2,073 research outputs found
Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication
Understanding the query complexity for testing linear-invariant properties
has been a central open problem in the study of algebraic property testing.
Triangle-freeness in Boolean functions is a simple property whose testing
complexity is unknown. Three Boolean functions , and are said to be triangle free if there is no such that . This property
is known to be strongly testable (Green 2005), but the number of queries needed
is upper-bounded only by a tower of twos whose height is polynomial in 1 /
\epsislon, where \epsislon is the distance between the tested function
triple and triangle-freeness, i.e., the minimum fraction of function values
that need to be modified to make the triple triangle free. A lower bound of for any one-sided tester was given by Bhattacharyya and
Xie (2010). In this work we improve this bound to .
Interestingly, we prove this by way of a combinatorial construction called
\emph{uniquely solvable puzzles} that was at the heart of Coppersmith and
Winograd's renowned matrix multiplication algorithm
New Lower Bounds for van der Waerden Numbers Using Distributed Computing
This paper provides new lower bounds for van der Waerden numbers. The number
is defined to be the smallest integer for which any -coloring
of the integers admits monochromatic arithmetic progression of
length ; its existence is implied by van der Waerden's Theorem. We exhibit
-colorings of that do not contain monochromatic arithmetic
progressions of length to prove that . These colorings are
constructed using existing techniques. Rabung's method, given a prime and a
primitive root , applies a color given by the discrete logarithm base
mod and concatenates copies. We also used Herwig et al's
Cyclic Zipper Method, which doubles or quadruples the length of a coloring,
with the faster check of Rabung and Lotts. We were able to check larger primes
than previous results, employing around 2 teraflops of computing power for 12
months through distributed computing by over 500 volunteers. This allowed us to
check all primes through 950 million, compared to 10 million by Rabung and
Lotts. Our lower bounds appear to grow roughly exponentially in . Given that
these constructions produce tight lower bounds for known van der Waerden
numbers, this data suggests that exact van der Waerden Numbers grow
exponentially in with ratio asymptotically, which is a new conjecture,
according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader
comment
Asymptotic entanglement transformation between W and GHZ states
We investigate entanglement transformations with stochastic local operations
and classical communication (SLOCC) in an asymptotic setting using the concepts
of degeneration and border rank of tensors from algebraic complexity theory.
Results well-known in that field imply that GHZ states can be transformed into
W states at rate 1 for any number of parties. As a generalization, we find that
the asymptotic conversion rate from GHZ states to Dicke states is bounded as
the number of subsystems increase and the number of excitations is fixed. By
generalizing constructions of Coppersmith and Winograd and by using monotones
introduced by Strassen we also compute the conversion rate from W to GHZ
states.Comment: 11 page
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