382 research outputs found
On the asymptotic minimum number of monochromatic 3-term arithmetic progressions
Let V(n) be the minimum number of monochromatic 3-term arithmetic
progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2
(1+o(1)) <= V(n) <= (117/2192) n^2(1+o(1)). As a consequence, we find that V(n)
is strictly greater than the corresponding number for Schur triples (which is
(1/22) n^2 (1+o(1)). Additionally, we disprove the conjecture that V(n) =
(1/16) n^2(1+o(1)), as well as a more general conjecture.Comment: 9 pages. Revised version fixes formatting errors (same text
Lower bounds for Max-Cut in -free graphs via semidefinite programming
For a graph , let denote the size of the maximum cut in . The
problem of estimating as a function of the number of vertices and edges
of has a long history and was extensively studied in the last fifty years.
In this paper we propose an approach, based on semidefinite programming (SDP),
to prove lower bounds on . We use this approach to find large cuts in
graphs with few triangles and in -free graphs.Comment: 21 pages, to be published in LATIN 2020 proceedings, Updated version
is rewritten to include additional results along with corrections to original
argument
Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities
Quantum theory imposes a strict limit on the strength of non-local
correlations. It only allows for a violation of the CHSH inequality up to the
value 2 sqrt(2), known as Tsirelson's bound. In this note, we consider
generalized CHSH inequalities based on many measurement settings with two
possible measurement outcomes each. We demonstrate how to prove Tsirelson
bounds for any such generalized CHSH inequality using semidefinite programming.
As an example, we show that for any shared entangled state and observables
X_1,...,X_n and Y_1,...,Y_n with eigenvalues +/- 1 we have | + <X_2
Y_1> + + + ... + - | <= 2 n
cos(pi/(2n)). It is well known that there exist observables such that equality
can be achieved. However, we show that these are indeed optimal. Our approach
can easily be generalized to other inequalities for such observables.Comment: 9 pages, LateX, V2: Updated reference [3]. To appear in Physical
Review
Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix
The positive semidefinite rank of a nonnegative -matrix~ is
the minimum number~ such that there exist positive semidefinite -matrices , such that S(k,\ell) =
\mbox{tr}(A_k^* B_\ell).
The most important, lower bound technique for nonnegative rank is solely
based on the support of the matrix S, i.e., its zero/non-zero pattern. In this
paper, we characterize the power of lower bounds on positive semidefinite rank
based on solely on the support.Comment: 9 page
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