Truncation Approximation Research Articles

How do mirror charge radii constrain density dependence of the symmetry energy?

It has recently been suggested that differences in the charge radii of mirror nuclei (ΔRchmirr) are strongly correlated with the neutron-skin thickness (Rskin) of neutron-rich nuclei and with the slope of the symmetry energy (L). To test this assumption, we present ab initio calculations of Rskin in 48Ca and 208Pb, ΔRchmirr in 36Ca−36S, 38Ca−38Ar, 41Sc−41Ca, 48Ni−48Ca, 52Ni−52Cr, and 54Ni−54Fe mirror pairs, and L. Employing the recently developed 34 chiral interaction samples, identified by the history matching approach, we conduct rigorous statistical analysis of correlations among ΔRchmirr, Rskin and L, accounting for quantified uncertainties from low-energy constants of chiral interaction, chiral effective field theory truncation and many-body method approximation. The ab initio results reveal an appreciable ΔRchmirr−L correlation in fp-shell mirror pairs. However, contrary to previous studies, the present calculation finds that the studied sd-shell mirror pairs do not exhibit any ΔRchmirr−L correlation.

On the balanced truncation error bound and sign parameters from arrowhead realizations

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study.

On the spectrum around numerical solutions in Siegel gauge in open string field theory

Abstract In bosonic open string field theory, the spectrum around the numerical tachyon vacuum solution in Siegel gauge was investigated by Giusto and Imbimbo. Using their numerical method, we study the mass spectrum around two other solutions, which are “double brane” and “single brane” solutions in Siegel gauge constructed by the level truncation approximation. The “double brane” solution was constructed by Kudrna and Schnabl and its energy might correspond to a double brane. On the other hand, the “single brane” solution was constructed by Takahashi and the author in the theory around the identity-based solution for the tachyon vacuum and its energy corresponds to the perturbative vacuum, namely, a single brane. From the eigenvalues of the matrix for the kinetic term in Siegel gauge, we find a tachyon state and a massless vector state in the ghost number g = 1 sector around the “single brane” solution, which is consistent with the perturbative vacuum, although the mass spectrum around the “double brane” solution is obscure up to truncation level L = 10 and within scalar and vector states.

On kinematics of one‐dimensional radially symmetric shocks in non‐ideal reacting gases

In this manuscript, the advancement of one‐dimensional shock waves propagating through a non‐ideal reacting gas is examined. Considering the shock wave as a propagating discontinuity surface, a pair of evolution equations describing the strength of the shock and the first‐order discontinuity for variable initial data are obtained. The truncation approximations are used to obtain the exponent of shock velocity for the case of strong shock with constant initial data and compared with those exponents obtained from Guderley's scheme and Chisnell–Chester–Whitham (CCW) rule. The effects of non‐ideal and reaction parameters on the shock evolution in the case of arbitrary shock strength are examined and compared with those obtained from the CCW rule. Furthermore, the global behaviors of shock strength and induced discontinuity for different values of non‐ideal and reaction parameters are analyzed.

A geometric convergence formula for the level-increment-truncation approximation of M/G/1-type Markov chains

This paper considers an approximation usually used when implementing Ramaswami's recursion for the stationary distribution of the M/G/1-type Markov chain. The approximation is called the level-increment-truncation approximation because it truncates level increment at a given threshold. The main contribution of this paper is to present a geometric convergence formula of the level-wise difference between the respective stationary distributions of the original M/G/1-type Markov chain and its LI truncation approximation under the assumption that the level-increment distribution is light-tailed.

Boundary states in the SU(2)k WZW model from open string field theory

We analyze boundary states in the SU(2)k WZW model using open string field theory in the level truncation approximation. We develop algorithms that allow effective calculation of the action in this model and we search for classical solutions of the equations of motion, which are conjectured to describe boundary states. We find three types of solutions. First, there are real solutions that represent maximally symmetric Cardy boundary states and we show that they satisfy certain selection rules regarding their parameters. Next, we find complex solutions that go beyond the SU(2) model and describe maximally symmetric SL(2, ℂ) boundary conditions. Finally, we find exotic solutions that correspond to symmetry-breaking boundary states. Most of real exotic solutions describe the so-called B-brane boundary states, but some may represent yet unknown boundary states.

Generic two-phase coexistence in a type-2 Schloegl model for autocatalysis on a square lattice: Analysis via heterogeneous master equations.

Schloegl's second model (also known as the quadratic contact process) on a square lattice involves spontaneous annihilation of particles at lattice sites at rate p, and their autocatalytic creation at unoccupied sites with n≥2 occupied neighbors at rate k_{n}. Kinetic Monte Carlo (KMC) simulation reveals that these models exhibit a nonequilibrium discontinuous phase transition with generic two-phase coexistence: the p value for equistability of coexisting populated and vacuum states, p_{eq}(S), depends on the orientation or slope, S, of a planar interface separating those phases. The vacuum state displaces the populated state for p>p_{eq}(S), and the opposite applies for p<p_{eq}(S) for 0<S<∞. The special "combinatorial" rate choice k_{n}=n(n-1)/12 facilitates an appealing simplification of the exact master equations for the evolution of spatially heterogeneous states in the model, which aids analytic investigation of these equations via hierarchical truncation approximations. Truncation produces coupled sets of lattice differential equations which can describe orientation-dependent interface propagation and equistability. The pair approximation predicts that p_{eq}(max)=p_{eq}(S=1)=0.09645 and p_{eq}(min)=p_{eq}(S→∞)=0.08827, values deviating less than 15% from KMC predictions. In the pair approximation, a perfect vertical interface is stationary for all p<p_{eq}(S=∞)=0.08907, a value exceeding p_{eq}(S→∞). One can regard an interface for large S→∞ as a vertical interface decorated with isolated kinks. For p<p_{eq}(S=∞), the kink can move in either direction along this otherwise stationary interface depending upon p, but for p=p_{eq}(min) the kink is also stationary.

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

AbstractTopological insulators (TIs) are renowned for their remarkable electronic properties: quantized bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations of these physical properties generally rely on artificial periodicity (the supercell approximation, which struggles in the presence of edges), or unphysical boundary conditions (artificial truncation). In this work, we build on recently developed methods for computing spectral properties of infinite-dimensional operators. We apply these techniques to develop efficient and accurate computational tools for computing the physical properties of TIs. These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinite-dimensional operator directly, even in the presence of material defects, edges and disorder. Our methods permit computation of spectra, approximate eigenstates, spectral measures, spectral projections, transport properties and conductances. Numerical examples are given for the Haldane model, and the techniques can be extended similarly to other TIs in two and three dimensions.

Fractional derivative truncation approximation for real-time applications

Fractional derivatives are non local operators as they are well-suited for modeling long-memory phenomena such as heat transfers or fluid dynamics. However, such non-locality implies a constant knowledge of the full past of the functions to differentiate. In the context of real-time (online) system identification, such a global property may limit the implementation as calculations can become slower as time grows. Also, instead of using the full data length to compute fractional order derivatives, truncated fractional derivatives are used. However, such a truncation windowing will bring inaccuracy on the computation of fractional derivatives. This study deals with the relationship between the signal frequency content, the fractional derivative truncated approximation as well as the fractional system relaxation in order to get a good approximation of the truncated fractional derivative and consequently, to help providing real-time exploitable system identification algorithms that uses such truncated fractional derivatives.

Instability analysis and regularization approximation to the forward/backward problems for fractional damped wave equations with random noise

Considered herein is the forward/backward problems for fractional damped wave equations utt+α(−Δ)s1ut+β(−Δ)s2u=F(u,x,t) subject to the random Gaussian white noise initial and final data. First of all, we construct the mild solutions to forward/backward problems under all cases for parameters α, β, s1 and s2, and then investigate their stability and instability properties in the sense of Hadamard. Secondly, we propose the regularized solutions by using the Fourier truncation method and establish well-posedness for regularized solutions in above unstable cases. Thirdly, we derive some error estimates between the exact solutions and their regularized solutions in E‖⋅‖22-norm, and theoretically characterize the Fourier truncation approximation effect from regularized solutions to exact solutions. Finally, we give a series of numerical examples used to illuminate the regularization approximation effect of above method.

Spatial spin-spin correlations of the single-impurity Anderson model with a ferromagnetic bath

We investigate the interplay between the Kondo effect and the ferromagnetism by an one dimension Anderson impurity model with a spin partially polarized bath, using the projective truncation approximation under Lacroix basis.The equal-time spatial spin-spin correlation function (SSCF) is calculated. For the case of spin-unpolarized conduction electrons, it agrees qualitatively with the results from density matrix renormalization group (DMRG). For system with partially spin-polarized conduction electrons, an oscillation in the envelope of SSCF emerges due to the beating of two Friedel oscillations associated to two spin-split Fermi surfaces of conduction electrons. The period is proportional to the inverse of magnetic field $h$. A fitting formula is proposed to perfectly fits the numerical results of SSCF in both the short- and long-range regions. For large enough bath spin polarization, a bump appears in the curve of the integrated SSCF. It marks the boundary between the suppressed Kondo cloud and the polarized bath sites.

Effective Hamiltonian and spin dynamics in fast MAS TRAPDOR-HMQC experiments involving spin-3/2 quadrupolar nuclei

We present a theoretical and numerical description of the spin dynamics associated with TRAPDOR-HMQC (T-HMQC) experiment for a 1H (I) – 35Cl (S) spin system under fast magic angle spinning (MAS). Towards this an exact effective Hamiltonian describing the system is numerically evaluated with matrix logarithm approach. The different magnitudes of the heteronuclear and pure S terms in the effective Hamiltonian allow us to suggest a truncation approximation, which is shown to be in excellent agreement with the exact time evolution. Limitations of this approximation, especially at the rotary resonance condition, are discussed. The truncated effective Hamiltonian is further employed to monitor the buildup of various coherences during TRAPDOR irradiation. We observe and explain a functional resemblance between the magnitude of different terms in the truncated effective Hamiltonian and the amplitudes of various coherences during TRAPDOR irradiation, as function of crystallite orientation. Subsequently, the dependence of the sign (phase) of the T-HMQC signal on the coherence type generated is investigated numerically and analytically. We examine the continuous creation and evolution of various coherences at arbitrary times, i.e., at and between avoided level crossings. Behavior between consecutive crossings is described analytically and reveals ‘quadrature’ evolution of pairs of coherences and coherence interconversions. The adiabatic, sudden, and intermediate regimes for T-HMQC experiments are discussed within the approach established by A. J. Vega. Equations as well as numerical simulations suggest the existence of a driving coherence which builds up between consecutive crossings and then gets distributed at crossings among other coherences. In the intermediate regime, redistribution of the driving coherence to other coherences is almost uniform such that coherences involving S-spin double-quantum terms may be efficiently produced.

Projective-truncation-approximation study of the one-dimensional ϕ^{4} lattice model.

In this paper, we first develop the projective truncation approximation (PTA) in the Green's function equationof motion (EOM) formalism for classical statistical models. To implement PTA for a given Hamiltonian, we choose a set of basis variables and projectively truncate the hierarchical EOM. We apply PTA to the one-dimensional ϕ^{4} lattice model. Phonon dispersion and static correlation functions are studied in detail. Using one- and two-dimensional bases, we obtain results identical to and beyond the quadratic variational approximation, respectively. In particular, we analyze the power-law temperature dependence of the static averages in the low- and high-temperature limits, and we give exact exponents.

Nearly Exact Discrepancy Principle for Low-Count Poisson Image Restoration.

The effectiveness of variational methods for restoring images corrupted by Poisson noise strongly depends on the suitable selection of the regularization parameter balancing the effect of the regulation term(s) and the generalized Kullback–Liebler divergence data term. One of the approaches still commonly used today for choosing the parameter is the discrepancy principle proposed by Zanella et al. in a seminal work. It relies on imposing a value of the data term approximately equal to its expected value and works well for mid- and high-count Poisson noise corruptions. However, the series truncation approximation used in the theoretical derivation of the expected value leads to poor performance for low-count Poisson noise. In this paper, we highlight the theoretical limits of the approach and then propose a nearly exact version of it based on Monte Carlo simulation and weighted least-square fitting. Several numerical experiments are presented, proving beyond doubt that in the low-count Poisson regime, the proposed modified, nearly exact discrepancy principle performs far better than the original, approximated one by Zanella et al., whereas it works similarly or slightly better in the mid- and high-count regimes.

Efficient calculation of radiative intensity including the polarization effect in moderately thick atmospheres using a truncation approximation

The computational efficiency of solving the radiative transfer equation is an important issue for remote sensing and monitoring of aerosols and radiation. This study developed a novel calculation method for polarized radiance named Pn-IMS (Improved Multiple and Single scattering approximation by n-th order multiple scattering correction of the forward Peak Pn), consisting of P1-IMS, P2-IMS and P3-IMS. The P1- and P2-IMS methods are extended versions of the TMS (Truncated Multiple and Single scattering approximation) and IMS (Improved Multiple and Single scattering approximation) methods of Nakajima and Tanaka (1988), and are, respectively, the first and secondary order scattering corrections of the forward peak of the scattering phase function in the scalar approximation of the radiation field. We extended these methods to include the polarization effect based on the vector radiative transfer theory. We also developed the third order scattering correction named P3-IMS method and generalized the formula for any n-th order scattering. A series of numerical tests were performed to indicate that the P1-IMS method is accurate enough to reconstruct the Stokes parameters to within 0.2% with a hemispheric Gaussian stream number as low as N = 10 in the hemisphere, except for total radiance. The total radiance in the solar aureole region requires a higher order scattering correction by the P2- and P3-IMS methods. Numerical tests indicated that the P2- and P3-IMS methods have an accuracy within 5 and 1%, respectively, with N = 10 in the 340–1020 nm spectral region in a moderately thick atmosphere at an aerosol optical thickness at 500 nm of < 1. Thus, the P3-IMS method is an efficient calculation method more than the single scattering correction method required N > 20.

Augmented truncation approximations to the solution of Poisson’s equation for Markov chains

Poisson’s equation has a lot of applications in various areas, such as Markov decision theory, perturbation theory, central limit theorems (CLTs), etc. Usually it is hard to derive the explicit expression of the solution of Poisson’s equation for a Markov chain on an infinitely many state space. Here we will present a computational framework for the solution for both discrete-time Markov chains (DTMCs) and continuous-time Markov chains (CTMCs), by developing the technique of augmented truncation approximations. The censored Markov chain and the linear augmentation to some columns are shown to be effective truncation approximation schemes. Moreover, the convergence to the variance constant in CLTs are also considered. Finally the results obtained are applied to discrete-time single-birth processes and continuous-time single-death processes.

Interacting spinless fermions on the square lattice: Charge order, phase separation, and superconductivity

We investigate the phase diagram of spinless fermions on a square lattice with nearest-neighbor interaction, using the recently developed projective truncation approximation in Green's function equation of motion. For attractive interaction, the ground state is in an homogeneous p + ip superconducting (SC) phase at high or low electron densities. Near half filling is a phase separation (PS) between the SC phases. Allowing inhomogeneous solution, we obtain p-wave SC domains with positive interface energy. As temperature increases, the SC phases transit into normal phases above Tsc, generating an homogeneous normal phase (far away from n = 1/2), or a PS between normal phases with different densities (close to n = 1/2). Further increasing temperature to Tps, the PS disappears and the particle-hole symmetry of the Hamiltonian is recovered. For repulsive interaction, depending on electron filling, the ground state is in charge-ordered phase (half filling), charge-disordered phase (large hole/electron doping), or PS between them (weak doping). At finite temperature, the regime of charge order phase moves to finite V and extends to finite doping regime.

Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents

Abstract In this work, we consider the system of partial differential equations describing one-dimensional (1D) radially symmetric (i.e., cylindrical or spherical) flow of a nonideal gas with small solid dust particles. We analyze the implosion of cylindrical and spherical symmetric strong shock waves in a mixture of a nonideal gas with small solid dust particles. An evolution equation for the strong cylindrical and spherical shock waves is derived by using the Maslov technique based on the kinematics of 1D motion. The approximate value of the similarity exponent describing the behavior of strong shocks is calculated by applying a first-order truncation approximation. The obtained approximate values of similarity exponent are compared with the values of the similarity exponent obtained from Whitham’s rule and Guderley’s method. All the above computations are performed for the different values of mass fraction of dust particles, relative specific heat, and the ratio of the density of dust particle to the density of the mixture and van der Waals excluded volume.

Dynamics and enhanced stability properties of slender leaching tubings in salt cavern storage with a Y-type manifold fitted at free downstream end

Currently, the construction of salt cavern storages in China generally adopts solution mining technology with single-well convection and using a diesel cushion, however, leaching tubing is occasionally subjected to excessive bending and/or breaking during leaching process. Leaching tubing failure leads to the cavern shape being out of control or even scrapped. In this paper, regarding the leaching tubing as vertical hanging pipes conveying fluid, the dynamic equation of the leaching tubing with a Y-type manifold installed at the lower free end is established. The equation of motion is solved by means of Galerkin's truncation approximation method, and the dynamic response of the leaching tubing and the characteristics of the enhanced stability of the manifold are investigated. Since the downstream Y-type manifold has a select of the inclination angle, it is found through theoretical calculations that the flutter instability flow velocity of the fluid-conveying pipe can be conveniently controlled. The physical model experiment verifies that the Y-type manifold has a better enhancement effect on the stability of the leaching tubing. The proposed Y-type manifold may be helpful for the design and improvement of leaching tubing systems in the construction of salt cavern storage and salt mining.

NUMERICAL IMPLEMENTATION OF THE AUGMENTED TRUNCATION APPROXIMATION TO SINGLE-SERVER QUEUES WITH LEVEL-DEPENDENT ARRIVALS AND DISASTERS

This paper considers the computation of the stationary queue length distribution in single-server queues with level-dependent arrivals and disasters. We assume that service times follow a general distribution and therefore, we consider the stationary queue length distribution via an imbedded Markov chain. Because this imbedded Markov chain has infinitely many states, level dependence, and bidirectional jumps of levels, it is hard to compute the solution of the global balance equation exactly. We thus consider the augmented truncation approximation. In particular, we focus on the computation of the truncated state transition probability matrix of the imbedded Markov chain, assuming that the underlying continuous-time absorbing Markov chain during a service time is not uniformizable. Under some stability conditions, we develop a computational procedure for the truncated transition probability matrix, where the upper bound of errors owing to truncation can be set in advance. We also provide some numerical examples and demonstrate that our procedure works well.