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1.
The only primitive trinomials of degree over are and its reciprocal.

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2.
All primitive trinomials over with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are and its reciprocal. Also two examples of primitive pentanomials over with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.

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3.
In the Laurent expansion


of the Riemann-Hurwitz zeta function, the coefficients are known as Stieltjes, or generalized Euler, constants. [When , (the Riemann zeta function), and .] We present a new approach to high-precision approximation of . Plots of our results reveal much structure in the growth of the generalized Euler constants. Our results when for , and when for (for such as 53/100, 1/2, etc.) suggest that published bounds on the growth of the Stieltjes constants can be much improved, and lead to several conjectures. Defining , we conjecture that is attained: for any given , for some (and similarly that, given and , is within of for infinitely many ). In addition we conjecture that satisfies for 1$">. We also conjecture that , a special case of a more general conjecture relating the values of and for . Finally, it is known that for . Using this to define for all real 0$">, we conjecture that for nonintegral , is precisely times the -th (Weyl) fractional derivative at of the entire function . We also conjecture that , now defined for all real arguments 0$">, is smooth. Our numerical method uses Newton-Cotes integration formulae for very high-degree interpolating polynomials; it differs in implementation from, but compares in error bounding to, Euler-Maclaurin summation based methods.

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4.
For , we consider the set . The polynomials are in , with only mild restrictions, and is the Weil height of . We show that this set is dense in for some effectively computable limit point .

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5.
In this paper we analyze the convergence and stability of the iterated Lubkin transformation and the -algorithm as these are being applied to sequences whose members behave like as , where and are complex scalars and is a nonnegative integer. We study the three different cases in which (i) , , and (logarithmic sequences), (ii) and (linear sequences), and (iii) (factorial sequences). We show that both methods accelerate the convergence of all three types of sequences. We show also that both methods are stable on linear and factorial sequences, and they are unstable on logarithmic sequences. On the basis of this analysis we propose ways of improving accuracy and stability in problematic cases. Finally, we provide a comparison of these results with analogous results corresponding to the Levin -transformation.

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6.
Let be the minimal length of a polynomial with coefficients divisible by . Byrnes noted that for each , and asked whether in fact . Boyd showed that for all , but . He further showed that , and that is one of the 5 numbers , or . Here we prove that . Similarly, let be the maximal power of dividing some polynomial of degree with coefficients. Boyd was able to find for . In this paper we determine for .

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7.
We report an error in our previous paper [#!K1!#], where we announced that we listed all the primitive trinomials over of degree 859433, but there is a bug in the sieve. We missed the primitive trinomial and its reciprocal, as pointed out by Richard Brent et al. We also report some new primitive pentanomials.

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8.
Let be the minimal positive integer , for which there exists a splitting of the set into  subsets, , , ..., , whose first moments are equal. Similarly, let be the maximal positive integer , such that there exists a splitting of into subsets whose first moments are equal. For , these functions were investigated by several authors, and the values of and have been found for and , respectively. In this paper, we deal with the problem for any prime . We demonstrate our methods by finding for any and for .

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9.
Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

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10.
We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.

In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.

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