Asymptotic Ratio Research Articles

On the biological interpretation of a definition for the parameter R 0 in periodic population models

An adaptation of the definition of the basic reproduction number R (0) to time-periodic seasonal models was suggested a few years ago. However, its biological interpretation remained unclear. The present paper shows that in demography, this R (0) is the asymptotic ratio of total births in two successive generations of the family tree. In epidemiology, it is the asymptotic ratio of total infections in two successive generations of the infection tree. This result is compared with other recent work.

A necessary and sufficient condition for Ricci shrinkers to have positive AVR

In this short paper we observe that a recent result of C.-W. Chen meshes well with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L. Ni, and S.-J. Zhang. We give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci solitons to have positive asymptotic volume ratio.

Analytic Study of Performance of Error Estimators for Linear Discriminant Analysis

We derive double asymptotic analytical expressions for the first moments, second moments, and cross-moments with the actual error for the resubstitution and leave-one-out error estimators in the case of linear discriminant analysis in the multivariate Gaussian model under the assumption of a common known covariance matrix and a fixed Mahalanobis distance as dimensionality approaches infinity. Sample sizes for the two classes need not be the same; they are only assumed to reach a fixed, but arbitrary, asymptotic ratio with the dimensionality. From the asymptotic moment representations, we directly obtain double asymptotic expressions for the bias, variance, and RMS of the error estimators. The asymptotic expressions presented here generally provide good small sample approximations, as demonstrated via numerical experiments. The applicability of the theoretical results is illustrated by finding the minimum sample size to bound the RMS in gene-expression classification.

Yamabe 流下完备Riemannian 流形在无穷远的性质

We study complete locally conformally flat Riemannian manifolds <italic>Mⁿ</italic>.We prove that the flatness of the in nity geometry can be preserved under the Yamabe flow. As an immediate application, the asymptotic volume ratio of the manifold is a constant.

MMSE Dimension

If N is standard Gaussian, the minimum mean square error (MMSE) of estimating a random variable X based on √( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">snr</i> ) <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> vanishes at least as fast as 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">snr</i> as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">snr</i> → ∞. We define the MMSE dimension of X as the limit as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">snr</i> → ∞ of the product of snr and the MMSE. MMSE dimension is also shown to be the asymptotic ratio of nonlinear MMSE to linear MMSE. For discrete, absolutely continuous or mixed distribution we show that MMSE dimension equals Rényi's information dimension. However, for a class of self-similar singular X (e.g., Cantor dis tribution), we show that the product of snr and MMSE oscillates around information dimension periodically in snr (dB). We also show that these results extend considerably beyond Gaussian noise under various technical conditions.

Two Dimensional Strip Packing with Unloading Constraints

Abstract In this paper we present approximation algorithms for the two dimensional strip packing problem with unloading constraints. In this problem, we are given a strip S of width 1 and unbounded height, and n items of C different classes, each item a i of height h ( a i ) , width w ( a i ) and class c ( a i ) . As in the strip packing problem with rotations, we have to pack all items minimizing the used height, but now we have the additional constraint that items of higher classes cannot block the way out of lower classes items. We design a bin packing based algorithm with asymptotic approximation ratio of 5.745. These problems have practical applications on routing problems with loading/unloading constraints.

Dynamical properties of a trapped dipolar Fermi gas at finite temperature

We investigate the dynamical properties of a trapped finite-temperature normal Fermi gas with dipole-dipole interaction. For the free expansion dynamics, we show that the expanded gas always becomes stretched along the direction of the dipole moment. In addition, we present the temperature and interaction dependences of the asymptotical aspect ratio. We further study the collapse dynamics of the system by suddenly increasing the dipolar interaction strength. We show that, in contrast to the anisotropic collapse of a dipolar Bose-Einstein condensate, a dipolar Fermi gas always collapses isotropically when the system becomes globally unstable. We also explore the interaction and temperature dependences for the frequencies of the low-lying collective excitations.

SIGNATURE PAIRS FOR GROUP-INVARIANT HERMITIAN POLYNOMIALS

We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of SU(2). We introduce the asymptotic positivity ratio and compute it for cyclic subgroups of U(2). We calculate the signature pair for dihedral subgroups of U(2).

Parametric Poincaré-Perron theorem with applications

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations, where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application, we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.

Uniform unweighted set cover: The power of non-oblivious local search

We are given n base elements and a finite collection of subsets of them. The size of any subset varies between p to k ( p < k ). In addition, we assume that the input contains all possible subsets of size p . Our objective is to find a subcollection of minimum-cardinality which covers all the elements. This problem is known to be NP-hard. We provide two approximation algorithms for it, one for the generic case, and an improved one for the special case of ( p , k ) = ( 2 , 4 ) . The algorithm for the generic case is a greedy one, based on packing phases: at each phase we pick a collection of disjoint subsets covering i new elements, starting from i = k down to i = p + 1 . At a final step we cover the remaining base elements by the subsets of size p . We derive the exact performance guarantee of this algorithm for all values of k and p , which is less than H k , where H k is the k ’th harmonic number. However, the algorithm exhibits the known improvement methods over the greedy one for the unweighted k -set cover problem (in which subset sizes are only restricted not to exceed k ), and hence it serves as a benchmark for our improved algorithm. The improved algorithm for the special case of ( p , k ) = ( 2 , 4 ) is based on non-oblivious local search: it starts with a feasible cover, and then repeatedly tries to replace sets of size 3 and 4 so as to maximize an objective function which prefers big sets over small ones. For this case, our generic algorithm achieves an asymptotic approximation ratio of 1.5 + ϵ , and the local search algorithm achieves a better ratio, which is bounded by 1.458333 … + ϵ .

Two-machine flowshop scheduling with intermediate transportation under job physical space consideration

In this paper, we study a coordinated production–transportation scheduling problem in a two-machine flowshop environment where a single transporter may carry several jobs simultaneously as a batch between the machines. Each job has its own physical-space requirement while being loaded into the transporter. The goal is to minimize the makespan. For the jobs with the same size of physical space during transportation, we present a heuristic algorithm with an absolute worst-case ratio of 2 and a polynomial-time optimal algorithm for a special case with given job sequence. For the jobs having different size of physical storage space, a heuristic algorithm is constructed with an absolute worst-case ratio of 7/3 and asymptotic worst-case ratio of 20/9. Computational experiments demonstrate that the two heuristic algorithms developed are capable of generating near-optimal solutions quickly.

Bootstrapping the likelihood ratio cointegration test in error correction models with unknown lag order

The finite-sample size and power properties of bootstrapped likelihood ratio system cointegration tests are investigated via Monte Carlo simulations when the true lag order of the data generating process is unknown. Recursive bootstrap schemes are employed which differ in the way in which the lag order is chosen. The order is estimated by minimizing different information criteria and by combining the corresponding order estimates. It is found that, in comparison to the standard asymptotic likelihood ratio test based on an estimated lag order, bootstrapping can lead to improvements in small samples even when the true lag order is unknown, while the power loss is moderate.

Bin packing with general cost structures

Following the work of Anily et al., we consider a variant of bin packing called bin packing with general cost structures (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construction of the AFPTAS requires novel techniques for dealing with small items, which are developed in this work. In addition, we develop a fast approximation algorithm which acts identically for all non-decreasing and concave functions, and has an asymptotic approximation ratio of 1.5 for all functions simultaneously.

Strain path controlled shear tests on an analogue granular material

A series of strain controlled tests have been performed on a two-dimensional analogue granular material, composed of three different sizes of rods, in a special laboratory apparatus which permits full control of plane deformations, including rotation of principal axes. These tests have been designed and used to explore certain constitutive conjectures concerning the mechanical behaviour of granular materials. The results, within the accuracy of the apparatus, show that irrespective of the initial stress history of the material, proportional strain paths (if continued for long enough) lead to proportional stress states (asymptotic states). The asymptotic states appear to be independent of the fixed imposed rotation of principal strain axes and at these states coaxiality between the principal stress direction and the principal incremental strain direction seems to be reached. A stress dilatancy relationship between the imposed dilatancy and the corresponding asymptotic stress ratio has been identified.

Polylog Space Compression, Pushdown Compression, and Lempel-Ziv Are Incomparable

The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation (Hariharan, S., & Shankar, P. in: Proceedings of the 2006 IEEE data compression conference, p. 453, 2006; League, C., & Eng, K. in: Proceedings of the 2007 IEEE data compression conference, pp. 272–282, 2007), and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. We also consider online compression algorithms that use at most polylogarithmic space (plogon). These algorithms correspond to compressors in the data stream model. We compare the performance of these two families of compressors with each other and with the general purpose Lempel-Ziv algorithm. This comparison is made without any a priori assumption on the data’s source and considering the asymptotic compression ratio for infinite sequences. We prove that in all cases they are incomparable.

Tight results for Next Fit and Worst Fit with resource augmentation

It is well known that the two simple algorithms for the classic bin packing problem, NF and WF both have an approximation ratio of 2 . However, WF seems to be a more reasonable algorithm, since it never opens a new bin if an existing bin can still be used. Using resource augmented analysis, where the output of an approximation algorithm, which can use bins of size b > 1 , is compared to an optimal packing into bins of size 1 , we give a complete analysis of the asymptotic approximation ratio of WF and of NF , and use it to show that WF is strictly better than NF for any 1 < b < 2 , while they have the same asymptotic performance guarantee for all b ≥ 2 , and for b = 1 .

Investigation of internal microstructure and thermo-responsive properties of composite PNIPAM/silica microcapsules

Composite microcapsules consisting of a thermo-responsive hydrogel poly- N-isopropylacrylamide (PNIPAM) and coated by silica (SiO 2) nanoparticles have been synthesized by the inverse Pickering emulsion polymerization method. The composite capsules, whose mean diameter is in the 25–86 μm range in the expanded state, were characterized by static light scattering, atomic force microscopy (AFM), scanning electron microscopy (SEM), and laser scanning confocal microscopy (LSCM). It is reported that the hydrogel surface is uniformly covered by a monolayer of silica nanoparticles and that depending on the capsule size and degree of polymer cross-linking, either hollow-core or partially-filled hydrogel-core microcapsules can be created. Equilibrium thermo-responsive behavior of the composite microcapsules is investigated and it is found that after heating the particles above the lower critical solution temperature (LCST) of PNIPAM, the shrinkage ratio V/ V max varies from 0.8 to 0.4 for a cross-linking ratio from 0.6% to 9% on a mass basis. Dynamic temperature cycling studies reveal no hysteresis in the shrinking and recovery phases, but a small measurable dependence of the asymptotic shrinkage ratio V/ V max on the rate of temperature change exists. The composite capsules are stable under long-term storage in both dried and hydrated states and easily re-dispersible in water.

Bootstrap Sequential Determination of the Co-Integration Rank in VAR Models

Determining the co-integrating rank of a system of variables has become a fundamental aspect of applied research in macroeconomics and finance. It is wellknown that standard asymptotic likelihood ratio tests for co-integration rank of Johansen (1996) can be unreliable in small samples with empirical rejection frequencies often very much in excess of the nominal level. As a consequence, bootstrap versions of these tests have been developed. To be useful, however, sequential procedures for determining the co-integrating rank based on these bootstrap tests need to be consistent, in the sense that the probability of selecting a rank smaller than (equal to) the true co-integrating rank will converge to zero (one minus the marginal significance level), as the sample size diverges, for general I(1) processes. No such likelihood-based procedure is currently known to be available. In this paper we fill this gap in the literature by proposing a bootstrap sequential algorithm which we demonstrate delivers consistent cointegration rank estimation for general I(1) processes. Finite sample Monte Carlo simulations show the proposed procedure performs well in practice.

Thin Film Limits for Ginzburg–Landau with Strong Applied Magnetic Fields

In this work, we study thin film limits of the full three-dimensional Ginzburg–Landau model for a superconductor in an applied magnetic field oriented obliquely to the film surface. We obtain $\Gamma$-convergence results in several regimes, determined by the asymptotic ratio between the magnitude of the parallel applied magnetic field and the thickness of the film. Depending on the regime, we show that there may be a decrease in the density of Cooper pairs. We also show that in the case of variable thickness of the film, its geometry will affect the effective applied magnetic field, thus influencing the position of vortices.

Bootstrap Sequential Determination of the Co-Integration Rank in VAR Models

Determining the co-integrating rank of a system of variables has become a fundamental aspect of applied research in macroeconomics and finance. It is wellknown that standard asymptotic likelihood ratio tests for co-integration rank of Johansen (1996) can be unreliable in small samples with empirical rejection frequencies often very much in excess of the nominal level. As a consequence, bootstrap versions of these tests have been developed. To be useful, however, sequential procedures for determining the co-integrating rank based on these bootstrap tests need to be consistent, in the sense that the probability of selecting a rank smaller than (equal to) the true co-integrating rank will converge to zero (one minus the marginal significance level), as the sample size diverges, for general I(1) processes. No such likelihood-based procedure is currently known to be available. In this paper we fill this gap in the literature by proposing a bootstrap sequential algorithm which we demonstrate delivers consistent cointegration rank estimation for general I(1) processes. Finite sample Monte Carlo simulations show the proposed procedure performs well in practice.