1,005 research outputs found
Gradual sub-lattice reduction and a new complexity for factoring polynomials
We present a lattice algorithm specifically designed for some classical
applications of lattice reduction. The applications are for lattice bases with
a generalized knapsack-type structure, where the target vectors are boundably
short. For such applications, the complexity of the algorithm improves
traditional lattice reduction by replacing some dependence on the bit-length of
the input vectors by some dependence on the bound for the output vectors. If
the bit-length of the target vectors is unrelated to the bit-length of the
input, then our algorithm is only linear in the bit-length of the input
entries, which is an improvement over the quadratic complexity floating-point
LLL algorithms. To illustrate the usefulness of this algorithm we show that a
direct application to factoring univariate polynomials over the integers leads
to the first complexity bound improvement since 1984. A second application is
algebraic number reconstruction, where a new complexity bound is obtained as
well
The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer
A quantum computer can efficiently find the order of an element in a group,
factors of composite integers, discrete logarithms, stabilisers in Abelian
groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already
known how to phrase the first four problems as the estimation of eigenvalues of
certain unitary operators. Here we show how the solution to the more general
Abelian `hidden subgroup problem' can also be described and analysed as such.
We then point out how certain instances of these problems can be solved with
only one control qubit, or `flying qubits', instead of entire registers of
control qubits.Comment: 16 pages, 3 figures, LaTeX2e, to appear in Proceedings of the 1st
NASA International Conference on Quantum Computing and Quantum Communication
(Springer-Verlag
Dynamics of modal power distribution in a multimode semiconductor laser with optical feedback
The dynamics of power distribution between longitudinal modes of a multimode
semiconductor laser subjected to external optical feedback is experimentally
analyzed in the low-frequency fluctuation regime. Power dropouts in the total
light intensity are invariably accompanied by sudden activations of several
longitudinal modes. These activations are seen not to be simultaneous to the
dropouts, but to occur after them. The phenomenon is statistically analysed in
a systematic way, and the corresponding delay is estimated.Comment: 3 pages, 4 figures, revte
Pattern formation driven by nematic ordering of assembling biopolymers
The biopolymers actin and microtubules are often in an ongoing
assembling/disassembling state far from thermal equilibrium. Above a critical
density this leads to spatially periodic patterns, as shown by a scaling
argument and in terms of a phenomenological continuum model, that meets also
Onsager's statistical theory of the nematic--to--isotropic transition in the
absence of reaction kinetics.
This pattern forming process depends much on nonlinear effects and a common
linear stability analysis of the isotropic distribution of the filaments is
often misleading. The wave number of the pattern decreases with the
assembling/disassembling rate and there is an uncommon discontinuous transition
between the nematic and the periodic state.Comment: 4 pages, 3 figure
On the String Consensus Problem and the Manhattan Sequence Consensus Problem
In the Manhattan Sequence Consensus problem (MSC problem) we are given
integer sequences, each of length , and we are to find an integer sequence
of length (called a consensus sequence), such that the maximum
Manhattan distance of from each of the input sequences is minimized. For
binary sequences Manhattan distance coincides with Hamming distance, hence in
this case the string consensus problem (also called string center problem or
closest string problem) is a special case of MSC. Our main result is a
practically efficient -time algorithm solving MSC for sequences.
Practicality of our algorithms has been verified experimentally. It improves
upon the quadratic algorithm by Amir et al.\ (SPIRE 2012) for string consensus
problem for binary strings. Similarly as in Amir's algorithm we use a
column-based framework. We replace the implied general integer linear
programming by its easy special cases, due to combinatorial properties of the
MSC for . We also show that for a general parameter any instance
can be reduced in linear time to a kernel of size , so the problem is
fixed-parameter tractable. Nevertheless, for this is still too large
for any naive solution to be feasible in practice.Comment: accepted to SPIRE 201
Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields
Let be a finite Galois extension of fields with abelian Galois group
. A self-dual normal basis for is a normal basis with the
additional property that for .
Bayer-Fluckiger and Lenstra have shown that when , then
admits a self-dual normal basis if and only if is odd. If is an
extension of finite fields and , then admits a self-dual normal
basis if and only if the exponent of is not divisible by . In this
paper we construct self-dual normal basis generators for finite extensions of
finite fields whenever they exist.
Now let be a finite extension of \Q_p, let be a finite abelian
Galois extension of odd degree and let \bo_L be the valuation ring of . We
define to be the unique fractional \bo_L-ideal with square equal to
the inverse different of . It is known that a self-dual integral normal
basis exists for if and only if is weakly ramified. Assuming
, we construct such bases whenever they exist
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