15,744 research outputs found
Optimal Staged Self-Assembly of General Shapes
We analyze the number of tile types , bins , and stages necessary to
assemble squares and scaled shapes in the staged tile assembly
model. For squares, we prove stages suffice and
are necessary for almost all .
For shapes with Kolmogorov complexity , we prove
stages suffice and are necessary to
assemble a scaled version of , for almost all . We obtain similarly tight
bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201
Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture
Most, if not all, unconditional results towards the abc-conjecture rely
ultimately on classical Baker's method. In this article, we turn our attention
to its elliptic analogue. Using the elliptic Baker's method, we have recently
obtained a new upper bound for the height of the S-integral points on an
elliptic curve. This bound depends on some parameters related to the
Mordell-Weil group of the curve. We deduce here a bound relying on the
conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable
quantities. We then study which abc-type inequality over number fields could be
derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2,
three references added ([Mas75], [MB90] and [Yu94]) and one reference updated
[BS12]. Accepted in Bull. Brazil. Mat. So
Exploring pure quantum states with maximally mixed reductions
We investigate multipartite entanglement for composite quantum systems in a
pure state. Using the generalized Bloch representation for n-qubit states, we
express the condition that all k-qubit reductions of the whole system are
maximally mixed, reflecting maximum bipartite entanglement across all k vs. n-k
bipartitions. As a special case, we examine the class of balanced pure states,
which are constructed from a subset of the Pauli group P_n that is isomorphic
to Z_2^n. This makes a connection with the theory of quantum error-correcting
codes and provides bounds on the largest allowed k for fixed n. In particular,
the ratio k/n can be lower and upper bounded in the asymptotic regime, implying
that there must exist multipartite entangled states with at least k=0.189 n
when . We also analyze symmetric states as another natural class
of states with high multipartite entanglement and prove that, surprisingly,
they cannot have all maximally mixed k-qubit reductions with k>1. Thus,
measured through bipartite entanglement across all bipartitions, symmetric
states cannot exhibit large entanglement. However, we show that the permutation
symmetry only constrains some components of the generalized Bloch vector, so
that very specific patterns in this vector may be allowed even though k>1 is
forbidden. This is illustrated numerically for a few symmetric states that
maximize geometric entanglement, revealing some interesting structures.Comment: 10 pages, 2 figure
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
We formulate and give partial answers to several combinatorial problems on
volumes of simplices determined by points in 3-space, and in general in
dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by
points in \RR^3 is at most , and there are point sets
for which this number is . We also present an time
algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby
extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for
every k,d\in \NN, , the maximum number of -dimensional
simplices of minimum (nonzero) volume spanned by points in \RR^d is
. (ii) The number of unit-volume tetrahedra determined by
points in \RR^3 is , and there are point sets for which this
number is . (iii) For every d\in \NN, the minimum
number of distinct volumes of all full-dimensional simplices determined by
points in \RR^d, not all on a hyperplane, is .Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings
of the ACM-SIAM Symposium on Discrete Algorithms, 200
Bounding sequence extremal functions with formations
An -formation is a concatenation of permutations of letters.
If is a sequence with distinct letters, then let be
the maximum length of any -sparse sequence with distinct letters which
has no subsequence isomorphic to . For every sequence define
, the formation width of , to be the minimum for which
there exists such that there is a subsequence isomorphic to in every
-formation. We use to prove upper bounds on
for sequences such that contains an alternation
with the same formation width as .
We generalize Nivasch's bounds on by showing that
and for every and , such that denotes the inverse Ackermann function.
Upper bounds on have been used in other
papers to bound the maximum number of edges in -quasiplanar graphs on
vertices with no pair of edges intersecting in more than points.
If is any sequence of the form such that is a letter,
is a nonempty sequence excluding with no repeated letters and is
obtained from by only moving the first letter of to another place in
, then we show that and . Furthermore we prove that
and for every .Comment: 25 page
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