Is there some g for which g(x mod 1) is not periodic? If so, please tell me what it is.
I want to know all sets of single-variate functions that fulfill the following qualities: $\forall_{f:R^2 \to R^2} \forall_{g \in CAF} \forall_{h \in CAF} \exists_{k\in CAF} k(x) = f(g(x),h(x))$ where CAF is the set I am wishing to find. I know that the set of constants and the set of piece wise constants each satisfy that equation. Do any others?
strictly positive elements in $C^*$-algebra Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to find the following:a)What are the strictly positive elements of $C_0(\Omega)$ where $\Omega$ is locally compact Hausdorff space, and $C_0(\Omega)$ is the space of continuous complex valued functions that vanishes at $\infty$.b)If $A$ is unital, then $a\in A_+$ is strictly positive iff $a$ is invertible.c)if $(e_n)$ is an approximate identity of $A$, then $a:=\sum_{n=}^{\infty}\frac{1}{2^n}e_n$is strictly positive.What I tried and know:a) I know that positive elements in this space are functions with positive images, and thus, strictly positive elements are functions with positive images and now real root. is this correct? and how can I show that the positive elements are these kind of functions.b) I figures out that if $a$ in invertible then $a$ is strictly positive, but no idea how to do the other direction.c)I did this but using another characterization of strictly positive elements but want to do it with this one, and I don't know how.Thank you for your help. When considering the Doppler shift, the 'canonical equation' is  $$f=\frac{c+vr}{c+vs}f_0$$  However, this equation seems to run into trouble in the following situation: A light source inside water is moving at a speed $v$ towards a receiver outside the water. Can we modify the above equation to deal with a transition between two media? Or is there a completely different formula that applies here? Why is my iPhone screen stuck on the 'connect to iTunes' screen? I need help with my iPhone. It's stuck on the 'connect to iTunes' screen.   iTunes does recognize the iPhone, but when I select 'restore', it goes through a few progress bars, then errors again. I've already tried to hard reset the phone, but that doesn't work either. Family name pluralization What is the 'molecular' origin of the viscosity? The molecular origin of elasticity is almost clear for me: at the very bottom the 'elasticity' comes from the attraction and repulsion between atoms when they are reasonably far from each other or very close (like Lennard-Jones potential). Is there a similar description for the viscosity as well? Where does it come from? Is it true that the order of $ab$ is always equal to the order of $ba$? Why do some bad traits evolve, and good ones don't? What's the difference between $(stuff) and `stuff`? How to draw commutative diagrams? Can anyone help me in giving commands for commutative diagrams in LaTeX ? It doesn't seem to appear properly in the LaTeX guides that I have. A command for a simple triangular diagram with arrows associated to maps would be good enough, or maybe some appropriate reference. How to sort this JSON I installed or upgraded to Ubuntu 16.04, but I'm having some problems with my graphics card.   The display is blank The display flickers Items on the screen do not display correctly Certain programs complain of graphics issues The computer boots to low-graphics mode Your specific issue may not be here, but try the solutions described in the answers anyway.   This question is very broad, but each one of these issues likely has one solution.    If you are experiencing a login loop, check the last part of my answer.    If you experience one of the listed issues and have your own solution (mine didn't work, or there's a faster way), feel free to post another POLITE answer, describing your problem and how you fixed it.    Those attempting to install Ubuntu, please refer to the section titled: If you are unable to enter a TTY or are trying to install Ubuntu. Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence? Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$. In D&amp;D 4e there was an option to Delay your initiative:     Perform your actions as desired and adjust your initiative to your new   position in the order.   Does this rule to change initiative order still exist in 5th edition? I cannot find it in the PHB. what operation repeated $n$ times results in the addition operator? $\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$ I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to  $\mathbb  Z_5$.   The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \}  \to  \mathbb N$ defined as $N(a+bi)=a^2+b^2$, then I can express any element $z$ in $\mathbb Z[i]$ as $z=(1+2i)q+r$ with $N(r)=1,2,3,4$ or $r=0$.   If I define $f:\mathbb Z[i] \to \mathbb  Z_5$ as $f(x=(1+2i)q_x+r_x)=\overline{r_x}$, then it is clear that $f(x)=0$ if and only if $x \in  \langle 1+2i \rangle$. The problem with this function is that it doesn't satisfy $f(x+y)=f(x)+f(y)$ and it is not surjective for if $r_x=f(x)=3 (5)$, then if $r_x=a+bi$, we have $a^2+b^2=3$, which is absurd.  I don't know what else to do, any suggestions would be appreciated. Integer solutions for $x^2-y^3 = 23$. As the title stated, I am wondering the integers $x,y$ that satisfy the equation $x^2-y^3 = 23$. I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the smoothing-out happen so relatively quickly?  $Z_2$:     $Z_3$:    (images shamelessly stolen from John D. Cook's blog: )  Why doesn't it take, say, four uniforms? Or five? Or...? Property of $\{0^n, 1^n, \ldots\}$ The differences between adjacent squares are the odd numbers:  0 1 4 9 16 25 …  1 3 5 7  9 …  The differences between adjacent odd numbers are 2 = 2!  I found that this truth is more general: If the differences between adjacent powers of $n$ are written out, and the differences of those differences etc, the $(n+1)$th sequence is {n!, n!, …}  Is this a well-known theorem? Does it have an easy proof? Enumerations on PHP I know that PHP doesn't yet have native Enumerations. But I have become accustomed to them from the Java world. I would love to use enums as a way to give predefined values which IDEs' auto-completion features could understand. Constants do the trick, but there's the namespace collision problem and (or actually because) they're global. Arrays don't have the namespace problem, but they're too vague, they can be overwritten at runtime and IDEs rarely know how to autofill their keys without additional static analysis annotations or attributes. Are there any solutions/workarounds you commonly use? Does anyone recall whether the PHP guys have had any thoughts or decisions around enumerations?