Asymptotic Ratio Research Articles

Heat Kernel Asymptotics for Scaling Limits of Isoradial Graphs

AbstractWe consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.

The Tail Characteristics of Several Asymmetric Distributions and Their Applications in the Field of Financial Data Modelling and Machine Learning

Asymmetric distributions play a pivotal role in various fields, including finance, machine learning, and artificial intelligence. In finance, asymmetric distributions, characterized by negative skewness, peakedness, heavy tails, and asymmetric fluctuations, provide a more accurate representation of financial data. These distributions are particularly useful in capturing the complex behavior of assets and risk management. In the field of machine learning, data disturbance, noise, outliers and other disturbances will have a negative impact on the performance and reliability of the model, asymmetric distributions offer a valuable tool for describing the tail behavior of noise and interference. This is crucial in designing more robust machine learning models that can handle outliers and noise effectively. Additionally, when dealing with unbalanced data classification, asymmetric distributions can be leveraged by adjusting the decision threshold of classifiers or employing specialized loss functions. In this paper, the asymptotic tail ratio behavior of probability density function(pdf) and cumulative distribution function (cdf) of three asymmetric distributions (asymmetric Laplace, generalized Logistic and asymmetric normal distribution) and Student-t distribution were considered respectively under some regular conditions. The detailed characteristics of these asymmetric tails will have a profound impact on the fields of financial data analysis, artificial intelligence, and unbalanced classification.

Soft independence guided filter pruning

Filter pruning (FP) is an effective method for reducing the computational costs of convolutional neural networks, and herein, the most critical task involves evaluating the significance of each convolutional filter and eliminating the less important ones while minimizing performance degradation. Most existing FP methods consider only local information, which may prevent them from accurately recognizing the most important filters. To address this limitation, we propose the soft filter independence (SFI) method, which leverages global information to identify the most important filters using their magnitude and correlation information in different functional layers. The SFI criterion measures the replaceability of filters from a global perspective in a network. Filters with low independence can be represented effectively by others, so their information can be accurately conveyed by other filters. In addition, we introduce a novel SFI-based asymptotic pruning ratio, which improves training and pruning stability. Compared to the most advanced FP methods, our method enables CNNs to achieve higher pruning rates and better classification performance.

The Near Exact Bin Covering Problem

AbstractWe present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant $$\varDelta $$ Δ , and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and $$1+\varDelta $$ 1 + Δ . Our goal is to maximize the number of near exact covered bins. If $$\varDelta =0$$ Δ = 0 or $$\varDelta >0$$ Δ > 0 is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that $$P\ne NP$$ P ≠ N P ). However, for the case where $$\varDelta >0$$ Δ > 0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.

Commodity Market Efficiency: New Evidence From the Russia-Ukraine War

We re-evaluate the efficiency of selected commodity markets during the Russia-Ukraine crisis using the asymptotic normal variance ratio test. We find that there is a substantial difference in the commodity prices before and after the war’s commencement. We also show that the markets are weak-form inefficient before and after the invasion except for gasoline and oats which became efficient during the war. We highlight the investment implications of the findings.

Inference on order restricted means of inverse Gaussian populations under heteroscedasticity

The hypothesis testing problem of homogeneity of k(≥2) inverse Gaussian means against ordered alternatives is studied when nuisance or scale-like parameters are unknown and unequal. The maximum likelihood estimators (MLEs) of means and scale-like parameters are obtained when means satisfy some simple order restrictions and scale-like parameters are unknown and unequal. An iterative algorithm is proposed for finding these estimators. It has been proved that under a specific condition, the proposed algorithm converges to the true MLEs uniquely. A likelihood ratio test and two simultaneous tests are proposed. Further, an algorithm for finding the MLEs of parameters is given when means are equal but unknown. Using the estimators, the likelihood ratio test is developed for testing against ordered alternative means. Using the asymptotic distribution, the asymptotic likelihood ratio test is proposed. However, for small samples, it does not perform well. Hence, a parametric bootstrap likelihood ratio test (PB LRT) is proposed. Therefore, the asymptotic validity of the bootstrap procedure has been shown. Using the Box-type approximation method, test statistics are developed for the two-sample problem of equality of means when scale-like parameters are heterogeneous. Using these, two PB-based heuristic tests are proposed. Asymptotic null distributions are derived and PB accuracy is also developed. Two asymptotic tests are also proposed using the asymptotic null distributions. To get the critical points and test statistics of the three PB tests and two asymptotic tests, an ‘R’ package is developed and shared on GitHub. Applications of the tests are illustrated using real data.

Evaluation of the Effects of Different Strain Paths on the Behavior of Sands Using Direct Simple Shear Tests

Abstract Many researchers have evaluated different parameters that can affect the mechanical behavior of sands. However, most of the conducted investigations considered drained and undrained stress–strain paths as the limiting boundaries of the behavior; recently applied studies have indicated that this assumption cannot be accurate for all situations. In this paper, by using expansive and contractive strain paths (in addition to the conventional constant volume path), partially drained responses of sands are simulated. Influences of different grain size distributions on the behavior of sands are studied by controlled coupling between volumetric to shear strain ratios. The results show that the asymptotic stress ratio (steady state stress ratio), phase transformation stress ratio, and instability stress ratio depend on the experienced strain paths and cannot be considered material constant (inherent) parameters. It is shown that coarse sands can be affected by the various strain paths more than finer sands, and loose sands are more sensitive against the partially drained strain paths compared to dense sands.

Исследование особенностей медленной динамики и эффектов старения в трехпленочных наноструктурах с намагниченностью перпендикулярной плоскости плёнок

The results of a Monte Carlo study of trilayer nanostructure behavior with magnetization perpendicular to plane of films with thickness = 3 ML = 21 ML are given. At the calculated value of the critical temperature, as well as for temperatures < investigation of nonequilibrium behavior of this structure is carried out with calculation and analysis of time dependence for magnetization, autocorrelation function and dynamic susceptibility with evolution from different initial states. Aging properties are revealed. The value of the asymptotical fluctuation ratio 𝑋∞(𝑁 = 3, 𝑐) = 0.329(3), 𝑋∞(𝑁 = 21, 𝑐) = 0.324(3),𝑋∞(𝑁 = 3, 0.5𝑇 𝑐) = 0.9208(24), 𝑋∞(𝑁 = 21, 0.5𝑇 𝑐) = 0.9528(21) is obtained with demonstration of violation for the fluctuation-dissipation theorem in nonequilibrium behavior both at the critical temperature and in the low-temperature phase of trilayer nanostructure.

Sharp log-Sobolev inequalities in [formula omitted] spaces with applications

Given p,N>1, we prove the sharp Lp-log-Sobolev inequality on noncompact metric measure spaces satisfying the CD(0,N) condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in CD(0,N) spaces, symmetrization, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf–Lax semigroup in CD(0,N) spaces. The proof of this result uses Hamilton–Jacobi inequality and Sobolev regularity properties of the Hopf–Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Moreover, a sharp Gaussian-type L2-log-Sobolev inequality and a hypercontractivity estimate are obtained in RCD(0,N) spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (2004) [55] on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.

Testing on the Quantiles of a Single Normal Population in the Presence of Several Normal Populations with a Common Variance

We consider the problem of testing a hypothesis on the qth quantile ξ q = ξ ( say ) = μ 1 + ν σ (for known ν ≠ 0 , and dependent on q) of the first population when independent samples are available from k ( ≥ 2 ) normal populations with a common unknown standard deviation σ , and unknown as well as possibly unequal means μi’s ( 1 ≤ i ≤ k ). We discuss several test methods, such as the ones based on the computational approach test (CAT), the asymptotic likelihood ratio test (ALRT), the parametric bootstrap likelihood ratio test (PBLRT), and tests based on the generalized p value approach. The performance of these tests, in terms of size and power, have been studied through a comprehensive simulation study, and recommendations have been made about their applicability. Finally, usage of these tests have demonstrated by using three real-life datasets.

There is no APTAS for 2-dimensional vector bin packing: Revisited

We study the Vector Bin Packing and the Vector Bin Covering problems, multidimensional generalizations of the Bin Packing and the Bin Covering problems, respectively. In the Vector Bin Packing, we are given a set of d-dimensional vectors from [0,1]d and the aim is to partition the set into the minimum number of bins such that for each bin B, each component of the sum of the vectors in B is at most 1. Woeginger [17] claimed that the problem has no APTAS for dimensions greater than or equal to 2. We note that there was a slight oversight in the original proof. In this work, we give a revised proof using some additional ideas from [3,8]. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 600599.An instance of Vector Bin Packing is called δ-skewed if every item has at most one dimension greater than δ. As a natural extension of our general d-Dimensional Vector Bin Packing result we show that for ε∈(0,12500) it is NP-hard to obtain a (1+ε)-approximation for δ-Skewed Vector Bin Packing if δ>20ε.In the Vector Bin Covering problem given a set of d-dimensional vectors from [0,1]d, the aim is to obtain a family of disjoint subsets (called bins) with the maximum cardinality such that for each bin B, each component of the sum of the vectors in B is at least 1. Using ideas similar to our Vector Bin Packing result, we show that for Vector Bin Covering there is no APTAS for dimensions greater than or equal to 2. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 998997.

Long-Term Response of Sand Subjected to Repetitive Simple Shear Loading: Shakedown, Ratcheting, and Terminal Void Ratio

Low-amplitude repetitive drained loading may hinder the long-term performance of engineered and natural systems. This study examines the volumetric and shear response of a uniform quarzitic sand subjected to repetitive drained simple shear loading under constant vertical stress while tracking the evolution of the secant stiffness and the small-strain shear modulus. We explore the effects of initial density, initial shear stress and cyclic shear stress amplitude to identify criteria that can be used to anticipate asymptotic volumetric and shear states. We analyze experimental results in reference to the sand response under monotonic simple shear loading. All specimens evolved toward some asymptotic terminal void ratio eT when subjected to simple shear cycles. Contractive specimens exhibited unceasing shear strain accumulation and ratcheting when the normalized shear stress exceeded τ*=(τo+Δτ)/τult>0.85; on the other hand, dense-dilative specimens exhibited ratcheting only when the normalized shear stress exceeded τ*=(τo+Δτ)/τult>1.25. The small-strain Gmax and the secant Gpp shear moduli increased during repetitive shear cycles to reflect early fabric changes followed by abrasion/fretting among enduring contacts. Results obtained in this study allow us to propose simple guidelines to predict the asymptotic shear and volumetric response of uniform sands subjected to repetitive simple shear loading for first-order engineering analyses.

Asymptotic subadditivity/superadditivity of Value‐at‐Risk under tail dependence

AbstractThis paper presents a new method for discussing the asymptotic subadditivity/superadditivity of Value‐at‐Risk (VaR) for multiple risks. We consider the asymptotic subadditivity and superadditivity properties of VaR for multiple risks whose copula admits a stable tail dependence function (STDF). For the purpose, a marginal region is defined by the marginal distributions of the multiple risks, and a stochastic order named tail concave order is presented for comparing individual tail risks. We prove that asymptotic subadditivity of VaR holds when individual risks are smaller than regularly varying (RV) random variables with index −1 under the tail concave order. We also provide sufficient conditions for VaR being asymptotically superadditive. For two multiple risks sharing the same copula function and satisfying the tail concave order, a comparison result on the asymptotic subadditivity/superadditivity of VaR is given. Asymptotic diversification ratios for RV and log regularly varying (LRV) margins with specific copula structures are obtained. Empirical analysis on financial data is provided for highlighting our results.

Approximation algorithms for scheduling single batch machine with incompatible deteriorating jobs

Motivated by the soaking process under separate heating mode in iron and steel enterprises, we study the parallel batch machine scheduling problem with incompatible deteriorating jobs. The objective is to minimize makespan. A soaking furnace can be seen as a parallel batch processing machine. In order to avoid the thermal stress caused by excessive temperature difference, initial temperature is needed for the ingot before processing. With the increasing of waiting time, the ingot temperature decreases and the soaking time increases. This property is called deterioration. Setup time is needed between incompatible jobs. We show that if jobs have the same sizes, an optimal solution can be found withinO(nlogn) time. If jobs have identical processing times, the problem is proved to be NP-hard in the strong sense. We propose an approximate algorithm whose absolute and asymptotic worst-case ratios are less than 2 and 11/9, respectively. When the jobs have arbitrary sizes and arbitrary processing times, the model is also NP-hard in the strong sense. An approximate algorithm with an absolute and asymptotic worst-case ratio less than 2 is proposed. The time complexity isO(nlogn).

The asymptotic central-raw ratios for binomial, Poisson and normal distributions

The n-th central-raw ratios of random variable with non-zero expectation are considered in this article. Hayman’s method for expansion of the entire function is used to derive the asymptotic formulas of the n-th raw and central moments for binomial and Poisson distributions. Laplace’s expansion of integral is used to obtain the asymptotic formula of the n-th raw moment for normal distribution. Consequently, the n-th central-raw ratios are shown to be infinitesimal with different patterns for binomial Poisson and normal distributions, respectively.

Effect of Sea Waves on Vertical Underwater Visible Light Communication Links

Underwater visible light communication (VLC) has been proposed to deal with emerging high bandwidth underwater applications. Initial research works on underwater VLC are based on the assumption that both transmitter and receiver are submerged, creating a horizontal link. In most of the vertical communication links, one of the transceiver nodes takes the form of a buoy and requires taking into the effect of the sea surface, which is inherently unsteady due to wind and waves. In this article, we consider a vertical underwater VLC link where the transmitter is in the form of a buoy at the sea surface, and the receiver is a submerged node at a certain depth. We assume sinusoidal waves and consider the fact that the buoy will fluctuate and oscillate, during drifting up and down, around its vertical axis. This effectively results in a 3-D displacement at the suspended transmitter. Building upon these assumptions of practical relevance, we propose an aggregate channel model, which includes a random path loss due to periodic changes of the transmission distance and a fading term induced by pointing errors with periodic changes of relative movement. Based on the proposed statistical model, we derive closed-form expressions for the exact and asymptotic bit error ratio and investigate the achievable diversity orders. We further present numerical results to confirm the analytical findings.

Testing equality of quantiles for several logistic populations with some real-life applications

One of the novel methods described by the researchers for comparing several independent populations is to compare their quantiles. Researchers have previously considered comparing quantiles for normal and exponential model setups. In some cases, the datasets could be modeled using a different distribution, such as a logistic distribution. In this study, we consider the problem of comparing the quantiles for several () logistic populations. First, the conventional likelihood ratio test is applied, and based on its asymptotic property, the cutoff point is determined. We propose two modifications of the likelihood ratio test- a standardized likelihood ratio test and a parametric bootstrap likelihood ratio test. Further, we propose a computational approach test to test the equality of quantiles by leveraging technology. The sizes and powers of all proposed tests are compared using an extensive simulation study. Except for the asymptotic likelihood ratio test, all the tests achieve the nominal level which has been revealed from our simulation study. Finally, we illustrate the importance of our model problem using two real-life situations.

Hall Fields and Current Systems of Magnetic Reconnection under Asymmetric Conditions

The Hall effect is a key component of collisionless reconnection. A previous study showed that a quadrupolar Hall pattern under symmetric conditions degenerates into a bipolar pattern under highly asymmetric conditions. We study the properties of Hall magnetic fields and current systems during three reconnection events observed by the Magnetospheric Multiscale mission at the dayside magnetopause. Although the asymptotic density ratio between the magnetosheath and magnetospheric plasmas is very high for all three events, clear quadrupolar Hall field patterns are observed in all three events. The quadrupolar Hall magnetic fields in the three events display their respective properties on the intensity asymmetry and the distributing location. Among these events, a quadrupolar Hall field pattern is observed for the first time in in situ observations: the magnetosheath Hall pattern occupies the whole midplane region, while the magnetosphere Hall pattern still exists under highly asymmetric conditions. Observations show that the plasma mixture modulates the density asymmetry in the Hall region, which can be very different from the asymptotic density asymmetry in the magnetosheath and magnetospheric inflow plasma. The analyses indicate that the different density asymmetry inside the Hall region, but not the asymptotic density asymmetry, is an exact indicator that explains the different observed Hall patterns. Based on the observed facts, we suggest that the three reconnection events studied here are in different phases of their development after they are triggered under highly asymmetric conditions. Our results provide new insights into how the Hall effect works with the evolution of asymmetry during reconnection.

Exponential quasi-ergodicity for processes with discontinuous trajectories

This paper tackles the issue of establishing an upper-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack’s inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property that we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications, notably for processes with jumps on a multidimensional state-space. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on ℝd that do not satisfy the classical Harnack’s inequality, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps.

Speed of Evolution and Correlations in Multi-Mode Bosonic Systems.

We employ an exact solution of the thermal bath Lindblad master equation with the Liouvillian superoperator that takes into account both dynamic and environment-induced intermode couplings to study the speed of evolution and quantum speed limit (QSL) times of a open multi-mode bosonic system. The time-dependent QSL times are defined from quantum speed limits, giving upper bounds on the rate of change of two different measures of distinguishability: the fidelity of evolution and the Hilbert-Schmidt distance. For Gaussian states, we derive explicit expressions for the evolution speed and the QSL times. General analytical results are applied to the special case of a two-mode system where the intermode couplings can be characterized by two intermode coupling vectors: the frequency vector and the relaxation rate vector. For the system initially prepared in a two-mode squeezed state, dynamical regimes are generally determined by the intermode coupling vectors, the squeezing parameter and temperature. When the vectors are parallel, different regimes may be associated with the disentanglement time, which is found to be an increasing (a decreasing) function of the length of the relaxation vector when the squeezing parameter is below (above) its temperature-dependent critical value. Alternatively, we study dynamical regimes related to the long-time asymptotic behavior of the QSL times, which is characterized by linear time dependence with the proportionality coefficients defined as the long-time asymptotic ratios. These coefficients are evaluated as a function of the squeezing parameter at varying temperatures and relaxation vector lengths. We also discuss how the magnitude and orientation of the intermode coupling vectors influence the maximum speed of evolution and dynamics of the entropy and the mutual information.