825 research outputs found
Root finding with threshold circuits
We show that for any constant d, complex roots of degree d univariate
rational (or Gaussian rational) polynomials---given by a list of coefficients
in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a
uniform family of constant-depth polynomial-size threshold circuits). The basic
idea is to compute the inverse function of the polynomial by a power series. We
also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur
Evaluating Matrix Circuits
The circuit evaluation problem (also known as the compressed word problem)
for finitely generated linear groups is studied. The best upper bound for this
problem is , which is shown by a reduction to polynomial
identity testing. Conversely, the compressed word problem for the linear group
is equivalent to polynomial identity testing. In
the paper, it is shown that the compressed word problem for every finitely
generated nilpotent group is in . Within
the larger class of polycyclic groups we find examples where the compressed
word problem is at least as hard as polynomial identity testing for skew
arithmetic circuits
Conjugacy in Baumslag's group, generic case complexity, and division in power circuits
The conjugacy problem belongs to algorithmic group theory. It is the
following question: given two words x, y over generators of a fixed group G,
decide whether x and y are conjugated, i.e., whether there exists some z such
that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word
problem, in general. We investigate the complexity of the conjugacy problem for
two prominent groups: the Baumslag-Solitar group BS(1,2) and the
Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is
TC^0-complete. To the best of our knowledge BS(1,2) is the first natural
infinite non-commutative group where such a precise and low complexity is
shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that
the conjugacy problem is decidable (which has been known before); but our
results go far beyond decidability. In particular, we are able to show that
conjugacy in G(1,2) can be solved in polynomial time in a strongly generic
setting. This means that essentially for all inputs conjugacy in G(1,2) can be
decided efficiently. In contrast, we show that under a plausible assumption the
average case complexity of the same problem is non-elementary. Moreover, we
provide a lower bound for the conjugacy problem in G(1,2) by reducing the
division problem in power circuits to the conjugacy problem in G(1,2). The
complexity of the division problem in power circuits is an open and interesting
problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 =
{\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the
base group H if and only if A \neq H \neq
Theories for TC0 and Other Small Complexity Classes
We present a general method for introducing finitely axiomatizable "minimal"
two-sorted theories for various subclasses of P (problems solvable in
polynomial time). The two sorts are natural numbers and finite sets of natural
numbers. The latter are essentially the finite binary strings, which provide a
natural domain for defining the functions and sets in small complexity classes.
We concentrate on the complexity class TC^0, whose problems are defined by
uniform polynomial-size families of bounded-depth Boolean circuits with
majority gates. We present an elegant theory VTC^0 in which the provably-total
functions are those associated with TC^0, and then prove that VTC^0 is
"isomorphic" to a different-looking single-sorted theory introduced by
Johannsen and Pollet. The most technical part of the isomorphism proof is
defining binary number multiplication in terms a bit-counting function, and
showing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc
Bose-Einstein condensation in an optical lattice
In this paper we develop an analytic expression for the critical temperature
for a gas of ideal bosons in a combined harmonic lattice potential, relevant to
current experiments using optical lattices. We give corrections to the critical
temperature arising from effective mass modifications of the low energy
spectrum, finite size effects and excited band states. We compute the critical
temperature using numerical methods and compare to our analytic result. We
study condensation in an optical lattice over a wide parameter regime and
demonstrate that the critical temperature can be increased or reduced relative
to the purely harmonic case by adjusting the harmonic trap frequency. We show
that a simple numerical procedure based on a piecewise analytic density of
states provides an accurate prediction for the critical temperature.Comment: 10 pages, 5 figure
Disorder-induced topological change of the superconducting gap structure in iron pnictides
In superconductors with unconventional pairing mechanisms, the energy gap in
the excitation spectrum often has nodes, which allow quasiparticle excitations
at low energies. In many cases, e.g. -wave cuprate superconductors, the
position and topology of nodes are imposed by the symmetry, and thus the
presence of gapless excitations is protected against disorder. Here we report
on the observation of distinct changes in the gap structure of iron-pnictide
superconductors with increasing impurity scattering. By the successive
introduction of nonmagnetic point defects into BaFe(AsP)
crystals via electron irradiation, we find from the low-temperature penetration
depth measurements that the nodal state changes to a nodeless state with fully
gapped excitations. Moreover, under further irradiation the gapped state
evolves into another gapless state, providing bulk evidence of unconventional
sign-changing -wave superconductivity. This demonstrates that the topology
of the superconducting gap can be controlled by disorder, which is a strikingly
unique feature of iron pnictides.Comment: 5 pages, 4 figure
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