3,467 research outputs found
Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials
This paper analyzes to what extent it is possible to efficiently reduce the
number of clauses in NP-hard satisfiability problems, without changing the
answer. Upper and lower bounds are established using the concept of
kernelization. Existing results show that if NP is not contained in coNP/poly,
no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT
with d literals per clause, to equivalent instances with bits for
any e > 0. For the Not-All-Equal SAT problem, a compression to size
exists. We put these results in a common framework by analyzing
the compressibility of binary CSPs. We characterize constraint types based on
the minimum degree of multivariate polynomials whose roots correspond to the
satisfying assignments, obtaining (nearly) matching upper and lower bounds in
several settings. Our lower bounds show that not just the number of
constraints, but also the encoding size of individual constraints plays an
important role. For example, for Exact Satisfiability with unbounded clause
length it is possible to efficiently reduce the number of constraints to n+1,
yet no polynomial-time algorithm can reduce to an equivalent instance with
bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the
previous version did NOT satisfy requirement 4 of lemma 18 (the proof of
Claim 20 was incorrect
Raising the critical temperature by disorder in unconventional superconductors mediated by spin fluctuations
We propose a mechanism whereby disorder can enhance the transition
temperature Tc of an unconventional superconductor with pairing driven by
exchange of spin fluctuations. The theory is based on a self-consistent real
space treatment of pairing in the disordered one-band Hubbard model. It has
been demonstrated before that impurities can enhance pairing by softening the
spin fluctuations locally; here, we consider the competing effect of
pair-breaking by the screened Coulomb potential also present. We show that,
depending on the impurity potential strength and proximity to magnetic order,
this mechanism results in a weakening of the disorder-dependent Tc-suppression
rate expected from Abrikosov-Gor'kov theory, or even in disorder-generated Tc
enhancements.Comment: 6 pages, 4 figures + Supplementary Materia
Induction in stages for crossed products of C*-algebras by maximal coactions
Let B be a C*-algebra with a maximal coaction of a locally compact group G,
and let N and H be closed normal subgroups of G with N contained in H. We show
that the process Ind_(G/H)^G which uses Mansfield's bimodule to induce
representations of the crossed product of B by G from those of the restricted
crossed product of B by (G/H) is equivalent to the two-stage induction process:
Ind_(G/N)^G composed with Ind_(G/H)^(G/N). The proof involves a calculus of
symmetric imprimitivity bimodules which relates the bimodule tensor product to
the fibred product of the underlying spaces.Comment: 38 pages, LaTeX, uses Xy-pic; significant reorganization of previous
version; short section on regularity of induced representations adde
Full and reduced coactions of locally compact groups on C*-algebras
We survey the results required to pass between full and reduced coactions of
locally compact groups on C*-algebras, which say, roughly speaking, that one
can always do so without changing the crossed-product C*-algebra. Wherever
possible we use definitions and constructions that are well-documented and
accessible to non-experts, and otherwise we provide full details. We then give
a series of applications to illustrate the use of these techniques. We obtain
in particular a new version of Mansfield's imprimitivity theorem for full
coactions, and prove that it gives a natural isomorphism between
crossed-product functors defined on appropriate categories
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