Smooth Cutoff Research Articles

Fractional and tempered fractional models for Reynolds-averaged Navier–Stokes equations

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier–Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or nonlinear stress–strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative-based diffusion equations. Thus, in the paper, we formulate a fractional stress–strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model applied to channel, couette and pipe flow. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. An expression for the fractional order is also found, thereby solving the closure problem for the considered cases. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Within the context of this paper, we limit ourselves to wall bounded flow. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions, as truncating the domain has computational advantages. For the above investigations, we carefully designed algorithms, notably, the pointwise version of fractional physics-informed neural network to find the fractional order as an inverse problem.

Longitudinal magnetoconductance and the planar Hall effect in a lattice model of tilted Weyl fermions

The experimental verification of chiral anomaly in Weyl semimetals is an active area of investigation in modern condensed matter physics, which typically relies on the combined signatures of longitudinal magnetoconductance (LMC) along with the planar Hall effect (PHE). It has recently been shown that for weak nonquantizing magnetic fields, a sufficiently strong finite intervalley scattering drives the system to switch the sign of LMC from positive to negative. Here we unravel another independent source that produces the same effect. Specifically, a smooth lattice cutoff to the linear dispersion, which is ubiquitous in real Weyl materials, introduces nonlinearity in the problem and also drives the system to exhibit negative LMC for noncollinear electric and magnetic fields even in the limit of vanishing intervalley scattering. We examine longitudinal magnetoconductivity and the planar Hall effect semianalytically for a lattice model of tilted Weyl fermions within the Boltzmann approximation. We independently study the effects of a finite lattice cutoff and tilt parameters and construct phase diagrams in relevant parameter spaces that are relevant for diagnosing chiral anomaly in real Weyl materials.

On the Mathematical Foundations of Causal Fermion Systems in Minkowski Space

The emergence of the concept of a causal fermion system is revisited and further investigated for the vacuum Dirac equation in Minkowski space. After a brief recap of the Dirac equation and its solution space, in order to allow for the effects of a possibly nonstandard structure of spacetime at the Planck scale, a regularization by a smooth cutoff in momentum space is introduced, and its properties are discussed. Given an ensemble of solutions, we recall the construction of a local correlation function, which realizes spacetime in terms of operators. It is shown in various situations that the local correlation function maps spacetime points to operators of maximal rank and that it is closed and homeomorphic onto its image. It is inferred that the corresponding causal fermion systems are regular and have a smooth manifold structure. The cases considered include a Dirac sea vacuum and systems involving a finite number of particles and antiparticles.

Maximum Earthquake Size and Seismicity Rate from an ETAS Model with Slip Budget

ABSTRACTWe define a seismicity model based on (1) the epidemic-type aftershock sequence model that accounts for earthquake clustering, and (2) a closed slip budget at long timescale. This is achieved by not permitting an earthquake to have a seismic moment greater than the current seismic moment deficit. This causes the Gutenberg–Richter law to be modulated by a smooth upper cutoff, the location of which can be predicted from the model parameters. We investigate the various regimes of this model that more particularly include a regime in which the activity does not die off even with a vanishingly small spontaneous (i.e., background) earthquake rate and one that bears strong statistical similarities with repeating earthquake time series. Finally, this model relates the earthquake rate and the geodetic moment rate and, therefore, allows to make sense of this relationship in terms of fundamental empirical law (the Gutenberg–Richter law, the productivity law, and the Omori law) and physical parameters (seismic coupling, tectonic loading rate).

A Future Percent-level Measurement of the Hubble Expansion at Redshift 0.8 with Advanced LIGO

Abstract Simultaneous measurements of distance and redshift can be used to constrain the expansion history of the universe and associated cosmological parameters. Merging binary black hole (BBH) systems are standard sirens—their gravitational waveform provides direct information about the luminosity distance to the source. There is, however, a perfect degeneracy between the source masses and redshift; some nongravitational information is necessary to break the degeneracy and determine the redshift of the source. Here we suggest that the pair instability supernova (PISN) process, thought to be the source of the observed upper limit on the black hole mass in merging BBH systems at , imprints a mass scale in the population of BBH mergers and permits a measurement of the redshift–luminosity–distance relation with these sources. We simulate five years of BBH detections in the Advanced LIGO and Virgo detectors with a realistic BBH merger rate, mass distribution with smooth PISN cutoff, and measurement uncertainty. We show that after one year of operation at design sensitivity the BBH population can constrain H(z) to at a pivot redshift . After five years the constraint improves to . If the PISN cutoff is sharp, the uncertainty is smaller by about a factor of two. This measurement relies only on general relativity and the presence of a mass scale that is approximately fixed or calibrated across cosmic time; it is independent of any distance ladder. Observations by future “third-generation” gravitational wave detectors, which can see BBH mergers throughout the universe, would permit subpercent cosmographical measurements to z ≳ 4 within one month of observation.

Irregular parameter dependence of numerical results in tensor renormalization group analysis

Abstract We study the parameter dependence of numerical results obtained by the tensor renormalization group. We often observe irregular behavior as the parameters are varied with the method. Using the two-dimensional Ising model we explicitly show that the sharp cutoff used in the truncated singular value decomposition causes this unwanted behavior when the level crossing happens between singular values below and above the truncation order as the parameters are varied. We also test a smooth cutoff, instead of the sharp one, as a truncation scheme and discuss its effects.

Global in time Strichartz estimates for the fractional Schrödinger equations on asymptotically Euclidean manifolds

In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely e−itΛgσ with σ∈(0,∞)\\{1} and Λg:=−Δg where Δg is the Laplace–Beltrami operator on asymptotically Euclidean manifolds (Rd,g). Let f0∈C0∞(R) be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part (1−f0)(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥2 inside a compact set under non-trapping condition. On the other hand, under the moderate trapping assumption (1.12), the high frequency part also satisfies the global in time Strichartz estimates outside a compact set. We next prove that the low frequency part f0(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥3 without using any geometric assumption on g. As a byproduct, we prove global in time Strichartz estimates for the fractional Schrödinger and wave equations on (Rd,g), d≥3 under non-trapping condition.

Large-scale all-electron density functional theory calculations using an enriched finite-element basis

We present a computationally efficient approach to perform large-scale all-electron density functional theory calculations by enriching the classical finite element basis with compactly supported atom-centered numerical basis functions that are constructed from the solution of the Kohn-Sham (KS) problem for single atoms. We term these numerical basis functions as enrichment functions, and the resultant basis as the enriched finite element basis. The enrichment functions are compactly supported through the use of smooth cutoff functions, which enhances the conditioning and maintains the locality of the basis. The integrals involved in the evaluation of the discrete KS Hamiltonian and overlap matrix in the enriched finite element basis are computed using an adaptive quadrature grid based on the characteristics of enrichment functions. Further, we propose an efficient scheme to invert the overlap matrix by using a block-wise matrix inversion in conjunction with special reduced-order quadrature rules to transform the discrete Kohn-Sham problem to a standard eigenvalue problem. Finally, we solve the resulting standard eigenvalue problem by using a Chebyshev polynomial based filtering technique to compute the relevant eigenspectrum. We demonstrate the accuracy, efficiency and parallel scalability of the proposed method on semiconducting and heavy-metallic systems of various sizes, with the largest system containing 8694 electrons. We obtain accuracies in the ground-state energies that are within $\sim 1$mHa with ground-state energies obtained from classical finite element as well as gaussian basis sets. We observe a $50-300$ and $\sim 8$ fold reduction in the overall computational time when compared to classical finite element and gaussian basis, respectively. We also observe good parallel scalability up to 384 processors for a benchmark system comprising of 280-atom silicon nano-cluster.

The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta_p\in[0,\pi]$ to be the angle for which $a(p)=2p^{( k -1)/2}\cos \theta_p $. Let $I\subset[0,\pi]$ be a closed subinterval, and let $d\mu_{ST}=\frac{2}{\pi}\sin^2\theta d\theta$ be the Sato-Tate measure of $I$. Assuming that the symmetric power $L$-functions of $f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $x$ is sufficiently large, then \[ \left|\#\{p\leq x:\theta_p\in I\} -\mu_{ST}(I)\int_2^x\frac{dt}{\log t}\right|\ll\frac{x^{3/4}\log(N k x)}{\log x} \] with an implied constant of $3.34$. By letting $I$ be a short interval centered at $\frac{\pi}{2}$ and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers $n$ for which $a(n)\neq0$. In particular, if $\tau$ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that \[ \lim_{x\to\infty}\frac{\#\{n\leq x:\tau(n)\neq0\}}{x}>1-1.54\times10^{-13}. \] We also discuss the connection between the density of positive integers $n$ for which $a(n)\neq0$ and the number of representations of $n$ by certain positive-definite, integer-valued quadratic forms.

Two-loop RGE of a general renormalizable Yang-Mills theory in a renormalization scheme with an explicit UV cutoff

We perform a systematic one-loop renormalization of a general renormalizable Yang-Mills theory coupled to scalars and fermions using a regularization scheme with a smooth momentum cutoff $\Lambda$ (implemented through an exponential damping factor). We construct the necessary finite counterterms restoring the BRST invariance of the effective action by analyzing the relevant Slavnov-Taylor identities. We find the relation between the renormalized parameters in our scheme and in the conventional $\overline{\rm MS}$ scheme which allow us to obtain the explicit two-loop renormalization group equations in our scheme from the known two-loop ones in the $\overline{\rm MS}$ scheme. We calculate in our scheme the divergences of two-loop vacuum graphs in the presence of a constant scalar background field which allow us to rederive the two-loop beta functions for parameters of the scalar potential. We also prove that consistent application of the proposed regularization leads to counterterms which, together with the original action, combine to a bare action expressed in terms of bare parameters. This, together with treating $\Lambda$ as an intrinsic scale of a hypothetical underlying finite theory of all interactions, offers a possibility of an unconventional solution to the hierarchy problem if no intermediate scales between the electroweak scale and the Planck scale exist.

Thermal and structural properties of ionic fluids.

The electrostatic interaction in ionic fluids is well known to give rise to a characteristic phase behavior and structure. Sometimes its long range is proposed to single out the electrostatic potential over other interactions with shorter ranges. Here the importance of the range for the phase behavior and the structure of ionic fluids is investigated by means of grandcanonical Monte Carlo simulations of the lattice restricted primitive model (LRPM). The long-ranged electrostatic interaction is compared to various types of short-ranged potentials obtained by sharp and/or smooth cutoff schemes. Sharply cutoff electrostatic potentials are found to lead to a strong dependence of the phase behavior and the structure on the cutoff radius. However, when combined with a suitable additional smooth cutoff, the short-ranged LRPM is found to exhibit quantitatively the same phase behavior and structure as the conventional long-ranged LRPM. Moreover, the Stillinger-Lovett perfect screening property, which is well known to be generated by the long-ranged electrostatic potential, is also fulfilled by short-ranged LRPMs with smooth cutoffs. By showing that the characteristic phase behavior and structure of ionic fluids can also be found in systems with short-ranged potentials, one can conclude that the decisive property of the electrostatic potential in ionic fluids is not the long range but rather the valency dependence.

Gradual time reversal in thermo- and photo-acoustic tomography within a resonant cavity

Thermo- and photo-acoustic tomography require reconstructing initial acoustic pressure in a body from time series of pressure measured on a surface surrounding the body. For the classical case of free space wave propagation, various reconstruction techniques are well known. However, some novel measurement schemes place the object of interest between reflecting walls that form a de facto resonant cavity. In this case, known methods (including the popular time reversal algorithm) cannot be used. The inverse problem involving reflecting walls can be solved by the gradual time reversal method we propose here. It consists in solving back in time on the interval the initial/boundary value problem for the wave equation, with the Dirichlet boundary data multiplied by a smooth cutoff function. If T is sufficiently large one obtains a good approximation to the initial pressure; in the limit of large T such an approximation converges (under certain conditions) to the exact solution.

The Broken Ray Transform on the Square

We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for \(C_{0}^{2}\) perturbations of a vanishing absorption parameter \(\sigma \equiv 0\). Given an open subset \(E\) of the boundary, we measure the attenuation of all broken rays starting and ending at \(E\) with the standard optical reflection rule applied to \(\partial \Omega {\setminus } E\). Using the analytic microlocal approach of Frigyik et al. for the X-ray transform on generic families of curves, we show injectivity via a path unfolding argument under suitable conditions on the available broken rays. Then we show that with a suitable decomposition of the measurement operator via smooth cutoff functions, the associated normal operator is a classical pseudo differential operator of order \(-1\), which leads to the desired result.

‘Vacuum-like’ Hadamard states for quantum fields on curved spacetimes

We present a modification of the recently proposed Sorkin–Johnston states for scalar free quantum fields on a class of globally hyperbolic spacetimes possessing compact Cauchy hypersurfaces. The modification relies on a smooth cutoff of the commutator function and leads always to Hadamard states, in contrast to the original Sorkin–Johnston states. The modified Sorkin–Johnston states are, however, due to the smoothing no longer uniquely associated to the spacetime.

The Stationary Navier-Stokes System with No-Slip Boundary Condition on Polygons: Corner Singularity and Regularity

We study the stationary Navier-Stokes system with no-slip boundary condition on polygonal domains. Near each non-convex vertex the solution is shown solution. For showing the regularity we apply the Mellin transform to to have a unique decomposition by singular and regular parts. The singular part is defined by a linear combination of the corner singularity functions of the Stokes type and the regular part is shown to have the H 2 × H1-regularity. Precisely, near a non-convex vertex located at (0, 0), [u, p] = 𝒞1[Φ1, φ1] + 𝒞2[Φ2, φ2] + [u R , p R ], [u R , p R ] ∈ H s × H s−1 for s ∈ (λ2 + 1, 2], where Φ i = χ r λ i 𝒯 i (θ), φ i = χ r λ i −1ξ i (θ) with 1/2 < λ1 < λ2 < 1, χ is a smooth cutoff function, and 𝒞 i is the stress intensity factor. Hence the velocity vector is not Lipshitz continuous at non-convex vertices and the pressure value is infinite there. Also viscous stress tensor and vorticity values blow up near non-convex vertices.

Three-body-continuum Coulomb problem using a compact-kernel-integral-equation approach

We present an approach associated with the Jacobi matrix method to calculate a three-body wave function that describes the double continuum of an atomic two-electron system. In this approach, a symmetrized product of two Coulomb waves is used to describe the asymptotic wave function, while a smooth cutoff function is introduced to the dielectronic potential that enters its integral part in order to have a compact kernel of the corresponding Lippmann-Schwinger-type equation to be solved. As an application, the integral equation for the (${e}^{\ensuremath{-}},\phantom{\rule{4pt}{0ex}}{e}^{\ensuremath{-}},\phantom{\rule{4pt}{0ex}}{\mathrm{He}}^{2+}$) system is solved numerically; the fully fivefold differential cross sections (FDCSs) for ($e,3e$) processes in helium are presented within the first-order Born approximation. The calculation is performed for a coplanar geometry in which the incident electron is fast ($\ensuremath{\sim}6$ keV) and for a symmetric energy sharing between both slow ejected electrons at excess energy of 20 eV. The experimental and theoretical FDCSs agree satisfactorily both in shape and in magnitude. Full convergence in terms of the basis size is reached and presented.

Flux intensity functions for the Laplacian at axisymmetric edges

We derive explicit representation formulas for the computation of flux intensity functions for mixed boundary value problems for the Poisson equation in axisymmetric domains with edges. We rely on the decomposition of the boundary value problems in three dimensions by means of partial Fourier analysis with respect to the rotational angle into boundary value problems in the two‐dimensional meridian domain of . Utilizing smooth cutoff functions, the solutions of the reduced problems are analyzed semi‐analytically near corners of the plane meridian domain, and the edge flux intensity functions are constructed via Fourier synthesis and convergence analysis. The formulas are also applicable in the case of crack fronts. The constructive nature of the formulas provides in a straightforward way an efficient strategy for the accurate computation of edge flux intensity functions in axisymmetric domains. A demonstration example that illustrates the application of the formulas is presented. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

We consider C^2 families of C^4 unimodal maps f_t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of f_t depends differentiably on t, as a distribution of order 1. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of the acim for a Benedicks-Carleson map f_t, in terms of a single smooth function and the inverse branches of f_t along the postcritical orbit. Along the way, we prove that the twisted cohomological equation v(x)=\alpha (f (x)) - f'(x) \alpha (x) has a continuous solution \alpha, if f is Benedicks-Carleson and v is horizontal for f.

Stereo Acoustic Echo Cancellation Employing Frequency-Domain Preprocessing and Adaptive Filter

This paper proposes a windowing frequency domain adaptive filter and an upsampling block transform preprocessing to solve the stereo acoustic echo cancellation problem. The proposed adaptive filter uses windowing functions with smooth cutoff property to reduce the spectral leakage during filter updating, so the utilization of the independent noise introduced by preprocessing in stereo acoustic echo cancellation can be increased. The proposed preprocessing is operated in short blocks with low processing delay, and it uses frequency-domain upsampling to meet the minimal block length requirement given by the band limit of simultaneous masking. Therefore, the simultaneous masking can be well utilized to improve the audio quality. The acoustic echo cancellation simulations and the audio quality evaluation show that, the proposed windowing frequency domain adaptive filter performs better than the conventional frequency domain adaptive filter in both mono and stereo cases, and the upsampling block transform preprocessing provides better audio quality and stereo acoustic echo cancellation performance than the half-wave preprocessing at the same noise level.

Identification of a Residue in Helix 2 of Rice Plasma Membrane Intrinsic Proteins That Influences Water Permeability

Molecular selection, ion exclusion, and water permeation are well known regulatory mechanisms in aquaporin. Water permeability was found to be diverse in different subgroups of plasma membrane intrinsic proteins (PIPs), even though the residues surrounding the water holes remained the same across the subgroups. Upon homology modeling and structural comparison, a conserved Ala/Ile(Val) residue difference was identified in helix 2 that affected the conformation of the NPA region and consequently influenced the water permeability. The residue difference was found to be conservative within the two subgroups of PIPs in rice as well as in other plants. Functional tests further confirmed the prediction via site-directed mutagenesis where replacement of Ala(103) or Ala(102) in respective OsPIP1;1 or OsPIP1;3 with Val yielded 7.0- and 2.2-fold increases in water transportation, and substitution of Ile(98) or Val(95) in respective OsPIP2;3 or OsPIP2;7 with Ala resulted in 73 or 52% reduction of water transportation. Based on structural analyses and molecular dynamics simulations, we proposed that the difference in water permeability was attributed to the orientation variations of helix 2 that modified water-water and water-protein interactions.