Non-analytic Functions Research Articles

Coherent reflection recovery in scattering from the ocean surface using the frequency-difference autoproducta).

The coherence of rough sea-surface-scattered acoustic fields decreases with increasing frequency. The frequency-difference autoproduct, a quadratic product of acoustic fields at nearby frequencies, mimics a genuine field at the difference frequency. In rough-surface scattering, the autoproduct's lower effective frequency decreases the apparent surface roughness, restoring coherent reflection. Herein, the recovery of coherent reflection in sea surface scattering via the frequency-difference autoproduct is examined for data collected off the coast of New Jersey during the Shallow Water '06 (SW06) experiment. An acoustic source at depth 40 m and receiver at depth 24.3 m and range 200 m interrogated 160 independent realizations of the ocean surface. The root mean square surface height h was 0.167 m, and broadcast frequencies were 14-20 kHz, so that 2.5 ≤kh cos θ≤ 3.7 for acoustic wavenumber k and incidence angle θ. Measured autoproducts, constructed from scattered constituent fields, show significant coherent reflection at sufficiently low difference frequencies. Theoretical results, using the Kirchhoff approximation and a non-analytic surface autocorrelation function, agree with experimental findings. The match is improved using a numerical strategy, exploiting the relationship between autoproduct-based coherence recovery, the ocean-surface autocorrelation function, and the ocean-surface height spectrum. Error bars computed from Monte Carlo scattering simulations support the validity of the measured coherence recovery.

Correction to: A gradient-enhanced partition-of-unity surrogate model and adaptive sampling strategy for non-analytic functions

Correction to: A gradient-enhanced partition-of-unity surrogate model and adaptive sampling strategy for non-analytic functions

Theory of Certain Non-Univalent Analytic Functions

ABSTRACT We investigate the non-univalent function’s properties reminiscent of the theory of univalent starlike functions. Let the analytic function ψ ( z ) = ∑ i = 1 ∞ A i z i , A 1 ≠ 0 be univalent in the unitdisk. Non-univalent functions may be found in the class ℱ ( ψ ) of analytic functions f of the form f ( z ) = z + ∑ k = 2 ∞ a k z k satisfying (zf′ (z)/f (z) – 1) ≺ ψ(z). Such functions, like the Ma and Minda classes k=2 of starlike functions, also have nice geometric properties. For these functions, growth and distortion theorems have been established. Further, we obtain bounds for some sharp coefficient functionals and establish the Bohr and Rogosinki phenomenon for the class ℱ ( ψ ) . Non-analytic functions that share properties of analytic functions are known as poly-analytic functions. Moreover, we compute Bohr and Rogosinski’s radius for poly-analytic functions with analytic counterparts in the class ℱ ( ψ ) or classes of Ma-Minda starlike and convex functions.

Current fluctuations in stochastically resetting particle systems.

We consider a system of noninteracting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) Each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run-and-tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomized. We study the effects of resetting on the distribution P(Q,t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution P_{an}(Q,t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution P_{qu}(Q,t) attains its stationary form for Q<Q_{crit}(t), while it remains time dependent for Q>Q_{crit}(t). We show that Q_{crit}(t) increases linearly with t for large t. On the scale where Q∼Q_{crit}(t), we show that P_{qu}(Q,t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10^{-14000}, we were able to compute numerically this nonanalytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

A gradient-enhanced partition-of-unity surrogate model and adaptive sampling strategy for non-analytic functions

A gradient-enhanced partition-of-unity surrogate model and adaptive sampling strategy for non-analytic functions

Hyper-power series and generalized real analytic functions

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.

Gravitational wave signals in an Unruh–DeWitt detector

We firstly generalize the massive scalar propagator for planar gravitational waves propagating on Minkowski space obtained recently in van Haasteren and Prokopec (2022 arXiv:2204.12930 [gr-qc]). We then use this propagator to study the response of a freely falling Unruh–DeWitt detector to a gravitational wave background. We find that a freely falling detector completely cancels the effect of the deformation of the invariant distance induced by the gravitational waves, such that the only effect comes from an increased average size of scalar field vacuum fluctuations, the origin of which can be traced back to the change of the surface in which the gravitational waves fluctuate. The effect originates from the quantum interference between propagation on off-shell detector’s trajectories which probe different spatial gravitational potential induced by the gravitational backreaction from gravitational waves, and it is therefore purely quantum. When resummed over classical graviton insertions, gravitational waves generate cuts on the imaginary axis of the complex -plane (where denotes the difference of proper times), and the discontinuity across these cuts is responsible for a continuum of energy transitions induced in the Unruh–DeWitt detector. Not surprisingly, we find that the detector’s transition rate is exponentially suppressed with increasing energy and the mass of the scalar field. What is surprising, however, is that the transition rate is a non-analytic function of the gravitational field strain. This means that, no matter how small is the gravitational field amplitude, expanding in powers of the gravitational field strain cannot approximate well the detector’s transition rate. We present numerical and approximate analytical results for the detector’s transition rate both for circularly polarized and for polarized monochromatic, unidirectional, gravitational waves.

Nonanalytic behavior of the relativistic r -modes in slowly rotating neutron stars

An inconsistency between the theoretical analysis and numerical calculations of the relativistic $r$-modes puzzles the neutron star community since the Kojima's finding of the continuous part in the $r$-mode oscillation spectrum in 1997. In this paper, after a brief review of the Newtonian $r$-mode theory and of the literature devoted to the continuous spectrum of $r$-modes, we apply our original approach to the study of relativistic oscillation equations. Working within the Cowling approximation, we derive the general equations, governing the dynamics of discrete relativistic $r$-modes for both barotropic (isentropic) and nonbarotropic stars. A detailed analysis of the obtained equations in the limit of extremely slow stellar rotation rate reveals that, because of the effect of inertial reference frame-dragging, the relativistic $r$-mode eigenfunctions and eigenfrequencies become {\it non-analytic} functions of the stellar angular velocity, $\Omega$. We also derive the explicit expressions for the $r$-mode eigenfunctions and eigenfrequencies for very small values of $\Omega$. These expressions explain the asymptotic behavior of the numerically calculated eigenfrequencies and eigenfunctions in the limit $\Omega\to 0$. All the obtained $r$-mode eigenfrequencies take discrete values in the frequency range, usually associated with the continuous part of the spectrum. No indications of the continuous spectrum, at least in the vicinity of the Newtonian $l=m=2$ $r$-mode frequency $\sigma=-4/3 \ \Omega$, are found.

Perturbation theory without power series: Iterative construction of non-analytic operator spectra

It is well known that quantum-mechanical perturbation theory often gives rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place. Using the elementary technique of relaxed fixed-point iteration, I obtain convergent expressions for various challenging ground-states wave functions, including quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing Rayleigh-Schrödinger coefficients), all at arbitarily strong coupling. These results challenge the notion that non-analytic functions of coupling constants are intrinsically “non-perturbative”. A possible application to exact diagonalization is briefly discussed.

Nonanalytic Relativistic r-Modes of Slowly Rotating Nonbarotropic Neutron Stars

We show that the r-modes of slowly rotating nonbarotropic neutron stars are described by nonanalytic functions of stellar angular velocity, which makes the perturbation techniques, used so far in the r-mode theoretical studies, inapplicable. In contrast to those studies and in accordance with numerical calculations beyond the slow rotation approximation, the obtained r-mode spectrum is discrete, which resolves the continuous spectrum problem, lasting since 1997. Our findings imply that the relativistic r-modes in slowly rotating neutron stars dramatically differ from their Newtonian cousins, which may have important implications for the detectability of r-mode signatures in observations, in particular for the r-mode excitation efficiency during the neutron star inspirals.

Lyapunov Spectrum Scaling for Classical Many-Body Dynamics Close to Integrability.

We propose a novel framework to characterize the thermalization of many-body dynamical systems close to integrable limits using the scaling properties of the full Lyapunov spectrum. We use a classical unitary map model to investigate macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM regime. We perform our analysis in two fundamentally distinct long-range and short-range integrable limits which stem from the type of nonintegrable perturbations. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only diverging control parameter and the rescaled spectrum approaching an analytical function. Short-range limits result in a dramatic slowing down of thermalization which manifests through the rescaled Lyapunov spectrum approaching a non-analytic function. An additional diverging length scale controls the exponential suppression of all Lyapunov exponents relative to the largest one.

Some new rearrangements in sensitivity integrals and concerning inequalities with their application in control

In this paper, we extend the sensitivity integrals for linear feedback systems via a non-analytic sensitivity function in Poisson–Jensen’s terminology. First, we sketch a generalized sensitivity integral accounting for the NMP zeros and the RHP poles of the sensitivity function. Interestingly, the sensitivity integral accounting for the NMP zeros becomes a special case of the generalized integral. Then the concerning sensitivity inequality and the associated lower bound are achieved. We also sketch a sensitivity integral that exploits a modified sensitivity function-based approach and stabilizes the linear feedback system. Their concerning sensitivity inequality and the associated lower bound are achieved as well. An illustrative case study example is employed to demonstrate the idea of the ‘sensitivity lower bound’ to choose the ability of controllers for the given linear feedback systems. The rearranged sensitivity lower bound offers simplified analysis involving fewer computation efforts.

Rational Krylov methods for fractional diffusion problems on graphs

In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product f(L^T) varvec{b}, where f is a non-analytic function involving fractional powers and varvec{b} is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for f(L^T) varvec{b} to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

Hairy black-holes in shift-symmetric theories

Scalar hair of black holes in theories with a shift symmetry are constrained by the no-hair theorem of Hui and Nicolis, assuming spherical symmetry, time-independence of the scalar field and asymptotic flatness. The most studied counterexample is a linear coupling of the scalar with the Gauss-Bonnet invariant. However, in this case the norm of the shift-symmetry current J2 diverges at the horizon casting doubts on whether the solution is physically sound. We show that this is not an issue since J2 is not a scalar quantity, since Jμ is not a diffinvariant current in the presence of Gauss-Bonnet. The same theory can be written in Horndeski form with a non-analytic function G5∼ log X . In this case the shift-symmetry current is diff-invariant, but contains powers of X in the denominator, so that its divergence at the horizon is again immaterial. We confirm that other hairy solutions in the presence of non-analytic Horndeski functions are pathological, featuring divergences of physical quantities as soon as one departs from time-independence and spherical symmetry. We generalise the no-hair theorem to Beyond Horndeski and DHOST theories, showing that the coupling with Gauss-Bonnet is necessary to have hair.

Giant leaps and long excursions: Fluctuation mechanisms in systems with long-range memory.

We analyze large deviations of time-averaged quantities in stochastic processes with long-range memory, where the dynamics at time t depends itself on the value q_{t} of the time-averaged quantity. First we consider the elephant random walk and a Gaussian variant of this model, identifying two mechanisms for unusual fluctuation behavior, which differ from the Markovian case. In particular, the memory can lead to large-deviation principles with reduced speeds and to nonanalytic rate functions. We then explain how the mechanisms operating in these two models are generic for memory-dependent dynamics and show other examples including a non-Markovian simple exclusion process.

Emergent scale behaviour in the effective equations for an elastic metamaterial

Asymptotic homogenisation of an elastic metamaterial consisting of a series of plates interspaced by a fluid is considered. It is shown that the usual method must be extended by the inclusion of a second macroscale, the “emergent scale”, in order to correctly capture the behaviour of this metamaterial at low frequency. The leading order solutions for plane wave propagation are found and the effective constitutive equations derived. At leading order, the dispersion curves for plane wave propagation depend on the direction of propagation and the branch associated with the emergent scale is dispersive. The effective constitutive equations contain some terms, associated with the shear normal to the plates, for which stress is not proportional to strain but rather depends on the third derivative of the normal displacement with respect to transverse position. These terms are not present in the usual form of the Willis equations and the effective fields are non-analytic functions of frequency as the frequency tends to zero.

Numerical evaluation of inverse integral transforms: Dynamic response of elastic materials

This study discusses the use of numerical integration in evaluating the improper integrals appearing as inverse integral transforms of non-analytic functions. These transforms appear while studying the response of various sources in an elastic medium through integral transform method. In these studies, the inverse Fourier transforms are solved numerically without bothering about the singularities and branch points in the corresponding integrands. References on numerical integration cited in relevant papers do not support such an evaluation but suggest contrary. Approximation of inverse Laplace transform integral into a series is used without following the essential restrictions and assumptions. Volume of the published papers using these dubious procedures has reached to an alarming level. The discussion presented aims to draw the attention of researchers as well as journals so as to stop this menace at the earliest possible.Keywords: Inverse Fourier transforms, inverse Laplace transforms, Romberg integration, improper integral, elastic waves

An efficient holomorphic embedded based approach for available transfer capability evaluation

Available Transfer Capability (ATC) is the additional power that can be securely and reliably transacted over and above the already committed transactions in a power network. In the electricity market, where every player (power producer and consumer) tries to earn higher profit, quick and accurate assessment of ATC of the power network is a prime concern. Holomorphic Embedding (HE) is a novel and non-iterative way of solving the Power Flow (PF) equations, which guarantees to find the operational High Voltage (HV) solution, in case it exists. Analytic continuation extends the domain of convergence of non-analytic functions to have voltage solutions. This paper, thus, proposes an extremely efficient and accurate approach for evaluation of ATC using the Holomorphic Embedded PF (HEPF) considering the intact system condition and N-1 contingency situations. This approach is successfully tested on IEEE 30 bus, IEEE 118 bus, and IEEE 300 bus systems for various bilateral and multilateral transactions. The results, thus, obtained prove the efficacy of the proposed approach in comparison with those obtained from the repeated Newton-Raphson PF (NRPF), Fast Decoupled PF (FDPF) and Continuation PF (CPF) based methods.

A Perturbation Analysis for General Nonlinear Continuous Systems Using Fourier Series and Green’s Function Method

In this paper, a procedure based on renormalization group method with the aid of Fourier series and Green’s function method is proposed to deal with continuous systems with general linear and nonlinear operators. In contrast to previous works, nonlinear operator has a general form (not necessarily quadratic and cubic form) and can be function of displacement, velocity, acceleration and time. The nonlinear operator may be non-analytic function of spatial coordinates and may be in an integro-differential form. We present our results for a general continuous system with and without internal resonance and general systems with nonlinear and non-homogenous boundary conditions. To show the effectiveness of the procedure, some examples are presented.

Asymptotical stability contouring control of dual‐arm robot with holonomic constraints: modified distributed control framework

The authors analyse the contour error dynamics in contouring control of dual-arm robotic manipulators with holonomic constraints. With the modified dynamics of robot, the dynamics of dual-arm robot with holonomic constraints is symetrically transformed into dynamics error problem by considering the equivalent error method. It becomes the control problem of stabilisation such that it is suitable for robust control approach to improve control performance in terms of contouring control, i.e. contour accuracy with high speed. The proposed method can deal with both analytic and non-analytic functions of desired path in task space. The experimental results reveal that the proposed method yields excellent performance in comparison to conventional distributed control. As a result, the proposed method can lead to the development of a modified distributed control for high degree of freedom robot manipulators.