Which are the integer solutions of this function?
How to find solutions of linear Diophantine ax + by = c?
Can't change mouse influence in proportional editing mode I press O and I see it change to Proportional Editing mode, but when I press G and mouse wheel down/up, or arrow keys down/up, it doesn't change the influence area.    And I am, I press G and then I tried to scroll to change influence, but it's not working.    Done this aswell and it's not working Playing system sound without importing your own Why have the subjunctive and indicative converged in Modern English? So here is one I wish to know:   I use Tri-Gate second effect, which allows me to target and banish one card on the field. Then I link that with another monster to bring out another link. Tri-Gate would be in the graveyard but then I use transcodes effect (1 of the 3 co-links) to bring it back from the graveyard.   Can I use the second effect again? what exactly is mathematical rigor? I am a programmer moving up to computer science and even in my past life as a mechanical engineer I used sufficient maths to get by but I have always wondered what exactly is this thing called rigor.  Please make me understand it. Keeping an updated balance (Accounting) I have been struggling trying to find a way to properly and reliably maintain a balance for accounts in an order/payment system.  Currently, the process consists of three tables. Below example is very simplified but gets the point across.  Table one contains the account ID, name, total paid, total orders, balance etc.  Table two contains payments, date, the account ID (FK) and amount.  Table three contains orders, date, the account ID (FK) and amount.  Calculating the balance is of course as easy as  SELECT SUM(payments.AMOUNT) FROM payments WHERE ID = [account] SELECT SUM(orders.AMOUNT) FROM orders WHERE ID = [account]   Followed by some simple addition/subtraction math:  UPDATE accounts SET BALANCE = [balance] WHERE ID = [account]   All good - the problem is - how do I maintain this over let's assume years of activity and many, many transactions?  Currently the accounts table is updated every single time a transaction is made, meaning the above three statements run every time. There is of course an index on the ID column in all three tables, and everything is fast and neat for now, but I work with less than 100,000 transactions, meaning everything fits into the RAM and basically all I throw at the database gets executed immediately. Over time this will change, of course - so how do I handle this?  Is it better to mark the accounts as "dirty" and process balancing in a batch job later, or can I keep updating the balance real-time even when I reach the point where the orders and payments tables will not fit into the RAM?  Do I need to segment the orders and payments and "lock" everything before a given time, such as before last calculation? (I would rather not). I ask because then I could do a SUM WHERE DATE > [last lock timestamp] and update with UPDATE BALANCE = [balance] + [value for last lock] and prevent the problem from outgrowing resources.  I don't need a running balance for each transaction. That's all I can find results for when I try to research this problem. I just care about the total amount paid, the total amount ordered and the balance of accounts.  Why don't I calculate on the fly when I need the balance, you ask? Because one must be able to query the system and ask "Which of my (let's say) 50.000 accounts have a balance below 0?" - something we cannot do easily if we do not store the balance with every account. Is it possible to produce such design hints (on elements padding, spacing, width, height) automatically in photoshop? Is there any plugin that can potentially do this? why does IE need for me to click twice 32-bit, 64-bit CPU op-mode on Linux How can I tell if an ETF is expensive? For stocks, I can easily look at the price/earnings ratio, and, according to The Intelligent Investor, stand clear of those with a ratio higher than 20.  How can I tell if the current price of an ETF is a good price to buy it? Are there such simple metrics? Which Answer to Accept: Better Solution or Answer to Actual Question? Possible Duplicate:          I received two answers to a question I asked. One solution suggested I do things in a completely different manner than I have been. Another acknowledged that the first solution was correct but went on to explain how to do what I was asking.  Should I accept the first answer because it provides the better way of doing things or should I accept the second one because it answers the precise question that was asked? Full file date (without GNU utilities) I am trying to obtain the full date (created or modified) of a particular file for passing to another program. I have tried variations of options with the ls command but none provide a full date for files less than 6 months old and I have limit usage of the options. When I try certain options I have seen trying to research this I get the following message:  usage: ls [-1ACFHLNRabcdefgilmnopqrstuxEUX] [File...]   None of which seems to provide what I needs as far as I can tell so I tried to use the stat command but it is not available to me.  I am using the Korn shell on AIX 5.3 which has limited commands available, can anyone suggest another way that I might be able to obtain a file's created or modified date as a full date (either dd/mm/yyyy or yyyy/mm/dd). Regulatory bodies and authoritative dictionaries for English How can I read input from the console using the Scanner class in Java? I am a student who has taken the basic calculus courses but I am working through Calculus (the fourth edition) by Spivak in my spare time in order to both review the material and to gain a more rigorous understanding of the concepts. I am currently in Chapter 8 where he supplies a proof of the basis of the Intermediate Value Theorem, Theorem 7-1 (pp. 135-136 of the fourth edition). It seems mostly straightforward but I think there is some nuance of it that I don't grasp.  I am including in this post the statement Theorem 6-3 since he references it in the proof of Theorem 7-1 as well as part of Theorem 7-1 itself.     Theorem 6-3 Suppose $f$ is continuous at $a$, and $f(a) &gt; 0$. Then $f(x) &gt; 0$ for all $x$ in some interval containing $a$; more precisely, there is a number $\delta &gt; 0$ such that $f(x) &gt; 0$ for all $x$ satisfying $|x-a| &lt; \delta$. Similarly, if $f(a) &lt; 0$, then there is a number $\delta &gt; 0$ such that $f(x) &lt; 0$ for all $x$ satisfying $|x-a| &lt; \delta$.     Problem 6-16 is also referenced. However,it is the same thing as Theorem 6-3 for one-sided limits. The following proof is the Theorem 7-1 up through the paragraph which I don't fully understand.     Theorem 7-1: If $f$ is continuous on $[a,b]$ and $f(a) &lt; 0 &lt; f(b)$, then there is some number $z$ in $[a,b]$ such that $f(x) = 0$.  Proof: Define the set $A$ as follows:  $$A = \{x : a \le x\le b, \mbox{ and } f \mbox{ is negative on the interval } [a,x] \}.$$  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Clearly $A \ne \emptyset$, since $a$ is in $A$; in fact, there is some $\delta &gt; 0$ such that $A$ contains all points $x$ satisfying $a \le x &lt; a + \delta$; this follows from Problem 6-16, since $f$ is continuous on $[a,b]$ and $f(a)&lt;0$. Similarly, $b$ is an upper bound for $A$ and, in fact, there is a $\delta &gt; 0$ such that all points $x$ satisfying $b-\delta &lt; x \le b$ are upper bounds for $A$; this also follows from Problem 6-16, since $f(b) &gt; 0$. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; From these remarks it follows that $A$ has a least upper bound $\alpha$ and that $a &lt; \alpha &lt; b$. We now wish to show that $f(\alpha) = 0$, by eliminating the possibilities $f(\alpha) &lt; 0$ and $f(\alpha) &gt; 0$. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Suppose first that $f(\alpha) &lt; 0$. By Theorem 6-3, there is a $\delta &gt; 0$ such that $f(x) &lt; 0$ for $\alpha - \delta &lt; x &lt; \alpha + \delta$. Now there is some number $x_0$ in $A$ which satisfies $\alpha - \delta &lt; x_0 &lt; \alpha$ (because otherwise $\alpha$ would not be the least upper bound of $A$). This means that $f$ is negative on the whole interval $[a,x_0]$. But if $x_1$ is a number between $\alpha$ and $\alpha+\delta$, then $f$ is also negative on the whole interval $[x_0,x_1]$. Therefore $f$ is negative on the interval $[a,x_1]$, so $x_1$ is in $A$. But this contradicts the fact that $\alpha$ is an upper bound for $A$; our original assumption that $f(\alpha) &lt; 0$ must be false.    The point I don't understand is in the final paragraph of the excerpt. Why does Spivak split the interval $[a,b]$ up into the subintervals $\lbrack a,x_0 \rbrack$ and $[x_0,x_1]$? Isn't it true that, because $\alpha$ is a least upper bound of $A$ and $f$ is continuous on $[a,b]$, it must be true that if $a \le x &lt; \alpha$ then $f(x)&lt;0$? I understand the need to pick out the $x_1$ but not the $x_0$. Especially perplexing is the statement "there is some number $x_0$ in $A$ which satisfies $\alpha - \delta &lt; x_0 &lt; \alpha$ (because otherwise $\alpha$ would not be the least upper bound of $A$." Wouldn't it be true that if there were some $x_0$ such that $a \le x_0 &lt; \alpha$ and $f(x_0) \ge 0$ that $x_0$ would be an upper bound for $A$, contradicting that $\alpha$ is the least upper bound of $A$?  Thanks for any help. Suppose we view fluids classically, i.e., as a collection of molecules (with some finite size) interacting via e&amp;m and gravitational forces. Presumably we model fluids as continuous objects that satisfy some differential equation. What mathematical result says that modeling fluids as continuous objects can accurately predict the discrete behavior of the particles? I don't know anything about fluid mechanics, so my initial assumption may in itself be wrong. Let $\mathbb{S} = \{ (X,Y,Z) \in \mathbb{R}^3 :X^2 + Y^2+Z^2=1\}$ denote the Riemann sphere. The following problem is taken from Gamelin "Complex Analysis" chapter one:     What is the image of the spherical cap $ C := \{(X,Y,Z) \in \mathbb{S} : A \leq X \leq 1\}$, where $A$ is some fixed real number between $-1$ and $1$, under the stereographic projection $ \pi(X,Y,Z) = \frac{X+iY}{1-Z}$?   Here is my work so far:  ( The case $A=1$ is trivial, so assume $A &lt;1$)  We want to find $\pi(C) = \{z \in \mathbb{C} \mid \exists(X,Y,Z) \in C: \pi(X,Y,Z) = z \}= \{z \in \mathbb{C} \mid \pi^{-1}(z) \in C\}$, where the last equality holds by $\pi$ being a bijection. Write $z=x+iy$.  Here, $ \pi^{-1}(z) \in C\iff A\leq \frac{2x}{1+x^2+y^2} \leq 1 \iff (\star) :A \leq \frac{2x}{1+x^2+y^2}$, since the inequality $2x \leq 1+x^2+y^2$ holds for all $x,y \in \mathbb{R}$.  $A=0$: $( \star)$ reduces to $ 0 \leq x$, so in this case the image is $\{z \in \mathbb{C} \mid \Re(z) \geq 0 \}$.   $A &gt;0$: Since $A \neq 0$ in $(\star)$ we have,  $ \frac{2x}{1+x^2+y^2} \geq A \iff Ax^2-2x+Ay^2+A \leq 0 \iff (x-\frac{1}{A})^2 +y^2 \leq \frac{1}{A^2}-1$. Thus the image is the interior and boundary of the circle centered at $\frac{1}{A}$ with radius $\sqrt{\frac{1}{A^2}-1}$. Note $\frac{1}{A^2}-1 \geq 0$ as $A \in (0,1)$.  $A &lt;0$ : By symmetry and the above, here the image is $ \mathbb{C} / \{ $ interior and boundary of circle centered at  $  \frac{1}{A}$ with radius $\sqrt {\frac{1}{A^2}-1} \}$.  Is my working correct? Many Thanks! Recommended ways to upgrade AlwaysOn AG from 2014 to 2016 What is the recommended process of upgrading Availability Groups with one Sync Replica from SQL 2014 to 2016?  My understanding is that in-place upgrades are not really liked among database professionals. Is there a way to avoid in-place upgrades in Availability Groups? Is there a possibility to join a 2016 server into an existing 2014 availability group and to failover there, then "kill" one of the other instances and upgrade it? (e.g. by removing/re-installing SQL side-by-side on the same host)  Are there any other upgrade scenarios that we should consider?  The ideal solution would be that we always have 2 online replicas, and that we don't need to do any in-place upgrades. Shocking? - MacBook stubby earthed charger UK I have no idea why. But apple have not earthed the stubbly plug for the macbook pro transformer. So on the short lead the only way the charger is earthed is when I touch it or the macs unibody which gives you a sort of vibrating feeling on your palms.  When I went to the genius bar to complain about it they said that it was completely 'normal'. Where their definition of normal is that every MacBook does it. I have had 4 MacBooks and all have done it.    The genius did say that he though he has seen a few earthed 3pin plugs. So my question is where can I pick one up? or how could we force apple to earth the stubby plugs and recall them? Map of with object references as keys?