave me some new ideas for good keywords to search which lead me to finding that question.) The questions are, in my opinion, not exact duplicates since the other question asks specifically about combinatorial proofs and my question does not have that restriction. But I agree that this is a very minor distinction. In any case, if you think that one of them should be closed as a duplicate, then you can vote to close. I will refrain from voting to close/reopen on this question. (If one of the two questions is voted to be a duplicate of the other one, they probably cannot be merged, since the summation variables are off by one.)",
"topk_rank": 18
},
{
"id": "corpus-170794",
"score": 0.7602052688598633,
"text": "How to show that $\\sum_{k=1}^n k(n+1-k)=\\binom{n+2}3$? While thinking about I found out that this equality might be useful there: $$n\\cdot 1 + (n-1)\\cdot 2 + \\dots + 2\\cdot (n-1) + 1\\cdot n = \\frac{n(n+1)(n+2)}6$$ To rewrite it in a more compact way: $$\\sum_{k=1}^n k(n+1-k)=\\frac{n(n+1)(n+2)}6.$$ This equality is relatively easy to prove: $$\\sum_{k=1}^n k(n+1-k)= (n+1)\\sum_{k=1}^n k - \\sum_{k=1}^n k^2 = (n+1) \\frac{n(n+1)}2 - \\frac{n(n+1)(2n+1)}6 = n(n+1) \\left(\\frac{n+1}2-\\frac{2n+1}6\\right) = n(n+1)\\frac{3(n+1)-(2n+1)}6 = \\frac{n(n+1)(n+2)}6.$$ (We only used the known formulas for the and the .) Are there some other nice proofs of this equality? (Induction, combinatorial arguments, visual proofs, ...) EDIT: Now I found another question which asks about the same identity: (I have tried to search before posting. But the answers posted here so far gave me some new ideas for good keywords to search which lead me to finding that question.) The questions are, in my opinion, not exact duplicates since the other question asks specifically about combinatorial proofs and my question does not have that restriction. But I agree that this is a very minor distinction. In any case, if you think that one of them should be closed as a duplicate, then you can vote to close. I will refrain from voting to close/reopen on this question. (If one of the two questions is voted to be a duplicate of the other one, they probably cannot be merged, since the summation variables are off by one.)",
"topk_rank": 19
}
] |
query-10
|
How is the message 'Please avoid extended discussions in comments' triggered? Is it triggered after a number of comments within a short timeframe, or is there a moderator button you have to click on?
|
[
{
"id": "corpus-10",
"score": 0.8644224405288696,
"text": "How do you choose when to show the \"Please avoid extended discussions in comments\" notice? I have answered two questions: On the first one we've posted 12 comments, and on the second one only 7. But - Why am I only getting the Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat? notice in the SECOND answer, while the first one has more comments?"
}
] |
[
{
"id": "corpus-77575",
"score": 0.7499752640724182,
"text": "Disable chat migration notification if one of the users has insufficient rep? Simple request, really. Can we add a detection for the Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat? prompt, so that if one of the users in the \"extended discussion\" has under 20 rep it doesn't appear? I think a lot of the users this apply to will have low rep and it may take some back and forth to get good info out of them. New suggestion: how about if a sufficiently high-rep user is clicking on the link, the low-rep user gets a field promotion sufficient to chat? Low-rep users are particularly prone to post questions to lead in this direction of needing to move to chat.",
"topk_rank": 0
},
{
"id": "corpus-116748",
"score": 0.7499752640724182,
"text": "Disable chat migration notification if one of the users has insufficient rep? Simple request, really. Can we add a detection for the Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat? prompt, so that if one of the users in the \"extended discussion\" has under 20 rep it doesn't appear? I think a lot of the users this apply to will have low rep and it may take some back and forth to get good info out of them. New suggestion: how about if a sufficiently high-rep user is clicking on the link, the low-rep user gets a field promotion sufficient to chat? Low-rep users are particularly prone to post questions to lead in this direction of needing to move to chat.",
"topk_rank": 1
},
{
"id": "corpus-54943",
"score": 0.7499698400497437,
"text": "Disable chat migration notification if one of the users has insufficient rep? Simple request, really. Can we add a detection for the Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat? prompt, so that if one of the users in the \"extended discussion\" has under 20 rep it doesn't appear? I think a lot of the users this apply to will have low rep and it may take some back and forth to get good info out of them. New suggestion: how about if a sufficiently high-rep user is clicking on the link, the low-rep user gets a field promotion sufficient to chat? Low-rep users are particularly prone to post questions to lead in this direction of needing to move to chat.",
"topk_rank": 2
},
{
"id": "corpus-143173",
"score": 0.7499698400497437,
"text": "Disable chat migration notification if one of the users has insufficient rep? Simple request, really. Can we add a detection for the Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat? prompt, so that if one of the users in the \"extended discussion\" has under 20 rep it doesn't appear? I think a lot of the users this apply to will have low rep and it may take some back and forth to get good info out of them. New suggestion: how about if a sufficiently high-rep user is clicking on the link, the low-rep user gets a field promotion sufficient to chat? Low-rep users are particularly prone to post questions to lead in this direction of needing to move to chat.",
"topk_rank": 3
},
{
"id": "corpus-213050",
"score": 0.7499698400497437,
"text": "Disable chat migration notification if one of the users has insufficient rep? Simple request, really. Can we add a detection for the Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat? prompt, so that if one of the users in the \"extended discussion\" has under 20 rep it doesn't appear? I think a lot of the users this apply to will have low rep and it may take some back and forth to get good info out of them. New suggestion: how about if a sufficiently high-rep user is clicking on the link, the low-rep user gets a field promotion sufficient to chat? Low-rep users are particularly prone to post questions to lead in this direction of needing to move to chat.",
"topk_rank": 4
},
{
"id": "corpus-188516",
"score": 0.7419746518135071,
"text": "Use comments to ask for clarification or add more information, then what? Is it intended to hide the extended instructive text? It isn't visible at all now because when you enter something, it disappears.",
"topk_rank": 5
},
{
"id": "corpus-243653",
"score": 0.7263383865356445,
"text": "Moving comments to chat: is it just about length? Over time, I've noticed an increasing practice among moderators of the smaller sites (i.e. not in the Trilogy) to move extended comment chains to chat, just because they're too long. Is this a good idea? In the past, I've taken the position on many occasions that I believe that comments serve two purposes only: clarification, and asking for clarification. I take this position because as a mod I want the freedom to purge a comment trail that is clearly running off the rails without having to argue with anyone about politeness definitions. But that's not the same thing as two pedantic C++ language lawyers debating the finer points of "undefined behavior." I give wide latitude to such discussions, as long as the participants don't start throwing spears, because I believe they are instructive. The SE software even has accommodations for this. Long comment chains are automatically collapsed, so as not to disrupt the Q&A format of the post. And moving a comment chain to chat is not even a move, really; to do an actual move, you have to perform the conversion, and then delete all of the copied comments from the post. Here is a recent example: and it's accompanying . What I find amusing is that the OP's detailed POV about the term, where he floats a theory about why everyone thinks that "bugged" is not a thing (that conversation took place in the comments, and is now safely sequestered in the chat room) gets to stay prominently featured in his question, at the very top of the post. Now you could argue that the chat room is a better environment to have these discussions anyway, and you'd be right. I could argue that if we're going to do that, then why not just automatically convert long discussions to chat rooms? The problem is not so much where the comments reside. I don't understand why mods are taking it on themselves to fix something that isn't hurting the site (at least in my opinion). In addition, every time I see this happen, the participants in the conversation immediately lose interest in it. The only comment that was posted in that room after all the comments were moved is most likely this one: It wouldn't surprise me if the folks using "bugged" to refer to a certain software quality are the same folks who refer to their programs as "codes" and who believe textspeak is the pinnacle of modern English. Which nobody will ever see. :) Some mods are really aggressive about this. I discussed this with a mod on Physics, and he told me that they routinely move discussions when they think they might run off the rails.",
"topk_rank": 6
},
{
"id": "corpus-242632",
"score": 0.7240343689918518,
"text": "Error in generated comment when using the tool: Move comments to chat I used the moderator tool for the first time today, on , and the comment that was created did not work properly. The comment that was created: Comments are not for extended discussion; this conversation has been <a href=\"http://chat.stackexchange.com/rooms/17535/discussion-on-question-by-igor-rivin-a-more-perfect-union\">moved to chat</a>. Additional examples from other users: Not only does the link appear malformed but it is also broken. Note that the <a href= form is used rather than the [name](url) markdown that works in comments. I had to edit the comment to correct the problem.",
"topk_rank": 7
},
{
"id": "corpus-235130",
"score": 0.7196958065032959,
"text": "Dealing with what's perceived to be an offensive comment I recently responded to that appeared to post an extension to their answer in a comment, (what I thought was) kindly letting them know that comments may not last forever. However, it ended up in this kerfuffle: After posting the reply comment, I felt that it wasn't the best course of action to have taken, but to take it back also seemed inappropriate. What should I have done, and does my original comment seem appropriate in the given situation?",
"topk_rank": 8
},
{
"id": "corpus-6048",
"score": 0.7192077040672302,
"text": "Insert a \"Let us continue this discussion in chat\" comment before thread gets messy I've seen , but there seem to be no proper solution. I suggest an ability to insert a \"Let us continue this discussion in chat\" comment before the thread gets messy enough to fit the requirement. The point is to prevent the problem before it occurs. Perhaps this should become a privilege for users of X reputation or more? Your opinions on the subject?",
"topk_rank": 9
},
{
"id": "corpus-146554",
"score": 0.7124546766281128,
"text": "Please make the multiple comment timer less annoying (non-resetting etc.) As a little feature to prevent comment spam, you can only enter one comment every 15 seconds. That's a very reasonable limitation, since you very rarely have a good reason to write comments that quickly. I've only seen it twice at all, I think, but both times it was extremely annoying. Only 1 comment allowed per 15 seconds; timer reset. The reason is that every time you hit the limit, the 15 seconds begin anew. You have no indicator that tells you when you are allowed to comment again, and if you hit enter after just 14 seconds, you're back to zero. Yesterday, I went through four cycles of not quite 15 seconds, counting in my head, before I got a myself a coffee and tried again afterwards. I suggest that you do one or more of the following (in decreasing order): Make the counter non-restarting Add a visible countdown Make it clearly visible that you can't comment right now You can actually increase the waiting time, if it is clearly visible and there is no unnerving invisible countdown. Actually, just remove all comment fields after one comment, and tell the user to reload the page. Just please don't have UI elements that look like they would work, but feel like they are trolling you.",
"topk_rank": 10
},
{
"id": "corpus-146795",
"score": 0.7124546766281128,
"text": "Please make the multiple comment timer less annoying (non-resetting etc.) As a little feature to prevent comment spam, you can only enter one comment every 15 seconds. That's a very reasonable limitation, since you very rarely have a good reason to write comments that quickly. I've only seen it twice at all, I think, but both times it was extremely annoying. Only 1 comment allowed per 15 seconds; timer reset. The reason is that every time you hit the limit, the 15 seconds begin anew. You have no indicator that tells you when you are allowed to comment again, and if you hit enter after just 14 seconds, you're back to zero. Yesterday, I went through four cycles of not quite 15 seconds, counting in my head, before I got a myself a coffee and tried again afterwards. I suggest that you do one or more of the following (in decreasing order): Make the counter non-restarting Add a visible countdown Make it clearly visible that you can't comment right now You can actually increase the waiting time, if it is clearly visible and there is no unnerving invisible countdown. Actually, just remove all comment fields after one comment, and tell the user to reload the page. Just please don't have UI elements that look like they would work, but feel like they are trolling you.",
"topk_rank": 11
},
{
"id": "corpus-96120",
"score": 0.7117119431495667,
"text": "Prevent the OP commenting on their own post? Unsurprisingly, most long discussion threads under a post involve the OP — and preventing them from commenting on their own question or answer will help to reduce comments being used for discussion. If the OP wants to discuss someone's comment, they should have the immediate option to 'take discussion to chat' instead, and be able to @reply from there. Obviously this would change the commenting behaviour of the OP, but there may be a significant side benefit: a reduction is comments that solicit discussion from other users as we come to realise that we can't simply start a discussion thread on the post any more. All the would still work, as none usually require discussion: Request clarification from the author; Leave constructive criticism that guides the author in improving the post; Add relevant but minor or transient information to a post (e.g. a link to a related question, or an alert to the author that the question has been updated).",
"topk_rank": 12
},
{
"id": "corpus-45013",
"score": 0.7094151973724365,
"text": "Can we get a comments only lock? Currently moderators have the ability to lock posts, for any of the following reasons: Content dispute (~ edit wars), Historical significance (to prevent great but off topic questions from being deleted), and Offtopic comments Wiki Answer The locks can be applied for an hour, a day, a week, or permanently. This is all great, but: Locking a post also prevents voting and (more importantly) editing, and If the locked post is a question, there are other ramifications, such as preventing new answers from being posted. Those side-effects aren't always desirable for posts that generated a lot of \"offtopic comments\" and I think we could use a lock that would only lock the comment section. The lock message should point to chat, and ideally should also give the option of migrating the comment thread to chat automagically. Additionally the comment lock, being a lesser lock, could be available to the OP as well, not only to moderators. Perhaps only after a certain rep level (3K? 5K? 10K?) and perhaps without the option to lock permanently. Getting spammed with inbox notifications on an irrelevant discussion happening on one of your posts isn't fun, and it's perhaps the only thing I truly hate on Stack Exchange. Similar feature requests: (duplicate?) Relevant discussions:",
"topk_rank": 13
},
{
"id": "corpus-34573",
"score": 0.7088261246681213,
"text": "Why is the time limit for editing comments 5 minutes? What is the reasoning behind this?",
"topk_rank": 14
},
{
"id": "corpus-150010",
"score": 0.7088261246681213,
"text": "Why is the time limit for editing comments 5 minutes? What is the reasoning behind this?",
"topk_rank": 15
},
{
"id": "corpus-193672",
"score": 0.7088261246681213,
"text": "Why is the time limit for editing comments 5 minutes? What is the reasoning behind this?",
"topk_rank": 16
},
{
"id": "corpus-92839",
"score": 0.707818329334259,
"text": "Blog comment moderation rules Since my profile will be deleted in a few hours, this doesn't concern me, but the remaining people may be interested. We know the already famous post: . One may be surprised that it has too few comments, especially highly negative ones. It seems that the reason is simple: moderation. About 4 hours ago I've written the comment there, but it wasn't published. Since I don't see it \"awaiting moderation\" anymore and I see much later comments published, it seems to be removed. The comment was along these lines: There were a lot of things I didn't like in SO: too positively-skewed reputation changes, generally-accepted reputation farming via answering duplicates which were asked 10 times, extremely low quality threshold for questions, \"accepted answer\" feature implementation, etc. and etc. Looking at the recent changes, things will become only worse. I'm deleting my profile. Now, the question is: why was it removed? Is it unfriendly? Sure, but I don't see why it should be friendly: we express our opinions, and my opinion is negative. Is it non-constructive? Well, to some extent, but definitely more constructive than: Nice! I’m (hopefully) looking forward to passing the 10k mark If there are rules for commenting (like 2. You DO NOT TALK bad about SO), I think it makes sense to publish them (and give an easily-found link to them in the blog). It also would be interesting to know how comment moderation is organized.",
"topk_rank": 17
},
{
"id": "corpus-57766",
"score": 0.703851044178009,
"text": "Remove link preview in discussion dashboard It can be sometimes bothersome when, in the Discussion (Comments) dashboard, where the admin can see a list of comments, moving the mouse over a link will cause a preview of that link. Sometimes the preview gets in the way of looking at the comment. This is especially true with spam comments. And I am also concerned that a link to a page/site that had some 'bad' code would cause a compromise of my site. Is there a way to disable this 'feature'? Not sure where it is coming from.",
"topk_rank": 18
},
{
"id": "corpus-207033",
"score": 0.6972532868385315,
"text": "A guide to moderating comments Commenting is one of the unsung heroes of Stack Exchange. : Request clarification from the author; Leave constructive criticism that guides the author in improving the post; Add relevant but minor or transient information to a post (e.g. a link to a related question, or an alert to the author that the question has been updated). These are all essential for coordinating the creation of useful questions and answers, but originally comments were not part of the design for this system! During the beta period, it was observed that, without a way to annotate or discuss specific posts, answers became noisy and tangential. Therefore, a separate space was carved out for this purpose below each post, leaving answers free for content that, y'know, actually tries to answer the question. This created a problem, however: how do we expose useful comments inline with answers without adding back the same amount of noise comments were created to alleviate? . This simple strategy proved to be quite durable, working admirably in most common cases - but it suffers from a few drawbacks: Comments can become obsolete, addressing issues or concerns long ago addressed via edits to the post itself. Those one-liner contests Jeff alluded to don't always produce results as as one might like. With the addition of comment replies and real-time notifications, comments can be used for casual conversation and debate - fun and perhaps even enlightening for those involved, noise for those just looking for an answer. Thus the need for comment moderation. This involves three groups of people: The authors of the comments themselves can clean up after they're done, if they so desire. can flag comments in response to the issues noted above, bringing them to the attention of the moderators. (They can also vote on comments, controlling to a degree which ones are shown) can remove individual comments or entire comment threads, , and even posts if comments are getting unruly. This raises several questions, which I'll try to address below: These guidelines are based on the authors' experience moderating comments on various sites. Individual communities, particularly on meta sites, may benefit from additional guidelines - this guide is intended as a baseline.",
"topk_rank": 19
}
] |
query-11
|
Is it possible to run os x 10.8 on a modern mac? I would like to get a new computer, but I have a lot of software that I use which is discontinued and requires OS X 10.8. Is there a way on a modern mac to have a dual-boot situation so that I can have the latest OS, but also have the option to boot into 10.8 without any performance issues?
|
[
{
"id": "corpus-11",
"score": 0.7977316975593567,
"text": "Is it possible to install an old version of Mac OS on a new Mac? I know that Apple doesn't want me to run a version of Mac OS older than the one my Mac came with, but is there a way around this? For example: if I buy a new Retina MacBook Pro, is there a way to run OS 10.6 on it without a VM?"
}
] |
[
{
"id": "corpus-80463",
"score": 0.74989253282547,
"text": "Installing OS X not from App Store This question might have been answered somewhere but have not been able to find it. So, I have a Mid-2009 MacBook Pro where I am still running OS X Yosemite since I forgot to update to OS X El Capitan when it was on the App Store before they changed it to macOS Sierra. If I try to install macOS Sierra through the App Store it doesn't let me because it says that: \"This version of macOS 10.12 cannot be installed on this computer.\" I started looking to see if I can find OS X El Capitan but there is no result through the Mac App Store nor Apple Developers site. So my last resort was through Google, where I found several options on where to download OS X El Capitan version 10.11. So, my main question is if I download from these third party websites, can I install on top of my current OS like it is done through the App Store or do I have to do a clean install? Would appreciate any help.",
"topk_rank": 0
},
{
"id": "corpus-104832",
"score": 0.7477951645851135,
"text": "Is it legal to run windows on a mac? I understand that it is Illegal to run OSx on a windows machine or PC for that matter, but is it illegal to do the reverse? Why or why not?",
"topk_rank": 1
},
{
"id": "corpus-218113",
"score": 0.7439460754394531,
"text": "App Store only sells latest OS X version? Can't get Lion 10.7 anymore? I have a MacBook model A1181, 2GHz Intel Core 2 Duo, 2GB memory. It's running Snow Leopard, and running fine including speed wise. I want to upgrade to make use of iOS simulator and XCode 4. Saw that App Store has only the newest 10.8 Mountain Lion for upgrade, and saw a few reviews saying it slowed down their Macs, even w/ 16GB memory, wonder what it will do to my machine. I would prefer to get 10.7 Lion. Any way to get that anymore, maybe at Apple Stores? Or am I out of luck and have to go to 10.8 if I want to upgrade? Too bad I should have upgraded sooner when 10.8 wasn't out yet. Update: looking around online, looks like my Mac may not even quality for requirements for 10.8, but 10.7 should be ok.",
"topk_rank": 2
},
{
"id": "corpus-172722",
"score": 0.7433388829231262,
"text": "Will my Mac run slow if I update it to Mavericks? Today my Mac wanted me to upgrade it to Mavericks. I'm hesitating because it is a bit old (Mid 2009 MacBook Pro) and I don't want things to run slow. I'm currently using 10.8.5 and it is OK. I have a Core 2 duo (2.53 GHz) with 4 GB of RAM. What do you recommend? How was your experience?",
"topk_rank": 3
},
{
"id": "corpus-20461",
"score": 0.7378590703010559,
"text": "OS X El Capitan and older not available on App Store? So it turns out I can not install Sierra because I have a Mid2009 Macbook Pro. I wanted to update to El Capitan, but it turns out its no longer available on the app store? Any ideas, how to get El Capitan?",
"topk_rank": 4
},
{
"id": "corpus-60083",
"score": 0.731924831867218,
"text": "Can I run OSX applications? Just wondering, are there any projects like WINE, but for Mac software? That is, can we use a program that acts like a compatibility layer to run Mac apps?",
"topk_rank": 5
},
{
"id": "corpus-84670",
"score": 0.731924831867218,
"text": "Can I run OSX applications? Just wondering, are there any projects like WINE, but for Mac software? That is, can we use a program that acts like a compatibility layer to run Mac apps?",
"topk_rank": 6
},
{
"id": "corpus-93313",
"score": 0.731924831867218,
"text": "Can I run OSX applications? Just wondering, are there any projects like WINE, but for Mac software? That is, can we use a program that acts like a compatibility layer to run Mac apps?",
"topk_rank": 7
},
{
"id": "corpus-198866",
"score": 0.731924831867218,
"text": "Can I run OSX applications? Just wondering, are there any projects like WINE, but for Mac software? That is, can we use a program that acts like a compatibility layer to run Mac apps?",
"topk_rank": 8
},
{
"id": "corpus-231969",
"score": 0.731924831867218,
"text": "Can I run OSX applications? Just wondering, are there any projects like WINE, but for Mac software? That is, can we use a program that acts like a compatibility layer to run Mac apps?",
"topk_rank": 9
},
{
"id": "corpus-249329",
"score": 0.731924831867218,
"text": "Can I run OSX applications? Just wondering, are there any projects like WINE, but for Mac software? That is, can we use a program that acts like a compatibility layer to run Mac apps?",
"topk_rank": 10
},
{
"id": "corpus-209968",
"score": 0.7210038900375366,
"text": "Is it possible to run Snow Leopard on a LATE 2011 MacBook Pro? I want to buy a new MacBook Pro, from Apple. However I want to run Snow Leopard instead of Lion. I did some Google searches and found a few success stories of people who got SL to run on their early-2011 MBPs. I didn't find much of any experiences of people doing the same on a late-2011 MBP though, so I'm wondering (and really hoping!) if it's possible?",
"topk_rank": 11
},
{
"id": "corpus-245348",
"score": 0.7207223773002625,
"text": "How do I install OS X Yosemite without a working Mac App Store application? My installation of the Mac App Store on Mavericks . What alternatives are there for installing the new OS?",
"topk_rank": 12
},
{
"id": "corpus-103177",
"score": 0.7205651998519897,
"text": "Running OS X Apps On Ubuntu 14.04 Is there any way that still works to run OS X apps on Ubuntu 14.04? If Darling still works, please tell me how to install it. Thanks. :)",
"topk_rank": 13
},
{
"id": "corpus-108580",
"score": 0.7197928428649902,
"text": "How to re-download Lion from Mountain Lion I'm currently running Mountain Lion, but I'd like to redownload Lion (to create a bootable install drive). However, when I click the download button in the App Store, I receive a message telling me my current OS X is newer and the download will not start. Does anyone know a way to download Lion when already running Mountain Lion?",
"topk_rank": 14
},
{
"id": "corpus-226135",
"score": 0.7174281477928162,
"text": "OS X license on used Mac? I just got off a very long and very frustrating phone call with Apple support. I purchased a used Apple computer which came installed with Mountain Lion. Meaning that the computer was shipped by Apple with Mountain Lion installed, and received by me with that same installation. As a precaution, I tried to reinstall Mountain Lion via Recovery mode. This failed because my Apple ID hasn't purchased Mountain Lion. Apple insisted that because I didn't purchase the computer new, I don't have a license for OS X. I don't believe this is correct, or even legal. This means that anyone using a used computer is doing so illegally, unless they've purchased a separate OS license. Am I way off-base here? I'm pretty sure I now \"own\" a copy of Mountain Lion, and Apple is in the wrong. If this is correct, what are my options?",
"topk_rank": 15
},
{
"id": "corpus-51948",
"score": 0.7160789370536804,
"text": "What should I be aware of when installing Linux on a Mac? I have a Macbook Pro and I am loving it, though I still miss my Linux box, there are many things I need which are not completely compatible with Mac OS X. I heard many stories about installing Linux on a Mac OS, some say it's not a problem, but some others tend to say differently. My question is, is it or is not fine to install Linux on a Mac OS machine? What are the pros and cons? I am very well aware about virtual machines, but let's be honest, they do not run quite as well when running on a physical hardware.",
"topk_rank": 16
},
{
"id": "corpus-109792",
"score": 0.7149782180786133,
"text": "Is there a way to virtualize Mac OSX in Windows? Possible Duplicate: I have heard rumors that there is a way to run Mac OSX Server in a Virtual Machine in windows. I know this is not exactly Legal. I want to do this to try out what it is like to develop applications for the iPhone but without the huge investment of buying a MAC.",
"topk_rank": 17
},
{
"id": "corpus-67613",
"score": 0.7132933735847473,
"text": "Windows 98 on a modern computer? Is it possible to install Windows 98 in a modern computer and run modern software? If not, why?",
"topk_rank": 18
},
{
"id": "corpus-230123",
"score": 0.7116985321044922,
"text": "How can I run Windows applications on a MacBook Pro? I have a friend that wants to buy a Mac Book Pro and he has some windows applications that he needs to run. He doesn't want to buy the applications again. Can he run the Windows applications on his MacBook Pro? What ways to accomplish this are available?",
"topk_rank": 19
}
] |
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