Sparse polynomials, redundant bases, gauss periods, and efficient exponentiation of primitive elements for small characteristic finite fields

Gauss periods give an exponentiation algorithm that is fast for many finite fields but slow for many other fields. The current paper presents a different method for construction of elements that yield a fast exponentiation algorithm for finite fields where the Gauss period method is slow or does not work. The basic idea is to use elements of low multiplicative order and search for primitive elements that are binomial or trinomial of these elements. Computational experiments indicate that such primitive elements exist, and it is shown that they can be exponentiated fast.     

Ratio methods to the mean estimation with known quantiles

We address the problem of estimating the finite population mean in survey sampling, by exploiting any available auxiliary information in order to increase the precision of classical estimators. The idea is to use any population quantiles of the available auxiliary variables which are known in many real situation from census, administrative files, etc. This is achieved using these known quantities in the construction of the estimators, by modifying the usual ratio estimation methods and afterwards defining a general class of exponentiation ratio estimators. The advantages of the proposed estimators are demonstrated using theoretical asymptotic tools and through a simulation study.     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

L-模糊自然数的乘法运算和幂运算

在L是完全分配格时,定义了L-模糊自然数的乘法运算和幂运算,研究了乘法运算、幂运算的交换律、结合律以及乘法对加法的分配律等性质。     

Efficient parallel exponentiation in using normal basis representations

Von zur Gathen proposed an efficient parallel exponentiation algorithm in finite fields using normal basis representations. In this paper we present a processor-efficient parallel exponentiation algorithm in GF(qⁿ) which improves upon von zur Gathen's algorithm. We also show that exponentiation in GF(qⁿ) can be done in O((log₂n)²/log_qn) time using n/(log₂n)² processors. Hence we get a processor-time bound of O(n/log_qn), which matches the best known sequential algorithm. Finally, we present an efficient on-line processor assignment scheme which was missing in von zur Gathen's algorithm.     

On String Replacement Exponentiation

The string replacement (SR) method was recently proposed as a methodfor exponentiation a ^e in a group G. The canonicalk-SR method operates by replacing a run of i onesin a binary exponent,0k, with i-1 zeroes followedby the single digit b=2 ⁱ -1. After recoding, it was shown in[5] that the expected weight of e tends to n/4 forn-bit exponents. In this paper we show that the canonicalk-SR recoding process can be described as a regular language andthen use generating functions to derive the exact probability distribution ofrecoded exponent weights. We also show that the canonical 2-SR recodingproduces weight distributions very similar to (optimal) signed-digitrecodings, but no group inversions are required.     

Exponentiation problem in the construction of an effective low-momentum Hamiltonian for bosons

The problem of exponentiation of connected-graph contributionsC, when one carries out only a partial trace of the density matrix of an assembly of bosons in order to construct an effective, low-momentum Hamiltonian, is examined. It is found that besides accounting for the exponentiation of connected graphs, disconnected graphs contribute certain termsD to connected-graph contributions. TheD-terms diminish as the number of iterations increases in the Singh’s renormalization-group theory for the present system. Therefore, these terms play no role in determining critical behaviour of the system.