Asymptotic Ratio Research Articles

Fragmentation of water by heavy ions

The connection between single and double electron removal and the fragmentation yields of water by heavy ions is analyzed. The contribution from the Auger-like decay of vacancies in the 2a1 molecular orbital to the asymptotic branching ratios for water fragmentation by fast particles is obtained. A model assuming the independence of double and single electron removal due to collisional processes is proposed, from which double electron removal cross sections can be obtained from the fragmentation pattern obtained in single coincidence experiments. The proposed scheme helps in the comparison between theoretical calculations for electron removal from ionization, electron capture and electron loss and the observed fragmentation pattern of water.

Selfish Bin Packing

Following recent interest in the study of computer science problems in a game theoretic setting, we consider the well known bin packing problem where the items are controlled by selfish agents. Each agent is charged with a cost according to the fraction of the used bin space its item requires. That is, the cost of the bin is split among the agents, proportionally to their sizes. Thus, the selfish agents prefer their items to be packed in a bin that is as full as possible. The social goal is to minimize the number of the bins used. The social cost in this case is therefore the number of bins used in the packing. A pure Nash equilibrium is a packing where no agent can obtain a smaller cost by unilaterally moving his item to a different bin, while other items remain in their original positions. A Strong Nash equilibrium is a packing where there exists no subset of agents, all agents in which can profit from jointly moving their items to different bins. We say that all agents in a subset profit from moving their items to different bins if all of them have a strictly smaller cost as a result of moving, while the other items remain in their positions. We measure the quality of the equilibria using the standard measures PoA and PoS that are defined as the worst case worst/best asymptotic ratio between the social cost of a (pure) Nash equilibrium and the cost of an optimal packing, respectively. We also consider the recently introduced measures SPoA and SPoS, that are defined similarly to the PoA and the PoS, but consider only Strong Nash equilibria. We give nearly tight lower and upper bounds of 1.6416 and 1.6428, respectively, on the PoA of the bin packing game, improving upon previous result by Bilo. We study the Strong Nash equilibria of the bin packing game, and show that a packing is a Strong Nash equilibrium iff it is produced by the Subset Sum algorithm for bin packing. This characterization implies that the SPoA of the bin packing game equals the approximation ratio of the Subset Sum algorithm, for which an almost tight bound is known. Moreover, the fact that any lower bound instance for the Subset Sum algorithm can be converted by a small modification of the item sizes to a lower bound instance on the SPoS, implies that in the bin packing game SPoA=SPoS. Finally, we address the issue of complexity of computing a Strong Nash packing and show that no polynomial time algorithm exists for finding Strong Nash equilibria, unless P=NP.

Ratios of double-to-single photoionization cross sections of helium atom embedded in dense quantum plasmas at high nonrelativistic photon energies

An investigation on the asymptotic ratios of double-to-single photoionization cross sections of the helium atom (He) embedded in dense quantum plasmas at very high nonrelativistic photon energies has been made. Exponential cosine-screened Coulomb potential has been used to represent the effective potential of a test charge in a dense quantum plasma. Making use of highly correlated wave functions for the ground states of He, results of the asymptotic ratios (R) of double-to-single photoionization cross sections for the screening parameter in the range [0,0.5] a.u. are reported. The reported value of R for the unscreened case confirms its established value. A comparative study of the behavior of R in weakly coupled plasmas and in dense quantum plasmas has also been made.

Comparison of Bayesian sample size criteria: ACC, ALC, and WOC

A challenge for implementing performance-based Bayesian sample size determination is selecting which of several methods to use. We compare three Bayesian sample size criteria: the average coverage criterion (ACC) which controls the coverage rate of fixed length credible intervals over the predictive distribution of the data, the average length criterion (ALC) which controls the length of credible intervals with a fixed coverage rate, and the worst outcome criterion (WOC) which ensures the desired coverage rate and interval length over all (or a subset of) possible datasets. For most models, the WOC produces the largest sample size among the three criteria, and sample sizes obtained by the ACC and the ALC are not the same. For Bayesian sample size determination for normal means and differences between normal means, we investigate, for the first time, the direction and magnitude of differences between the ACC and ALC sample sizes. For fixed hyperparameter values, we show that the difference of the ACC and ALC sample size depends on the nominal coverage, and not on the nominal interval length. There exists a threshold value of the nominal coverage level such that below the threshold the ALC sample size is larger than the ACC sample size, and above the threshold the ACC sample size is larger. Furthermore, the ACC sample size is more sensitive to changes in the nominal coverage. We also show that for fixed hyperparameter values, there exists an asymptotic constant ratio between the WOC sample size and the ALC (ACC) sample size. Simulation studies are conducted to show that similar relationships among the ACC, ALC, and WOC may hold for estimating binomial proportions. We provide a heuristic argument that the results can be generalized to a larger class of models.

Tail Behavior of the General Error Distribution

An inequality on the tail behavior of the general error distribution and an asymptotic Mills-type ratio are established. Two applications are provided. The first application considers the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the general error and normal distributions. The second application establishes the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the general error distribution.

A note on the minimum bounded edge-partition of a tree

Minimum bounded edge-partition divides the edge set of a tree into the minimum number of disjoint connected components given a maximum weight for any component. It is an adaptation of the uniform edge-partition of a tree. An optimization algorithm is developed for this NP-hard problem, based on repeated bin packing of inter-related instances. The algorithm has linear running time for the class of ‘balanced trees’ common for the stochastic programming application which motivated investigation of this problem. Fast 2-approximation algorithms are formed for general instances by replacing the optimal bin packing with almost any bin packing heuristic. The asymptotic worst-case ratio of these approximation algorithms is never better than the absolute worst-case ratio of the bin packing heuristic used.

A note on minimizing makespan on a single batch processing machine with nonidentical job sizes

In a relatively recent paper (G. Zhang, X. Cai, C.Y. Lee, C.K. Wong, Minimizing makespan on a single batch processing machine with nonidentical job sizes, Naval Research Logistics 48 (2001) 226–240), authors considered minimizing makespan on a single batching machine having unit capacity. For the restricted version of the problem in which the processing times of the jobs with sizes greater than 1 / 2 are not less than those of jobs with sizes not greater than 1 / 2 , they proposed an O ( n log n ) algorithm with absolute worst-case ratio 3 / 2 . We propose an algorithm with absolute worst-case ratio 3 / 2 and asymptotic worst-case ratio ( m + 1 ) / m ( m ≥ 2 and integer) for a more general version in which the processing times of the jobs of sizes greater than 1 / m are not less than the remaining (the case of m = 2 has been considered by Zhang et al.). This general assumption is particularly held for those problem instances in which the job sizes and job processing times are agreeable. We obtain an O ( n log n ) algorithm with asymptotic worst-case ratio 4 / 3 for these problems leading to a more dependable algorithm than that of Zhang et al.

Hardness of approximation for orthogonal rectangle packing and covering problems

Bansal and Sviridenko [N. Bansal, M. Sviridenko, New approximability and inapproximability results for 2-dimensional bin packing, in: Proceedings of the 15th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, 2004, pp. 189–196] proved that there is no asymptotic PTAS for 2- dimensional Orthogonal Bin Packing (without rotations), unless P = NP . We show that similar approximation hardness results hold for several 2- and 3-dimensional rectangle packing and covering problems even if rotations by ninety degrees are allowed. Moreover, for some of these problems we provide explicit lower bounds on asymptotic approximation ratio of any polynomial time approximation algorithm. Our hardness results apply to the most studied case of 2-dimensional problems with unit square bins, and for 3-dimensional strip packing and covering problems with a strip of unit square base.

Inference for Continuous Semimartingales Observed at High Frequency

The econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities. We here study this local-constancy approximation as a general approach to estimation in such data. We show that the technique yields asymptotic properties (consistency, normality) that are correct subject to an ex post adjustment involving asymptotic likelihood ratios. These adjustments are derived and documented. Several examples of estimation are provided: powers of volatility, leverage effect, and integrated betas. The first order approximations based on local constancy can be over the period of one observation or over blocks of successive observations. It has the advantage of gaining in transparency in defining and analyzing estimators. The theory relies heavily on the interplay between stable convergence and measure change, and on asymptotic expansions for martingales.

Sharp logarithmic Sobolev inequalities on gradient solitons and applications

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding solitons with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof, various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the asymptotic volume ratio of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.

An asymptotic approximation scheme for the concave cost bin packing problem

Abstract We consider a generalized one-dimensional bin packing model in which the cost of a bin is a nondecreasing concave function of the utilization of the bin. We show that for any given positive constant ϵ , there exists a polynomial-time approximation algorithm with an asymptotic worst-case performance ratio of no more than 1 + ϵ .

Curvature integrals under the Ricci flow on surfaces

In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.

Ratio of double-to-single photoionization cross sections of plasma-embedded helium atoms at x-ray energies

The asymptotic ratios of double-to-single photoionization cross sections of plasma-embedded helium atoms at very high but nonrelativistic photon energies are calculated for different shielding parameters that are based on the Debye model for weakly coupled plasmas. The ratio of double-to-single photoionization for free atom is determined as 1.644%, in good agreement with the synchrotron measurements. The asymptotic ratio of double-to-single photoionization cross sections for different screening parameters are reported.

Packing d-Dimensional Bins in d Stages

We consider the d-dimensional bin-packing problem, the most relevant generalization of classical bin packing, and show a general result about the asymptotic worst-case ratio of a wide class of approximation algorithms that construct solutions in d stages, containing many heuristics previously considered in the literature. Moreover, we give the exact value of the asymptotic worst-case ratio between the optimal solution and the best solution obtainable by packing the items in d stages, showing how to achieve such a ratio efficiently. The key property behind our result is the asymptotic optimality of the fractional relaxation of (one-dimensional) bin packing, an important by-product of the approximation schemes for the problem from the 1980s, which, to the best of our knowledge, is used here for the first time only for the sake of the analysis. For the two-dimensional case, we push the approximablity threshold below 2 after more than 20 years (reaching 1.691…), but also improve and widely simplify the analysis of the previous best method from the 1980s. Moreover, for d ≥ 3 we improve the previous approximability threshold by a factor of 1.691…, a notable improvement when d is small.

Elemental fractionation and stoichiometric sampling in femtosecond laser ablation

Elemental fractionation in femtosecond laser ablation is studied by ICP-MS by applying successive single laser shots to binary metallic and semiconductor samples as well as to multi-component glasses. Fractionation can be observed in the first laser shots in particular if the laser fluence is near the ablation threshold of the sample. However, the element ratio in the laser-sampled masses changes from shot to shot until it reaches an asymptotic fluence-independent value representing stoichiometric sampling. The asymptotic stoichiometric ratios can be obtained with fewer shots if higher laser fluences are applied. It is shown by electron probe X-ray analysis that different elemental ablation probabilities modify the element compositions in the surface layers of the laser craters until equilibrium conditions are obtained. These conditions can be reached by applying many shots of low laser fluence or with one high-fluence laser shot only. The experimental data reveal that in most cases the elemental ablation probability can be correlated with the respective ionization energies of the elements, i.e., the elements with lower first ionization energy have higher ablation probability. No or only very weak fractionation was observed when elements with nearly the same ionization energies were sampled.

Inference for Continuous Semimartingales Observed at High Frequency: A General Approach

The econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities. We here study this local-constancy approximation as a general approach to estimation in such data. We show that the technique yields asymptotic properties (consistency, normality) that are correct subject to an ex post adjustment involving asymptotic likelihood ratios. These adjustments are given. Several examples of estimation are provided: powers of volatility, leverage eect, integrated betas, bipower, and covariance under asynchronous observation. The first order approximations in this study can be over the period of one observation, or over blocks of successive observations. The advantage of blocking is a gain in transparency in defining and analyzing estimators. The theory relies heavily on the interplay between stable convergence and measure change, and on asymptotic expansions for martingales.

Lower Bound for the Online Bin Packing Problem with Restricted Repacking

In 1996 Ivkovič and Lloyd [A fundamental restriction on fully dynamic maintenance of bin packing, Inform. Process. Lett., 59 (1996), pp. 229–232] gave the lower bound $\frac{4}{3}$ on the asymptotic worst-case ratio for so-called fully dynamic bin packing algorithms, where the number of repackable items in each step is restricted by a constant. In this paper we improve this result to about $1.3871$. We present our proof for a semionline case of the classical bin packing, but it works for fully dynamic bin packing as well. We prove the lower bound by analyzing and solving a specific optimization problem. The bound can be expressed exactly using the Lambert W function.

Investigation of nodal domains in a chaotic three-dimensional microwave rough billiard with the translational symmetry

Abstract We show that using the concept of the two-dimensional level number N ⊥ one can experimentally study of the nodal domains in a three-dimensional (3D) microwave chaotic rough billiard with the translational symmetry. Nodal domains are regions where a wave function has a definite sign. We found the dependence of the number of nodal domains ℵ N ⊥ lying on the cross-sectional planes of the cavity on the two-dimensional level number N ⊥ . We demonstrate that in the limit N ⊥ → ∞ the least squares fit of the experimental data reveals the asymptotic ratio ℵ N ⊥ / N ⊥ ≃ 0.059 ± 0.029 that is close to the theoretical prediction ℵ N ⊥ / N ⊥ ≃ 0.062 . This result is in good agreement with the predictions of percolation theory.

On the effectiveness of the Harmonic Shelf Algorithm for on-line strip packing

In [J. Csirik, G.J. Woeginger, An on-line algorithm for multidimensional bin packing, Inform. Process. Lett. 63 (1997) 171–175] the authors study the asymptotic worst case ratio between the height of the strip needed to on-line pack a list of boxes by means of the Harmonic Shelf Algorithm and the height of the strip used by an optimal algorithm. In this note we analyze the effectiveness of the former algorithm in terms of the ratio between the unused area inside the strip and the total size of this strip, and we show that the Harmonic Shelf Algorithm is also capable of packing items so that the asymptotic worst case value of this ratio comes arbitrarily close to 1 2 .

Statistical tests to compare motif count exceptionalities.

BackgroundFinding over- or under-represented motifs in biological sequences is now a common task in genomics. Thanks to p-value calculation for motif counts, exceptional motifs are identified and represent candidate functional motifs. The present work addresses the related question of comparing the exceptionality of one motif in two different sequences. Just comparing the motif count p-values in each sequence is indeed not sufficient to decide if this motif is significantly more exceptional in one sequence compared to the other one. A statistical test is required.ResultsWe develop and analyze two statistical tests, an exact binomial one and an asymptotic likelihood ratio test, to decide whether the exceptionality of a given motif is equivalent or significantly different in two sequences of interest. For that purpose, motif occurrences are modeled by Poisson processes, with a special care for overlapping motifs. Both tests can take the sequence compositions into account. As an illustration, we compare the octamer exceptionalities in the Escherichia coli K-12 backbone versus variable strain-specific loops.ConclusionThe exact binomial test is particularly adapted for small counts. For large counts, we advise to use the likelihood ratio test which is asymptotic but strongly correlated with the exact binomial test and very simple to use.