1,428 research outputs found
Permutations destroying arithmetic progressions in finite cyclic groups
A permutation \pi of an abelian group G is said to destroy arithmetic
progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that
is c-b=b-a and a,b,c are not all equal, then (\pi(a),\pi(b),\pi(c)) is not an
AP. In a paper from 2004, the first author conjectured that such a permutation
exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case
of a more general result, that such a permutation exists for all n >= n_0, for
some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct
such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).Comment: 11 pages, no figure
The Arithmetic of Word Ladders
The relationship between word games and mathematical recreations is well-known. Martin Gardner has often described them in his Mathematical Games column in the Scientific American. The editor of WORD WAYS, in his recent book BEYOND LANGUAGE, has derived logological structures from mathematical ones
The Meaning of Memory Safety
We give a rigorous characterization of what it means for a programming
language to be memory safe, capturing the intuition that memory safety supports
local reasoning about state. We formalize this principle in two ways. First, we
show how a small memory-safe language validates a noninterference property: a
program can neither affect nor be affected by unreachable parts of the state.
Second, we extend separation logic, a proof system for heap-manipulating
programs, with a memory-safe variant of its frame rule. The new rule is
stronger because it applies even when parts of the program are buggy or
malicious, but also weaker because it demands a stricter form of separation
between parts of the program state. We also consider a number of pragmatically
motivated variations on memory safety and the reasoning principles they
support. As an application of our characterization, we evaluate the security of
a previously proposed dynamic monitor for memory safety of heap-allocated data.Comment: POST'18 final versio
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