Optimizing Kohana-based Websites for Speed and Scalability

A site I built with Kohana was slammed with an enormous amount of traffic yesterday, causing me to take a step back and evaluate some of the design. I'm curious what are some standard techniques for optimizing Kohana-based applications?

Fastest way to determine if an integer"s square root is an integer

I'm looking for the fastest way to determine if a long value is a perfect square (i.e. its square root is another integer). I've done it the easy way, by using the built-in Math.sqrt() function, but I'm wondering if there is a way to do it faster by restricting yourself to integer-only domain. Maintaining a lookup table is impratical (since there are about 2 ^31.5 integers whose square is less than 2 ⁶³ ).

Numerical library for Scala

I'm looking for a library to do numerical computing in Scala (or Java, although something that can use scala functions would be way nicer!) with at least the following capabilities:

Is using inline JavaScript preferred to an external include if the script is really short?

I use External JavaScripts in a website as I always try to keep JavaScript at bottom and external.

Improving MySQL tables with Indexes

I am very new to Indexes in MySQL. I know, I should probably have leart it earlier, but most projects been small enough for me to get away with out it ;)

Obsolete Java Optimization Tips

There are number of performance tips made obsolete by Java compiler and especially Profile-guided optimization . For example, these platform-provided optimizations can drastically (according to sources) reduces the cost of virtual function calls. VM is also capable of method inlining, loop unrolling etc.