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91.
Using multiscale analysis and methods of statistical physics, we show that a solution to the N-atom Liouville equation can be decomposed via an expansion in terms of a smallness parameter , wherein the long scale time behavior depends upon a reduced probability density that is a function of slow-evolving order parameters. This reduced probability density is shown to satisfy the Smoluchowski equation up to O(2) for a given range of initial conditions. Furthermore, under the additional assumption that the nanoparticle momentum evolves on a slow time scale, we show that this reduced probability density satisfies a Fokker–Planck equation up to O(2). This approach has applications to a broad range of problems in the nanosciences. 相似文献
92.
We discuss martingales, detrending data, and the efficient market hypothesis (EMH) for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). Beginning with x-independent drift coefficients R(t) we show that martingale stochastic processes generate uncorrelated, generally non-stationary increments. Generally, a test for a martingale is therefore a test for uncorrelated increments. A detrended process with an x-dependent drift coefficient is generally not a martingale, and so we extend our analysis to include the class of (x,t)-dependent drift coefficients of interest in finance. We explain why martingales look Markovian at the level of both simple averages and 2-point correlations. And while a Markovian market has no memory to exploit and presumably cannot be beaten systematically, it has never been shown that martingale memory cannot be exploited in 3-point or higher correlations to beat the market. We generalize our Markov scaling solutions presented earlier, and also generalize the martingale formulation of the EMH to include (x,t)-dependent drift in log returns. We also use the analysis of this paper to correct a misstatement of the ‘fair game’ condition in terms of serial correlations in Fama's paper on the EMH. We end with a discussion of Levy's characterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process. 相似文献
93.
It is shown that for every closed, convex and nowhere dense subset of a superreflexive Banach space there exists a Radon probability measure on so that for all . In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a closed convex nowhere dense set which is not Haar null.
94.
In this paper we shall make a further study on the the equivalence problem, i.e., equivalent conditions to the Riemann hypothesis in terms of Farey series by developing a rather analytical (-arithmetical) method to establish unexpected short interval results, namely
results, therewith simultaneously clarifying the underlying reasons for
results obtained in Part I. 相似文献
95.
V. I. Malykhin 《Mathematical Notes》1998,63(4):522-526
The maximal resolvability of totally bounded groups (and, under the assumption that the generalized continuum hypothesis holds, of 0-bounded groups) is proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 593–598, April, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-01374. 相似文献
96.
Dimitar K. Dimitrov 《Proceedings of the American Mathematical Society》1998,126(7):2033-2037
The celebrated Turán inequalities , where denotes the Legendre polynomial of degree , are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities , which hold for the Maclaurin coefficients of the real entire function in the Laguerre-Pólya class, .
97.
Sunil K. Chebolu J. Daniel Christensen Já n Miná c 《Proceedings of the American Mathematical Society》2008,136(4):1171-1179
A ghost in the stable module category of a group is a map between representations of that is invisible to Tate cohomology. We show that the only non-trivial finite -groups whose stable module categories have no non-trivial ghosts are the cyclic groups and . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups the second power of the Jacobson radical of is stably isomorphic to a suspension of .
98.
Karim Belabas Francisco Diaz y Diaz Eduardo Friedman. 《Mathematics of Computation》2008,77(262):1185-1197
Assuming the Generalized Riemann Hypothesis, Bach has shown that the ideal class group of a number field can be generated by the prime ideals of having norm smaller than . This result is essential for the computation of the class group and units of by Buchmann's algorithm, currently the fastest known. However, once has been computed, one notices that this bound could have been replaced by a much smaller value, and so much work could have been saved. We introduce here a short algorithm which allows us to reduce Bach's bound substantially, usually by a factor 20 or so. The bound produced by the algorithm is asymptotically worse than Bach's, but favorable constants make it useful in practice.
99.
Peter Borwein Ron Ferguson Michael J. Mossinghoff. 《Mathematics of Computation》2008,77(263):1681-1694
The Liouville function is the completely multiplicative function whose value is at each prime. We develop some algorithms for computing the sum , and use these methods to determine the smallest positive integer where . This answers a question originating in some work of Turán, who linked the behavior of to questions about the Riemann zeta function. We also study the problem of evaluating Pólya's sum , and we determine some new local extrema for this function, including some new positive values.
100.
Piermarco Cannarsa Genni Fragnelli Dario Rocchetti 《Journal of Evolution Equations》2008,8(4):583-616
We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval.
In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason,
we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally,
in a standard way, we deduce null controllability also for semilinear equations.
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