Quotes About Invariants

Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or "invariant," patterns.

Relativity physics is especially interested in invariants, and it has discovered and named a few more. It is a common mistake to suppose that Einstein's theory of relativity asserts that everything is relative. Actually, it says: 'There are absolute things in the world but you must look deeply for them. The things that first present themselves to your notice are for the most part relative'.

You can use volatile variables only when all the following criteria are met: Writes to the variable do not depend on its current value, or you can ensure that only a single thread ever updates the value; The variable does not participate in invariants with other state variables; and Locking is not required for any other reason while the variable is being accessed.

When designing thread-safe classes, good object-oriented techniques—encapsulation, immutability, and clear specification of invariants—are your best friends.

An immutable object is one whose state cannot be changed after construction. Immutable objects are inherently thread-safe; their invariants are established by the constructor, and if their state cannot be changed, these invariants always hold.

If you could write down the solution to a differential equation," Yorke said, "then necessarily it's not chaotic, because to write it down, you must find regular invariants, things that are conserved, like angular momentum. You find enough of these things, and that lets you write down a solution. But this is exactly the way to eliminate the possibility of chaos.

Furthermore, biological functioning, sensorimotor intelligence, and rational thought are based on the same self-organizing processes (i.e., functional invariants).

In science there is and will remain a Platonic element which could not be taken away without ruining it. Among the infinite diversity of singular phenomena science can only look for invariants.