The text features a wealth of problems and exercises requiring students to prove or verify a statement or example, fill in certain details in a proof, or provide an intermediate step or counterexample
. (Ringgold, Inc., Portland, OR)
: The following passage is taken from Terry Pratchett's Making Money (2007: 307):
With Britain already planning its exit from Europe, Merkel was left with the task of keeping the continent's liberal democracies together, shoring up its weaker economies, taking in Syrian war refugees (and in so doing, expressing long-standing Western humanitarian values) and, importantly, providing a counterweight and counterexample
to a belligerent Russia and an increasingly influential China.
One of the major advantages of model checking over other formal methods its ability to generate a counterexample
when the model falsifies the specification.
Now we can add the counterexample
of oil-rich Venezuela, which shows the dark side of this paradox.
We proceed by simply giving a counterexample
. Take the following case:
# 1 to Young: Perpetual Growth Despite Diminishing Returns and Constant Savings Rate
It only takes one counterexample
to disprove a theory, but in the case of simple network effects our counterexample
cup "runneth over." Systematic research on online platforms by several authors, including one of us, shows considerable churn in leadership for online platforms over periods shorter than a decade.
I also suggest that some of these difficulties are due to the lack of a shared paradigm governing the impact of a counterexample
"GJ 1132b provides a hopeful counterexample
of an atmosphere that has endured for billion of years (that is, long enough for us to detect it).
1, the accurate solution roughly remains as a bang-bang form before the system enters steady state.