Given $f$ is a bounded continuous function on $\mathbf{R}$ and $\mu$ is a probability measure such that for all $x \in \mathbf{R}$ $$ f(x) = \int_\mathbf{R}f(x+y)\mu(dy) $$  Please help to show that $f(x+s) = f(x)$ for all $s$ in the support of $\mu$. (The support of $\mu$ is the smallest set $E$ s.t. $\mu(E)=1$.) The hint is that we can try to use martingales to show $E[f(x+X_0)-f(x)]^2=0$, where $X_0$ has distribution $\mu$.  So far I've constructed a martingale $M_n=f(x+S_n)$. Here $S_n = \sum_{j=1}^nX_j$ and $\{ X_j\}$ are i.i.d. with distribution $\mu$. But I've no idea on how to connect this martingale with the previous hint.  I came cross another post about the same question on the forum, but it ends with the martingale that I stated here. So I'm wondering if anyone can provide some further insights about how to finish the proof. Since this is a hw problem, I've been working on it for three days only to get to this point. I sincerely appreciate any inputs that people will give.  Thanks a ton!
Assume f is a bounded continuous function on $\mathbf{R}$ and $X$ is a random variable with distribution $F$.  Assume for all $x \in \mathbf{R}$ that  $$ f(x) = \int_\mathbf{R}f(x+y)F(dy) $$  Please help conclude that $f(x+s) = f(x)$ where $s$ is any value in the support of $F$.  The hints that I have come across are to use Martingale theory and consider $\{ X_n\}$ to be i.i.d. with distribution $F$ and make a martingale with some function of $S_n = \sum_{j=1}^nX_j$.    Thanks!
I am new to the telecom simulations system specially multisim, which I am having a project that I want to simulate.  My project in (Analog Communications) is consisting to have FM Transmitter (Modulator) that accepts a generated signal (Sin, Cos) and get me the modulated output as a Graph.  I've found this FM transmitter circuit (3 km range):   ![two muppets][1]  and I am building that circuit on Multisim, and I failed. Because I faced many problems, first of, the variable capacitor symbol is weird here in the multisim, and the regular capacitor doesn't have variable options. Secondly, when I Use signal generator as (Sin, Cos), the output gets weird after 5 second of the simulation.  I want to show the input signal (before modulation) and the output signal (transmitted or modulated).  I guess my output result where not the as expected, as the signal (Sin with phase 90&deg;) is giving weird result of few seconds of simulating.  Below are screenshots of the simulation results within time.          Any ideas how to edit or fix the above circuit to have this output is similar to what it should be? Is it (always) true that $$\mathrm{Var}\left(\sum\limits_{i=1}^m{X_i}\right) = \sum\limits_{i=1}^m{\mathrm{Var}(X_i)} \&gt;?$$ Do computer viruses affect all user accounts on a computer? Do computer viruses affect all user accounts on a computer?  IF a virus affects a local user will that effect administrator account and vise-versa? Is it possible with GEE to export all the images in an image collection? Tangent bundle : is a manifold I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector is a derivation on the germs of differentiable functions at a point.  Now I am trying to see that the tangent bundle is indeed a differentiable manifold. I get that for an open set $U$ in Eucliden space, it takes the form $U \times \mathbb R^n$ with the vector $\sum a_i x_i$ giving the tangent vector $\sum a_i \partial /\partial x_i$. Now the book says that it is "obvious" that these glue up and give a differentiable manifold.  I may be stupid; but I will be grateful if somebody can hold my hand through this "obvious" statement. I typed up an assignment with a lot of numbers in LaTeX and turned it in to my adviser.  When he returned it to me with some corrections, one of which was a lot of $ around my numbers.  I understand that $ are necessary around some mathematical formulas, but I was just using numbers like -2 and 4.  I did not see any change in the font of the numbers when I added the $ either.  Since my adviser did not want to give me an explanation, what is the rationale? How do I test a nonlinear association? For plot 1, I can test the association between x and y by doing a simple correlation.   For plot 2, where the relationship is nonlinear yet there is a clear relation between x and y, how can I test the association and label its nature? Do harmonic numbers have a “closed-form” expression? Show that if $f(z)=\sum\limits_{n\geq0}a_nz^n$ is analytic in $\{z\in\mathbb{C}:|z|&lt;1\}\cup\{1\}$ and $\forall n\geq0:a_n\geq0$ then the radius of convergence of the power series is strictly larger than $1$.  My approach:   Let $D=\{z\in\mathbb{C}:|z|\leq1\}$.  We have that $f(1)$ converges, so $\sum\limits_{n\geq 0} a_n&lt;\infty$. Moreover since the $a_n$'s are all positive we have that $|a_n|=a_n$, therefore $\sum\limits_{n\geq 0} |a_n|=\sum\limits_{n\geq 0} a_n&lt;\infty$.  Let $f_n(z)=a_nz^n$, then $\forall z\in D, n\geq 0$ we have:  $$|f_n(z)|=|a_nz^n|=|a_n|\cdot|z^n|=a_n\cdot |z|^n\leq a_n\cdot 1^n\leq a_n$$  Then, applying Weierstrass M-test we get that the power series is convergent in $D$.  Now, I need to "find" an open ball $B$ such that $D\subset B$ and the power series is convergent in it.  Edit  $f$ is analytic in $1$, so there is an open set $U$ that contains $1$ such that   $$f(z)=\sum\limits_{n\geq 0} b_n(z-1)^n$$  Does it mean that the series is convergent in $U$?   I can see that we have that the two series coincide in $D\cap U$. How to add home directory path to be discovered by Unix which command? Where can I download Ubuntu 16.04? I am unable to find the link for downloading Ubuntu 16.04 onto my   desktop/laptop. I am able to see the link only for 18.04, but I need 16.04. Height and minimal number of generators of an ideal How can I determine the height and the least number of generators of the ideal $ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $?      I tried to calculate the dimension of the vector space $I/I\mathfrak m$ with $\mathfrak m=(x,y,z)$, but I'm not able to find it.  I think the height is $2$, because the affine variety should be a curve, and also I know that the ideal is prime, so I need a chain $0\subset \mathfrak p \subset I$. Maybe $(xz-y^2)?$ Is it prime? Dobble card game - mathematical background In  there is a deck of 55 cards. Each of them has 8 symbols on it. There is a total of 50 different symbols and each two cards have one and only one in common.  I am wondering, given a total number of symbols N and a number of symbols on each card K, how do you find the biggest number of cards such that:   on each of those cards there are exactly K of the N symbols, every two cards have one and only one symbol in common. Can we use "there is" for plural nouns? Why do damaged hard drives freeze the entire system? App Store asks for old Apple ID when installing updates I got a MacBook that a colleague of mine had used previously. It runs OS X 10.10.2. I created a new administrator account for myself and turned off the "admin" flag on the old account.  I also entered my iCloud / Apple ID account, so that it syncs my calendars and contacts correctly, and even in the App Store, I am signed in with my correct account.  But when I try to install pending software updates (that is, OS X system updates!), the window that pops up has the the e-mail address of my colleague filled in and greyed out—and I can't change it.     Note that I'm signed in with a different (correct) account in the App Store. When I go to Featured and click Account on the right, it has my correct e-mail address filled in:    How do I make the App Store "forget" the old Apple ID?  I  where it just says "it's not possible", and that apps are glued to the original account the Mac was set up with, but that doesn't make a lot of sense, especially in a company where laptops may be handed over from one person to another. Iterate over all pairs of consecutive items in a list Case insensitive searching in Oracle surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism?  I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to preserve the property of being immersion.  I also noticed that if we replace $\mathbb{R}^2$ with $S^2$ or any other compact simply connected manifold, no such immersion exists, because then it would be a covering map (as a local diffeomorphism from compact space, thus proper), but since the manifold is simply connected, it must be a diffeomorphism. Can't mount an IDE HDD - "can't read superblock" error message I'm trying to mount my HDD,  /dev/sdb, which is connected via an enclosure but every time I try do so I get an error message   mount: /dev/sdb: can't read superblock   How do I solve this problem and mount my HDD?