421 research outputs found
Quantum Circuits for the Unitary Permutation Problem
We consider the Unitary Permutation problem which consists, given unitary
gates and a permutation of , in
applying the unitary gates in the order specified by , i.e. in
performing . This problem has been
introduced and investigated by Colnaghi et al. where two models of computations
are considered. This first is the (standard) model of query complexity: the
complexity measure is the number of calls to any of the unitary gates in
a quantum circuit which solves the problem. The second model provides quantum
switches and treats unitary transformations as inputs of second order. In that
case the complexity measure is the number of quantum switches. In their paper,
Colnaghi et al. have shown that the problem can be solved within calls in
the query model and quantum switches in the new model. We
refine these results by proving that quantum switches
are necessary and sufficient to solve this problem, whereas calls
are sufficient to solve this problem in the standard quantum circuit model. We
prove, with an additional assumption on the family of gates used in the
circuits, that queries are required, for any
. The upper and lower bounds for the standard quantum circuit
model are established by pointing out connections with the permutation as
substring problem introduced by Karp.Comment: 8 pages, 5 figure
A Classical Sequential Growth Dynamics for Causal Sets
Starting from certain causality conditions and a discrete form of general
covariance, we derive a very general family of classically stochastic,
sequential growth dynamics for causal sets. The resulting theories provide a
relatively accessible ``half way house'' to full quantum gravity that possibly
contains the latter's classical limit (general relativity). Because they can be
expressed in terms of state models for an assembly of Ising spins living on the
relations of the causal set, these theories also illustrate how
non-gravitational matter can arise dynamically from the causal set without
having to be built in at the fundamental level. Additionally, our results bring
into focus some interpretive issues of importance for causal set dynamics, and
for quantum gravity more generally.Comment: 28 pages, 9 figures, LaTeX, added references and a footnote, minor
correction
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Partial Covering Arrays: Algorithms and Asymptotics
A covering array is an array with entries
in , for which every subarray contains each
-tuple of among its rows. Covering arrays find
application in interaction testing, including software and hardware testing,
advanced materials development, and biological systems. A central question is
to determine or bound , the minimum number of rows of
a . The well known bound
is not too far from being
asymptotically optimal. Sensible relaxations of the covering requirement arise
when (1) the set need only be contained among the rows
of at least of the subarrays and (2) the
rows of every subarray need only contain a (large) subset of . In this paper, using probabilistic methods, significant
improvements on the covering array upper bound are established for both
relaxations, and for the conjunction of the two. In each case, a randomized
algorithm constructs such arrays in expected polynomial time
High degree graphs contain large-star factors
We show that any finite simple graph with minimum degree contains a
spanning star forest in which every connected component is of size at least
. This settles a problem of J. Kratochvil
Maximum union-free subfamilies
An old problem of Moser asks: how large of a union-free subfamily does every
family of m sets have? A family of sets is called union-free if there are no
three distinct sets in the family such that the union of two of the sets is
equal to the third set. We show that every family of m sets contains a
union-free subfamily of size at least \lfloor \sqrt{4m+1}\rfloor - 1 and that
this bound is tight. This solves Moser's problem and proves a conjecture of
Erd\H{o}s and Shelah from 1972. More generally, a family of sets is
a-union-free if there are no a+1 distinct sets in the family such that one of
them is equal to the union of a others. We determine up to an absolute
multiplicative constant factor the size of the largest guaranteed a-union-free
subfamily of a family of m sets. Our result verifies in a strong form a
conjecture of Barat, F\"{u}redi, Kantor, Kim and Patkos.Comment: 10 page
Experimental Treatments for Spinal Cord Injury: What you Should Know
Experiencing a spinal cord injury (SCI) is extremely distressing, both physically and psychologically, and throws people into a complex, unfamiliar world of medical procedures, terminology, and decision making. You may have already had surgery to stabilize the spinal column and reduce the possibility of further damage. You are understandably distressed about the functions you may have lost below the level of spinal injury. You wish to recover any lost abilities as soon as possible. You, your family, or friends may have searched the Internet for treatments and cures
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