Is the following relation true: $$ \max \{a_1+b_1, ..., a_n+b_n\}\leq \max \{a_1,...,a_n\}+\max \{b_1,...,b_n\} $$ $\{a_1,...,a_n\}$ and $\{b_1,...,b_n\}$ are real numbers?  question is similar, but it seems focused on non-negative numbers. Here $\{a_1,...,a_n\}$ and $\{b_1,...,b_n\}$ can also be negative.
I'm looking for an elegant way to show that, among non-negative numbers, $$ \max \{a_1 + b_1, \dots, a_n + b_n\} \leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\} $$  I can show that $\max \{a+b, c+d\} \leq \max \{a,c\} + \max \{b,d\}$ by exhaustively checking all possibilities of orderings among $a,c$ and $b,d$.  But, I feel like there should be a more intuitive/efficient way to show this property for arbitrary sums like the one above.
Find the equation of the circle passing through two points (4,4) and (-2.-2) with radius 4 I am trying to solve this question but i cannot seem to find the center (h,k). I know i have to use the equation of circle form '(x-h)^2+(x-k)^2=r^2' and after that form to solution of the 2 point given and simultaneous equations them. I got the answer for that which is h+k=2 and that is where i got stuck. Please anyone help me with this. i also post my working as well. Is there a hook that runs after a user logs in? I am writing a plugin that fetches some extended user info from a remote service and I need it to execute its function each time a user logs in.  Is there a hook that gets fired after login that I can add an action to? How to loop the particle simulation indefinitely? I've made a simple snowfall particle simulation (using gandalf3's solution: ). Is there a way to loop the animation, so the transition between last frame and first frame is not visible? Is there a setting to do it? I'm looking for ways to speed up the combat in my D&amp;D 3.5 game. It seems that turns - both player and non-player - simply take too long. I have thought about giving everyone a 30 second timer, but I'm afraid that might inhibit their ability to come up with strategy and tactics. An inequality for arctan$x$ Why it is true that $|\arctan x-\arctan y|\leq |x-y|$? I have tried to use $\tan x$ but I didn't conclude to something. When should I use "when" and "while"? But you can't find anything while you're crying.   But you can't find anything when you're crying.   I'll tell you about it while Frank saddles the horse.   I'll tell you about it when Frank saddles the horse.     When should I use when, and when while? Are these sentences correct? Replacing a font in many illustrator files I have several hundred Illustrator files that are formatted similarly, and for legal reasons I need to replace a font in all of them. Even with the 'Find font' tool, that's a hell of a lot of fonts to replace by hand.  Is there some way of automating it? For example, using Bridge? Illustrator (CS5) doesn't offer many options in Bridge, compared to the automation options offered by Photoshop. I also don't see any option for running actions on multiple Illustrator files.  Also, creating a repeatable action for replacing the font doesn't seem to work: instead of repeating my action, it brings up the dialog box an waits for me to tell it to replace them.  As a programmer I'm prepared to edit the files directly if it can be done without breaking them.    Update: The 'Batch...' menu item from Illustrator's actions panel menu allows the same action to be performed on a folder of files. It doesn't fix the Find Fonts action waiting for input though. Toggling the dialog off makes the action do nothing at all. cd in bash script without alias, function, source I've written a bash script which should cd me to a directory.   My problem is that only the directory in the subshell changes.  I've read of many similar problems to this, but I want to know if there is a solution besides using an alias, a function or sourcing the script.  If you don't understand what I mean, here's an example:  user@linux ~: cat ./myscript.sh #!/bin/bash cd /any/directory user@linux ~: ./myscript.sh user@linux ~:    Please note that my script is much longer, so I don't want to use a function! I have a Laars gas boiler that has 3 heating zones. I want to replace the old mercury honeywell tstat on the 1st floor (heat only) with a newer honeywell RTH9580 wifi model. I've taken pictures of my entire system, labeled the wires in the pics to indicate the associated connections, and included the Laars wiring diagram.  There is a transformer mounted on the wall near the boiler that is wired in-line to the system (power from the panel --> into the emergency boiler switch --> into the wall transformer --> then to the boiler (low water cut off and boiler circuitry).  There is also a transformer within the boiler circuitry itself.  Can someone tell me the best method (proper preferred, not necessarily easiest) to connect the C wire? The Weyl equidistribution theorem states that the sequence of fractional parts ${n \xi}$, $n = 0, 1, 2, \dots$ is uniformly distributed for $\xi$ irrational.   This can be proved using a bit of ergodic theory, specifically the fact that an irrational rotation is uniquely ergodic with respect to Lebesgue measure. It can also be proved by simply playing with trigonometric polynomials (i.e., polynomials in $e^{2\pi i k x}$ for $k$ an integer) and using the fact they are dense in the space of all continuous functions with period 1.  In particular, one shows that if $f(x)$ is a continuous function with period 1, then for any $t$,  $\int_0^1 f(x) dx = \lim \frac{1}{N} \sum_{i=0}^{N-1} f(t+i \xi)$. One shows this by checking this (directly) for trigonometric polynomials via the geometric series.  This is a very elementary and nice proof.  The general form of Weyl's theorem states that if $p$ is a monic integer-valued polynomial, then the sequence ${p(n \xi)}$ for $\xi$ irrational is uniformly distributed modulo 1.  I believe this can be proved using extensions of these ergodic theory techniques -- it's an exercise in Katok and Hasselblatt.  I'd like to see an elementary proof.  Can the general form of Weyl's theorem be proved using the same elementary techniques as in the basic version? What species of insect is this? I saw this insect near a beach in Corsica France: List of PDOStatement::bindParam data_type parameters Is there an integer $N>0$ such that $\varphi(n) = N$ has infinitely many solutions? Let $\varphi: \mathbb{N} \to \mathbb{N}$ be the totient function.   Is there an integer $N &gt; 0$ such that there are infinitely many integers $n &gt; 0$ such that $$\varphi(n) = N?$$ If $U$ is unitary operator then spectrum $\sigma(U)$ is contained inside the unit circle In a Hilbert space, let $U$ be a continuous operator which it unitary. Prove $\sigma (U)\subseteq \Bbb{S}^1$.   It is important for me to know how I am doing, and I didn't come by a clear explanation so it would be appreciated to have your correcting and guiding.  Attempt: $U^{*}U=UU^{*}=I$ and therefore $U^{-1}=U$. I have $$ \| U x \|^2 = \langle U x, U x \rangle = \langle U^* U x, x \rangle = \langle x, x \rangle = \|x\|^2 $$ and the same goes for $U^*$ and therefore $\| U \| = \| U^{-1} \| = 1$.   Looking at $U-\lambda$ I can take $$ -\lambda U(U^{-1}-{1\over \lambda}I) =U-\lambda I. $$ Clearly, for $\lambda \ne 0$, $-\lambda U$ is defined and is a linear operator. $(U^{-1}-{1\over \lambda}I)$ is invertible if $|\lambda|\ge 1$ and therefore $\sigma (A^{-1})=\sigma (A)\subseteq \{|z|\le1\}$. But from the RHS, $U-\lambda I$ is invertible if $|\lambda|\le 1$ and therefore $\sigma(A)\subseteq \Bbb{S}^1$. I have noticed my claims tend to have holes in them many times which I cannot spot. If it is substantially true, how can I make sure it is completely formal? Russian babel and chemformula What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$  I tried to factorize $n^3+100$, but $100$ is not a perfect cube. I wish it were $1000$. Structure of a C++ Object in Memory Vs a Struct If I have a class as follows     class Example_Class     {        private:          int x;           int y;         public:           Example_Class()           {               x = 8;              y = 9;          }        ~Example_Class()         { }     };   And a struct as follows  struct {    int x;    int y; } example_struct;   Is the structure in memory of the  example_struct simmilar to that in Example_Class  for example if I do the following  struct example_struct foo_struct; Example_Class foo_class = Example_Class();  memcpy(&amp;foo_struct, &amp;foo_class, sizeof(foo_struct));   will foo_struct.x = 8 and foo_struct.y = 9 (ie: the same values as the x,y values in the foo_class) ?  The reason I ask is I have a C++ library (don't want to change it) that is sharing an object with C code and I want to use a struct to represent the object coming from the C++ library. I'm only interested in the attributes of the object.  I know the ideal situation would be to have Example_class wrap arround a common structure between the C and C++ code but it is not going to be easy to change the C++ library in use. "The following" vs. "Following" I wrote      "Following examples show ..."   and my colleague revised it as      "The following examples show ..."   Why do we need "the" here even though the examples have not been mentioned before? We know that for a position variable $x$ and momentum $p$, the uncertainties of the two quantities are bounded by  $$\Delta x \Delta p \gtrsim \hbar$$  Now, this is usually first explained with $x$ being a simple linearly measured position and $p$ being linear momentum.  But it should apply to any good coordinate and its conjugate momentum.  It should, for instance, apply to angle $\phi$ about the $z$ axis, and angular momentum $L_z$:  $$\Delta \phi \Delta L_z \gtrsim \hbar$$  The thing is, $\Delta \phi$ can never be greater than $2\pi$.  I mean, you have to have some value of $\phi$ and $\phi$ only runs from 0 to $2\pi$.  Therefore  $$\Delta L_z \gtrsim \hbar/\Delta \phi \geq \hbar/2\pi$$  But, uh-oh!  This means it is impossible for $\Delta L_z$ to be zero, and we should never be able to have angular momentum states with definite $L_z$ values.  Of course, it doesn't mean that.  But I have never figured out how this is not in contradiction with the Schroedinger eqn. calculations that give us states with definite values of $L_z$.  Can anyone help me out?  One answer I anticipate is that $\phi$ is sort of "abstract" in that if you chose your origin at some other point you will get completely different values of $\phi$ and $L_z$, and ipso facto, usual considerations don't apply.  I don't think this will work, though.  Consider a "quantum bead" sliding around on a rigid circular ring and you get the exact same problem with no ambiguity in $\phi$ or $L_z$.  (Well, there will be some limited ambiguity in $\phi$, but still, there won't be in $L_z$.) I read this quite a while back, part of my parents library (they're in their 50s now) I think the book was probably published in the 70s or 80s, my copy was paperback. I don't recall the cover art, but I have vague recollections of a satellite on the cover and maybe blue with grid-lines below it?  I've made concerted efforts to find it again but to no avail. My parents don't remember it either, but mum's an avowed book-aholic and buys a lot of books, reads them once and forgets about them, so that's not so surprising.  The key parts I remember:   Both the Soviets and Americans were working on psychic techniques and technologies, with real and tangible results including Remote Viewing. MK-ULTRA was explicitly namedropped in relation to the American psychic program. The Soviets and Americans both had orbiting superweapons, One was a microwave laser (or something very similar) and the other used slivers of the brains of psychics to blow people's brains up in a wide area. A super-psychic (male) took offence to the state of the cold-war and took control over the orbital weapons by manipulating the onboard computers psychically. There's a gruesome description of a psychic weapon being tested on monkeys in a lab, the POV character throws up afterwards in horror. I think there might have been more than one POV character. There's an extended description of the psychic orbital weapon as experienced by people in its area of effect. Similar to the monkeys, it builds up, starting with alzheimer-esque hand-shaking and building up to seizures and screaming before their brain liquifies fully and either comes out the nose and ears or outright blows the skull open. (this is one of the more vivid recollections I have of the book) I remember that the super-psychic was working wholly alone and did most of his psychic stuff from motels halfway around the world from events. I believe the super-psychic also killed other psychics explicitly. The weapon satellites had code-names Both weapons were used on cities in the course of the story I vaguely recall an office building getting targeted specifically at one point.