## Finding an expression for the sum of n tems of the series $1^2 + 2^2 + 3^2 + … + n^2$ February 7

I know that if you have a non-arithmetic or geometric progression, you can find a sum $S$ of a series with the formula $S=f(n+1)-f(1)$ where the term $u_n$ is $u_n=f(n+1)-f(n)$. Then you can prove that with induction.What I don't understand is how I

## Preserving independence of random variables February 7

Suppose I have three random variables, $X,Y,Z$ with $X$ independent of $Z$, $Y$ independent of $Z$.Which transformation can I apply to $X,Y$ to that the result is again a random variable independent of $Z$? Or better, for which $f(x,y)$ is $f(X,Y)$ i

Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators. Suppose:$T_n$ is bounded$(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in [0,1]$$Tf\in C([0,1]) for every f\in C([0,1]) Show that ## Is there a T_6 space in which a sequentially (or countably) compact subset is not closed February 7 It is known that a T_2 space X is KC, i.e. every compact subset of X is closed. The space [0, \omega_1] is T_5 but not T_6 and the subset [0, \omega_1) is sequentially compact (and thus countably compact) but not closed. Is there an e ## Compare Complexity of Graph (Landau) February 7 Assume I know that there is an algorithm of complexity \mathcal{O}( \vert V \vert^2 \vert E \vert ) for a Graph G(E,V).How do I compare this for example to the complexity of \mathcal{O}( \vert V \vert^2 + \vert E \vert ) or other complexity a ## When does the following sequences of random variables follow SLLN February 7 Find the values of the constant c in each case so that the Strong Law of Large Numbers holds for each sequence of independent random variables \{X_n\}:(i) P(X_n=\pm c^n)=\left(\dfrac{2}{3}\right)^{n+1},\space P(X_n=0)=\left(\dfrac{2}{3}\right ## Is the expected value \mu in the WLLN a random variable February 7 Am I right in thinking that the weak law of large numbers, when stating that$$\bar{X_n} \to \mu$$in probability convergence is stating that the sequence of random variables \{X_n\} tends to the random variable (as opposed to the real number) M = ## SLLN when the expectation in infinite February 7 In a Post I found it says:"Whenever {\rm E}(X) exists (finite or infinite), the strong law of large numbers holds. That is, if X_1,X_2,\ldots is a sequence of i.i.d. random variables with finite or infinite expectation, letting S_n = X_1+\cdots ## A faithful positive Radon measure February 7 Let X be a locally comapct and Hausdorff space. We say a positive Radon Measure on X is faithful if$$0\leq f ~~~,~~~\int fd\mu=0\rightarrow f=0$$Q: True or false: If there is a faithful positive Radon measure on X then X has a countable dens ## Support of a Radon measure February 7 Let X be a locally compact and Hausdorff space. For given a Radon measure \mu on X, the support of \mu is the smallest closed subset of X with |\mu|(X)=||\mu||.Q: Let E be a closed subset in X. Does there exist any (positive) measure ## Question about angular velocity February 7 1 If z=r\cos\theta +ir\sin \theta, show that \dfrac{dz}{dt}=r\dfrac{d\theta}{dt}(-\sin\theta+i\cos \theta) and that if \dfrac{d\theta}{dt} is constant, then \arg\dfrac{dz}{dt}=\theta-\pi/2.I managed to do everything apart from the argument, whi ## Russell's paradox question February 7 3 Tao's analysis book uses following example for Russell's paradox:$$P(x) \Longrightarrow  x\text{ is a set, and }x \notin x"\\ \Omega := \{x : P(x)\text{ is true} \} = \{x : x\text{ is a set and }x \notin x\}$$then conclude that ­\Omega \in \Omeg ## separation and russell's paradox February 7 I just want to be sure that I understand the connection between "Naive Comprehension", the Axiom of Separation, Russell's Paradox, and the existence of a universal set. Is the following correct?The Naive Comprehension Axiom has as an instance th ## lim_{(x,y)\to(0,0)}\frac{x^2y^2}{sin(x)cos(y)} is this done correctly February 7 lim_{(x,y)\to(0,0)}\frac{x^2y^2}{sin(x)cos(y)} is it allowed to split a multi-variable limit into its component variables as in the next step?= (lim_{x\to0}\frac{x^2}{sin(x)})(lim_{x\to0}\frac{y^2}{sin(y)}) this is an indeterminate form and now I ## Physics Velocity Correction February 7 So, been up all night working on getting the velocity/angle of arrow simulations perfect. Running into an issue with the physics engine I'm using is ever so slightly off (probably a rounding issue)The Velocity is a fixed 55 "Units"Correction Req ## How to find the required differential equation February 7 How to find the differential equation of tangent lines to the parabola y=x^2?How to find the differential equation of all conics whose axes coincide with axes of co ordinates? ## Differential equation on \Bbb R February 7 1 We have a differential equation on \Bbb R of the form$$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$where \chi_{[0,1]} is the characteristic function of the interval [0, 1] ⊂ \Bbb R. I want to find a generalized solution for this differential equation ## y'+y=x, x \in \Bbb R and y(-1)=0. Then y(1)= February 7 I calculated the integrating factor to be e^x.Then e^x y'+ e^x y=e^x |x| \Rightarrow \frac {d(e^x y)}{dx}=e^x |x| \Rightarrow d(e^x y)=e^x|x|dx Integrating both sides,e^xy=e^x \int |x|dx- \int [(\int |x|dx)(e^x)]dx +c.From here how to proceed?C ## Nonlinear operator sends bounded set to relatively compact set February 7 Consider g a continuous function on [a,b]\times\mathbb{R}, and let z_0\in\mathbb{R}. Define the (nonlinear) operator on C[a,b]:$$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$for x\in[a,b]. Prove that:(i) M is continuous and (ii) maps bounded set i ## How much can I gauge about the domain of a differential equation without actually solving it February 7 Say I have the differential equation$$y' = \frac{3t^2 - 2ty}{4 - t^2} \text{, where }y(1)=-3$$Clearly the equation is undefined at t = \pm2, and a solution exists at t = 1. Can I conclude from this that the solution must at least be defined on th You Might Also Like • I want to find a fourier expansion of only sines representing g(x) = 1 on the interval [0, \pi]. So I ex ... • A circle passes through the vertex A of an eq ... • I'm trying to proof an equality about the generating function os a compoung Poisson process and I don't know ... • Let k be an nonzero integer and b>2 a real. Is it true that there exist only finitely many positive i ... • Suppose a square grid graph g of side length n can be colored with q colors. In how many unique colori ... • Let p be a prime. Let H_{i}, i=1,...,n be normal subgroups of a finite group G. I want to prove the fo ... • I was going through some of my notes in abstract algebra (group and ring theory) and came across this$$R/S$ ...
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