254 research outputs found
The Satisfiability Threshold for k-XORSAT
We consider "unconstrained" random -XORSAT, which is a uniformly random
system of linear non-homogeneous equations in over
variables, each equation containing variables, and also consider a
"constrained" model where every variable appears in at least two equations.
Dubois and Mandler proved that is a sharp threshold for satisfiability
of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform
hypergraph to extend this result to find the threshold for unconstrained
3-XORSAT.
We show that remains a sharp threshold for satisfiability of
constrained -XORSAT for every , and we use standard results on the
2-core of a random -uniform hypergraph to extend this result to find the
threshold for unconstrained -XORSAT. For constrained -XORSAT we narrow
the phase transition window, showing that implies almost-sure
satisfiability, while implies almost-sure unsatisfiability.Comment: Version 2 adds sharper phase transition result, new citation in
literature survey, and improvements in presentation; removes Appendix
treating k=
The Density Matrix Renormalization Group and the Nuclear Shell Model
We summarize recent efforts to develop an angular-momentum-conserving variant
of the Density Matrix Renormalization Group method into a practical truncation
strategy for large-scale shell model calculations of atomic nuclei. Following a
brief description of the key elements of the method, we report the results of
test calculations for Cr and Ni. In both cases we consider
nucleons limited to the 2p-1f shell and interacting via the KB3 interaction.
Both calculations produce a high level of agreement with the exact shell-model
results. Furthermore, and most importantly, the fraction of the complete space
required to achieve this high level of agreement goes down rapidly as the size
of the full space grows
Density Matrix Renormalization Group study of Cr and Ni
We discuss the development of an angular-momentum-conserving variant of the
Density Matrix Renormalization Group (DMRG) method for use in large-scale
shell-model calculations of atomic nuclei and report a first application of the
method to the ground state of Ni and improved results for Cr. In
both cases, we see a high level of agreement with the exact results. A
comparison of the two shows a dramatic reduction in the fraction of the space
required to achieve accuracy as the size of the problem grows.Comment: 4 pages. Published in PRC Rapi
How many random questions are necessary to identify n distinct objects?
AbstractSuppose that X and A are two finite sets of the same cardinality n ⩾ 2. Assume that there is a bijective mapping φ: X → A which is unknown to us, and we must determine it. We are allowed to ask a sequence of questions each posed as follows. For a given B ⊂ A what is φ−1(B)? In this paper we study a case when the subsets B are chosen uniformly at random. The main result is: if each subset has to split all the atoms of a field generated by the previous subsets, then the total number of questions (needed to determine the mapping completely) is log2 n + (1 + op(1))(2 log2 n)12. Here op(1) stands for a random term approaching 0 in probability as n → ∞
k-core organization of complex networks
We analytically describe the architecture of randomly damaged uncorrelated
networks as a set of successively enclosed substructures -- k-cores. The k-core
is the largest subgraph where vertices have at least k interconnections. We
find the structure of k-cores, their sizes, and their birth points -- the
bootstrap percolation thresholds. We show that in networks with a finite mean
number z_2 of the second-nearest neighbors, the emergence of a k-core is a
hybrid phase transition. In contrast, if z_2 diverges, the networks contain an
infinite sequence of k-cores which are ultra-robust against random damage.Comment: 5 pages, 3 figure
Systematic study of proton-neutron pairing correlations in the nuclear shell model
A shell-model study of proton-neutron pairing in shell nuclei using a
parametrized hamiltonian that includes deformation and spin-orbit effects as
well as isoscalar and isovector pairing is reported. By working in a
shell-model framework we are able to assess the role of the various modes of
proton-neutron pairing in the presence of nuclear deformation without violating
symmetries. Results are presented for Ti, Ti, Ti, V
and Cr to assess how proton-neutron pair correlations emerge under
different scenarios. We also study how the presence of a one-body spin-obit
interaction affects the contribution of the various pairing modes.Comment: 12 pages, 16 figure
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