Asymptotic Domain Research Articles

Accurate and Fast Estimation of the Continuum Limit in Path Integral Simulations of Quantum Oscillators and Crystals.

Convergence of imaginary-time path integral results requires a substantial number of beads (n) when quantum effects are significant. Traditional Trotter scaling approaches estimate the continuum limit (n → ∞) through extrapolation; however, their validity is restricted to the asymptotic domain of large n. We introduce an efficient extrapolation approach for quantum oscillators and crystals with harmonic character. The fitting function is inspired by the analytic solution of the harmonic oscillator (HO), or the Einstein crystal for solids. While the formulation is designed for first derivative properties (energy and pressure), its extension to second derivative properties (e.g., heat capacity) is straightforward. We apply the method to a one-dimensional HO and anharmonic oscillator (AO), as well as a three-dimensional Lennard-Jones crystal. Configurations are sampled using path integral molecular dynamics simulations in the canonical ensemble, at a low temperature of T = 0.1 (simulation units). Compared to Trotter extrapolation approaches, the new method demonstrates a substantial accuracy in estimating the continuum limit, using only a few simulations of relatively small system sizes. This efficiency significantly reduces computational cost, providing a powerful tool to facilitate computations for more complex and challenging systems, such as molecules and crystals modeled using ab initio methods.

Approximation of eigenvalues and eigenfunctions of the diffusion operator in a domain containing thin tubes by asymptotic domain decomposition method

We consider the spectral problem for the diffusion operator considered in a domain containing thin tubes. A new version of the method of partial asymptotic decomposition of the domain is introduced to reduce the dimension inside the tubes, getting a model of hybrid dimensions. The method truncates the tubes at some small distance from the ends of the tubes and replaces the longer part of the tubes with segments. At the interface of the three-dimensional and one-dimensional subdomains, special junction conditions are set: the pointwise continuity of the flux and the continuity of the average over a cross-section of the eigenfunctions. We obtain conditions on the ratio of the characteristic sizes in the transverse and longitudinal directions that ensure the closeness of two spectra, i.e. of the diffusion operator in the full-dimensional domain and the partially reduced one, keeping the conservation of the multiplicity, all up to a prescribed accuracy.

RIS Partitioning Based Scalable Beamforming Design for Large-Scale MIMO: Asymptotic Analysis and Optimization

In next-generation wireless networks, reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output (MIMO) systems are foreseeable to support a large number of antennas at the transceiver as well as a large number of reflecting elements at the RIS. To fully unleash the potential of RIS, the phase shifts of RIS elements should be carefully designed, resulting in a high-dimensional non-convex optimization problem that is hard to solve with affordable computational complexity. In this paper, we address this scalability issue by partitioning RIS into sub-surfaces, so as to optimize the phase shifts in sub-surface levels to reduce complexity. Specifically, each sub-surface employs a linear phase variation structure to anomalously reflect the incident signal to a desired direction, and the sizes of sub-surfaces can be adaptively adjusted according to channel conditions. We formulate the achievable rate maximization problem by jointly optimizing the transmit covariance matrix and the RIS phase shifts. Under the RIS partitioning framework, the RIS phase shifts optimization reduces to the manipulation of the sub-surface sizes, the phase gradients of sub-surfaces, as well as the common phase shifts of sub-surfaces. Then, we characterize the asymptotic behavior of the system with an infinitely large number of transceiver antennas and RIS elements. The asymptotic analysis provides useful insights on the understanding of the fundamental performance-complexity tradeoff in RIS partitioning design. We show that in the asymptotic domain, the achievable rate maximization problem has a rather simple form with an explicit physical meaning of optimization variables. We develop an efficient algorithm to find an approximately optimal solution to the asymptotic problem via a one-dimensional (1D) grid search. Moreover, we discuss the insights and impacts of the asymptotic result on finite-size system design. By applying the asymptotic result to a finite-size system with necessary modifications, we show by numerical results that the proposed design achieves a favorable tradeoff between system performance and computational complexity.

A survey of mean-square destabilization of multidimensional linear stochastic differential systems with non-normal drift

Mean-square stability analysis of linear stochastic differential systems obtained perturbing ordinary systems by linear terms driven by independent Wiener processes is investigated. The so obtained stochastic regions are contractions of the asymptotic stability domain of the linear ordinary system. In this work, the mean-square stability regions exact shape is provided by means of necessary and sufficient conditions in terms of the eigenvalues of the drift and the intensities of the noises. Special attention is paid to how different structures of the perturbation affect the mean-square stability of systems with non-normal drifts. In each case, the obtained explicit stability condition reveals the role played by the parameter that controls the non-normality.

The Slender Rigid Block: Archetype for the Seismic Analysis of Masonry Structures

ABSTRACT The elasto-plastic oscillator, representing the basic model for reinforced concrete and steel structures, shows its inadequacy when dealing with masonry structures. The column, resting on a rigid plane with friction, is shown to be particularly appropriate to model the dynamic behaviour of masonry structures. At this aim, this paper investigates the motion of a slender rigid block (SRB), initially resting in its vertical configuration and hit by a seismic input, represented through a single impulse constituted by a rectangular or sinusoidal half-wave or a simplified sequence of horizontal acceleration pulses, having rectangular or sinusoidal shapes. An accumulation of the rotation of the SRB exists under the action of sequences of opposite acceleration pulses. The dynamic collapse is shown to take place when the column is led to its critical configuration – where the restoring moment of the weight is lost – so that the last pulse can further increase the rotation of the column. Expressing the behaviour of the SRB by means of the force reduction factor, the ratio between the magnitude of the collapse horizontal acceleration and the static strength of the column, the difference between the responses of the column decreases with increasing the number of the pulses of the sequence. A simple definition of the asymptotic failure domain can be obtained with a good approximation with a limited number of pulses. It is possible to evaluate the minimum collapse acceleration intensity of seismic sequences of pulses of any assigned duration able to produce the collapse of the rigid block, representing the seismic strength of the SRB. Finally, the results obtained in the theoretical analysis are compared with the outcomes of more accurate and time-consuming non-linear dynamic or incremental analyses (IDA).

Optical theorem, crossing property, and derivative dispersion relations: implications on the asymptotic behavior of and

In this paper, we present some results on the behavior of the total cross section and ρ-parameter at asymptotic energies in proton–proton ( ) and antiproton–proton ( ) collisions. Hence, we consider three of the main theoretical results in high energy physics: the crossing property, derivative dispersion relation, and optical theorem. The use of such machinery facilitates the derivation of analytic formulas for a wide set of the measured global scattering parameters and some important relations between them. The suggested parameterizations approximate the energy dependence for the total cross section and ρ-parameter for and with a statistically acceptable quality in the multi-TeV region. Additionally, the qualitative description is obtained for important interrelations, namely difference, sum, and ratio of the antiparticle–particle and particle–particle total cross sections. Despite the reduced number of experimental data for the total cross section and ρ-parameter at the TeV-scale, which complicates any prediction for the beginning of the asymptotic domain, the fitting procedures indicates that asymptotia occur in the energy range 25.5–130 TeV. Moreover, in the asymptotic regime, we obtain . A detailed quantitative study of the energy behavior of the measured scattering parameters and their combinations in the ultra–high energy domain indicates that the scenario with the generalized formulation of the Pomeranchuk theorem is more favorable with respect to the original formulation of this theorem.

Rectifiability of entropy defect measures in a micromagnetics model

Abstract We study the fine properties of a class of weak solutions u of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in [T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 2001, 3, 294–338]. In particular, we prove that the entropy defect measure associated to u is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of u.

Conformation-dependent sequence design of polymer chains in melts

Conformation-dependent design of polymer sequences can be considered as a tool to control macromolecular self-assembly. We consider the monomer unit sequences created via the modification of polymers in a homogeneous melt in accordance with the spatial positions of the monomer units. The geometrical patterns of lamellae, hexagonally packed cylinders, and balls arranged in a body-centered cubic lattice are considered as typical microphase-separated morphologies of block copolymers. Random trajectories of polymer chains are described by the diffusion-type equations and, in parallel, simulated in the computer modeling, the probability distributions of block length k being calculated. The problem is similar to that of gambler’s ruin and first passage time in probability theory but the consideration is generalized to 3D and the domains of different shapes are considered. In any domain, the probability distribution can be described by the asymptote ∼k −3/2 at moderate values of k if the spatial size of the block is less than the smallest characteristic size of the domain. For large blocks, the exponential asymptote exp(−const ) is valid, d as being the asymptotic domain length (a is the monomer unit size). The number average block lengths and their dispersities change linearly with the domain size for lamellae, cylinders, and balls, when the domain is characterized by a single characteristic size. If the domain is described by more than one size, the number average block length can grow nonlinearly with the domain sizes and the length d as can depend on all of them.

A Lyapunov approach for attraction domain estimation of polynomial discrete nonlinear systems with quadratic and cubic terms

In this paper, we propose a Lyapunov method for estimating asymptotic stability domain of polynomial discrete nonlinear systems, drawn from an existing approach for attraction domain estimation of continuous nonlinear systems. Proceeding from two different forms of a parametric equation, we present two development methods with the aim of identifying a sufficient condition to obtain a part of the stability domain for quadratic and cubic n-dimensional polynomial discrete systems. The results consist in solving a polynomial equation and unconstrained nonlinear optimization problems, where in case of quadratic or cubic systems, significant simplifications are obtained. The main notable advantages of this approach are its applicability on high-dimensional systems and obtaining an ellipsoidal stability domain which approaches the exact attraction domain limits. Two application examples are illustrated to validate each development.

Analog black holes and energy extraction by super-radiance from Bose Einstein condensates (BEC) with constant density

This paper investigates the acoustic superradiance of the density and phase fluctuations from the single vortex state of a Bose-Einstein condensate, by employing full time-domain and asymptotic frequency domain numerical calculations. The draining bathtub model of an incompressible barotropic fluid is adopted to describe the vortex. The propagation of the axisymmetric density and phase fluctuations in the condensate are governed by the massless scalar Klein-Gordon wave equation, which establishes the rotating black-hole analogy. Hence, the amplified scattering of these fluctuations from the vortex comprise the superradiance effect. A particular coordinate transformation is applied to reveal the event horizon and the ergosphere termwise in the metric and the respective asymptotic spectral solutions. A comparative analysis of the time domain and asymptotic frequency domain results are given for a range of rotational speed of the vortex and the frequency of the impinging fluctuations. The agreement at low rotational speeds of the vortex is shown to be very good, which starts to deteriorate at higher rotational speeds due to increasing constraint violations of the time-domain calculations. We further demonstrate an asymptotic upper bound for the superradiance as a function of vortex rotational speed, provided that the vortex remains stable.

Design and Implementation of New Multilevel Inverter Topology for Trinary Sequence Using Unipolar Pulsewidth Modulation

This paper proposes a new multilevel inverter (MLI) topology that utilizes trinary sequence for the dc sources. It gives maximum output voltage level with minimum dc source and switch count when compared to other sequences, such as symmetric, natural, binary, and quasi-linear. This is due to the fact that the trinary sequence generates of all additive and subtractive combinations of input dc levels in the output voltage waveform. The concept is implemented on a 9-level asymmetric MLI using only four active devices. Multicarrier unipolar pulsewidth modulation technique is adopted to create the switching pulses. Theoretical calculation of total harmonic distortion in both voltage and current waveforms has been performed using asymptotic time domain formula. These values are compared with simulation and experimental values for different modulation indices. Power loss calculation for proposed topology is discussed with appropriate mathematical equations.

The Resolution Matrix in Tomographic Multiplexing: Optimization of Inter-Parameter Cross-Talk, Relative Quantitation, and Localization.

We use a resolution matrix-based Bayesian framework to compare inversion methods for tomographic fluorescence lifetime multiplexing in a diffuse medium, such as biological tissue. We consider three inversion methods; an asymptotic time domain (ATD) approach, based on a multiexponential analysis of time domain data, a direct time domain (DTD) approach, which is a minimum error solution, and a cross-talk constrained time domain (CCTD) inversion, which is a solution to an optimization problem that minimizes both error and cross-talk. We compare these methods using Monte Carlo simulations and time domain fluorescence measurements with tissue-mimicking phantoms. The ATD approach provides high accuracy of relative quantitation and spatial localization of two fluorophores embedded in a 18-mm thick turbid medium, with concentration ratios of up to 1:4.25. DTD leads to significant errors in relative quantitation and localization. CCTD provides improved quantitation accuracy over DTD, and better spatial resolution compared to ATD. We present a rigorous theoretical basis for these results and provide a complete derivation of the CCTD estimator. The Bayesian analysis also leads to a formula for rapid computation of the DTD inverse operator for large-scale tomography measurements. The ATD and CCTD inversion methods provide significant advantages over DTD for accurately estimating multiple overlapping fluorophores. Time domain fluorescence tomography, using zero cross-talk estimators, can serve as a powerful tool for quantifying multiple fluorescently labeled biological processes. The Bayesian framework presented here can be applied to general multiparameter inverse problems for the quantitative estimation of multiple overlapping parameters.

Thermodynamic and kinetic analysis of the melt spinning process of Fe-6.5 wt.% Si alloy

Abstract The microstructural evolution of Fe-6.5 wt.% Si alloy during rapid solidification was studied over a quenching rate of 4 × 104 K/s to 8 × 105 K/s. The solidification and solid-state diffusional transformation processes during rapid cooling were analyzed via thermodynamic and kinetic calculations. The Allen-Cahn theory was adapted to model the experimentally measured bcc_B2 antiphase domain sizes under different cooling rates. The model was calibrated based on the experimentally determined bcc_B2 antiphase domain sizes for different wheel speeds and the resulting cooling rates. Good correspondence of the theoretical and experimental data was obtained over the entire experimental range of cooling rates. Along with the asymptotic domain size value at the infinite cooling rates, the developed model represents a reliable extrapolation for the cooling rate > 106 K/s and allows one to optimize the quenching process.

Discretization of a Lyapunov Approach for Attraction Domain Estimation Through a BMI Optimization Problem

This paper proposes a Lyapunov method for estimating the asymptotic stability domain of polynomial discrete nonlinear systems. Based on an existing approach of attraction domain estimation for continuous nonlinear systems, we present an analogous technique focusing on discrete systems. The idea is to reformulate the problem as a bilinear matrix inequalities (BMI) optimization problem where the Lyapunov function is considered as a decision variable. The exploitation of polar coordinates and positive definite matrices properties defines the objective function and the constraint functions. One of the main advantages of this method is that the results are independent of the Lyapunov function. Furthermore, the obtained stability domain approaches the exact domain boundaries. An application example is illustrated to validate found results.

An alternative approach for stability analysis of discrete time nonlinear dynamical systems

The paper endeavours to solve the problem of the necessary and sufficient conditions for testing asymptotic stability of the equilibrium state without using a positive definite or semi-definite Lyapunov function for time-invariant nonlinear discrete-time dynamical systems. The solution is based on the concept of the -functions introduced in this paper. As a result, new necessary and sufficient conditions for asymptotic stability of such systems and an estimation or the exact determination of the asymptotic stability domain of the state are established. Examples are worked out to illustrate the results.

A nonparametric test for stationarity in functional time series

We propose a new measure for stationarity of a functional time series, which is based on an explicit representation of the $L^2$-distance between the spectral density operator of a non-stationary process and its best ($L^2$-)approximation by a spectral density operator corresponding to a stationary process. This distance can easily be estimated by sums of Hilbert-Schmidt inner products of periodogram operators (evaluated at different frequencies), and asymptotic normality of an appropriately standardized version of the estimator can be established for the corresponding estimate under the null hypothesis and alternative. As a result we obtain a simple asymptotic frequency domain level $\alpha$ test (using the quantiles of the normal distribution) for the hypothesis of stationarity of functional time series. Other applications such as asymptotic confidence intervals for a measure of stationarity or the construction of tests for "relevant deviations from stationarity", are also briefly mentioned. We demonstrate in a small simulation study that the new method has very good finite sample properties. Moreover, we apply our test to annual temperature curves.

Computation of the Effective Conductivity of Fibrous Composites with Imperfect Thermal Contact by Combination of Asymptotic Homogenization, Domain Decomposition and Finite Elements Methods

The methods of asymptotic homogenization, domain decomposition and finite elements are combined for the computation of the effective thermal conductivity of periodic biphasic fibrous composites with interfacial thermal resistance. The asymptotic homogenization method is used to obtain the so-called local problems on the periodic cell whose solution allows the calculation of the effective conductivity tensor. The numerical solution of the local problems requires a special treatment because of the temperature discontinuity on the interfaces due to the thermal barrier. In the present work these problems are decomposed into two, one for each phase, linked via a coupling condition. The finite element method, implemented in the software FreeFEM++, is employed to solve the resulting problems on each phase. FreeFEM++, which is based on the variational formulation of the problems, potentially allows to consider arbitrary shapes for both the fiber and the periodic cell cross sections. Numerical results for a square periodic cell with fibers of circular cross-section are presented and compared with results from other reported approaches.

Spatially Coupled Codes Optimized for Magnetic Recording Applications

Spatially coupled (SC) codes are a class of sparse graph-based codes known to have capacity-approaching performance. SC codes are constructed based on an underlying low-density parity-check (LDPC) code, by first partitioning the underlying block code and then putting replicas of the components together. Significant recent research efforts have been devoted to the asymptotic, ensemble-averaged study of SC codes, as these coupled variants of the existing LDPC codes offer excellent properties. While the asymptotic analysis is important, due to simplifying assumptions and averaging effects, results from the asymptotic domain are not readily translatable to the practical, finite-length setting. Despite this chasm, the finite-length analysis of SC codes is still largely unexplored. In this paper, we tackle the problem of optimized design of SC codes in the context of magnetic-recording (MR) applications. In particular, we identify combinatorial structures in the graphical representation of the code that are detrimental in the MR setting. An intriguing observation is that for the same SC code, the problematic objects for the MR channels are combinatorially different from the additive white Gaussian noise (AWGN) setting, thus necessitating a careful code design approach for the MR applications. We first demonstrate that the choice of the so-called cutting vector, which guides the code partitioning in the SC code design, directly affects the cardinality of these problematic objects. In particular, we show that the number of problematic objects is the highest - and consequently that the performance is the worst - in the case of the degenerate cutting vector, which precisely corresponds to uncoupled LDPC block codes. We, therefore, show that coupling always improves the performance and that the degree of improvement is dependent on the choice of the cutting vector. We then extend our analysis to different column weights and present SC codes that outperform block codes with unoptimized edge weights by more than 3.5 orders of magnitude and that also outperform both optimized block codes and unoptimized SC codes by more than two orders of magnitude. Through presented examples, we demonstrate high performance of the proposed code design methodology.

The Three-Body Coordinate Asymptotics with Explicitly Orthogonalized Channels

We construct the asymptotic boundary conditions for Faddeev components of the wave function with the explicitly orthogonalized binary and breakup channels for the three-body scattering problem in configuration space. This makes it possible to obtain an accurate solution of the scattering problem above the breakup threshold, calculating scattering amplitudes by matching the solution of the Faddeev equations with the asymptote of the Faddeev wave function component in the asymptotic domain.

Comparison of tomographic fluorescence spectral and lifetime multiplexing.

Multispectral and lifetime imaging in turbid media can be mathematically described in two steps, involving spectral or temporal mixing of the fluorophores and the diffuse light transport in the turbid medium. We show that the order of fluorophore mixing and diffuse propagation is reversed in spectral and lifetime multiplexing, resulting in a fundamental difference in their multiplexing capabilities, regardless of the measurement conditions. Using the resolution matrix to define a quantitative measure for inter-fluorophore cross-talk, we show that lifetime multiplexing, using the asymptotic time domain approach, provides zero cross-talk, while spectral multiplexing can achieve zero cross-talk under special conditions. We also compare the performance of spectral and lifetime multiplexing for tomographic inversion of two overlapping fluorophores in a heterogeneous digital mouse atlas.