129,945 research outputs found
How to determine linear complexity and -error linear complexity in some classes of linear recurring sequences
Several fast algorithms for the determination of the linear complexity of -periodic sequences over a finite
field \F_q, i.e. sequences with characteristic polynomial , have been proposed in the literature.
In this contribution fast algorithms for determining the linear complexity of binary sequences with characteristic
polynomial for an arbitrary positive integer , and are presented.
The result is then utilized to establish a fast algorithm for determining the -error linear complexity of
binary sequences with characteristic polynomial
Form factors and action of U_{\sqrt{-1}}(sl_2~) on infinite-cycles
Let be a sequence of
skew-symmetric polynomials in satisfying , whose coefficients are symmetric Laurent polynomials in . We
call an -cycle if
holds for all .
These objects arise in integral representations for form factors of massive
integrable field theory, i.e., the SU(2)-invariant Thirring model and the
sine-Gordon model. The variables are the integration
variables and are the rapidity variables. To each
-cycle there corresponds a form factor of the above models.
Conjecturally all form-factors are obtained from the -cycles.
In this paper, we define an action of
on the space of -cycles.
There are two sectors of -cycles depending on whether is even or
odd. Using this action, we show that the character of the space of even (resp.
odd) -cycles which are polynomials in is equal to the
level irreducible character of with lowest
weight (resp. ). We also suggest a possible tensor
product structure of the full space of -cycles.Comment: 27 pages, abstract and section 3.1 revise
A 3-Stranded Quantum Algorithm for the Jones Polynomial
Let K be a 3-stranded knot (or link), and let L denote the number of
crossings in K. Let and be two positive real
numbers such that is less than or equal to 1.
In this paper, we create two algorithms for computing the value of the Jones
polynomial of K at all points of the unit circle in the complex
plane such that the absolute value of is less than or equal to .
The first algorithm, called the classical 3-stranded braid (3-SB) algorithm,
is a classical deterministic algorithm that has time complexity O(L). The
second, called the quantum 3-SB algorithm, is a quantum algorithm that computes
an estimate of the Jones polynomial of K at within a precision of
with a probability of success bounded below by $1-\epsilon_{2}%.
The execution time complexity of this algorithm is O(nL), where n is the
ceiling function of (ln(4/\epsilon_{2}))/(2(\epsilon_{2})^2). The compilation
time complexity, i.e., an asymptotic measure of the amount of time to assemble
the hardware that executes the algorithm, is O(L).Comment: 19 pages, 10 figures, to appear in Proc. SPIE, 6573-29, (2007
Algorithms for determining integer complexity
We present three algorithms to compute the complexity of all
natural numbers . The first of them is a brute force algorithm,
computing all these complexities in time and space . The
main problem of this algorithm is the time needed for the computation. In 2008
there appeared three independent solutions to this problem: V. V. Srinivas and
B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune
[3]. All three are very similar. Only [11] gives an estimation of the
performance of its algorithm, proving that the algorithm computes the
complexities in time , where . The other two algorithms, presented in [7] and
[3], were very similar but both superior to the one in [11]. In Section 2 we
present a version of these algorithms and in Section 4 it is shown that they
run in time and space . (Here ).
In Section 2 we present the algorithm of [7] and [3]. The main advantage of
this algorithm with respect to that in [11] is the definition of kMax in
Section 2.7. This explains the difference in performance from
to .
In Section 3 we present a detailed description a space-improved algorithm of
Fuller and in Section 5 we prove that it runs in time and space
, where and
.Comment: 21 pages. v2: We improved the computations to get a better bound for
$\alpha
IST Austria Technical Report
We study algorithmic questions for concurrent systems where the transitions are labeled from a complete, closed semiring, and path properties are algebraic with semiring operations. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural properties that arise in program analysis.
We consider that each component of the concurrent system is a graph with constant treewidth, and it is known that the controlflow graphs of most programs have constant treewidth. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis problems (e.g., alias analysis). The study of multiple queries allows us to consider the tradeoff between the resource usage of the \emph{one-time} preprocessing and for \emph{each individual} query. The traditional approaches construct the product graph of all components and apply the best-known graph algorithm on the product. In the traditional approach, even the answer to a single query requires the transitive closure computation (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time.
Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time,
each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results that show that the worst-case running times of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (such as improving
the worst-case bounds for the shortest path problem in general graphs whose current best-known bound has not been improved in five decades). Finally, we provide a prototype implementation of our algorithms which significantly outperforms the existing algorithmic methods on several benchmarks
IST Austria Technical Report
We consider partially observable Markov decision processes (POMDPs) with a set of target states and every transition is associated with an integer cost. The optimization objective we study asks to minimize the expected total cost till the target set is reached, while ensuring that the target set is reached almost-surely (with probability 1). We show that for integer costs approximating the optimal cost is undecidable. For positive costs, our results are as follows: (i) we establish matching lower and upper bounds for the optimal cost and the bound is double exponential; (ii) we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for POMDPs with finite-horizon objectives. While the worst-case running time of our algorithm is double exponential, we also present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest
Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz
The relation between solutions to Helmholtz's equation on the sphere
and the [{\gr sl}(2)]^n Gaudin spin chain is clarified. The joint
eigenfuctions of the Laplacian and a complete set of commuting second order
operators suggested by the --matrix approach to integrable systems, based on
the loop algebra \wt{sl}(2)_R, are found in terms of homogeneous polynomials
in the ambient space. The relation of this method of determining a basis of
harmonic functions on to the Bethe ansatz approach to integrable
systems is explained.Comment: 14 pgs, Plain Tex, preprint CRM--2174 (May, 1994
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