Inverse Polynomial Research Articles

Simulating quantum circuits by adiabatic computation: Improved spectral gap bounds

Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L \gg 1$ is $\min_s\Delta \gtrsim \pi^2 / [8(L+1)^2]$, an improvement over previous estimates. Our first method is a direct approach based on an eigenstate ansatz, while the the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.

A Novel Fuzzy PI Control Approach for Nonlinear Processes

Classical fuzzy logic controllers fail to control nonlinear systems, which have strong nonlinearity around a certain operating point. In this case, if it is possible, the block-oriented modeling approach could be adopted to separate the highly nonlinear system into two subsystems as a linear time-invariant subsystem and a memoryless nonlinear subsystem. Thereby, a control operation could be carried out on the linear time-invariant block to control the whole system. Respecting this approach, here, a highly nonlinear system (neutralization process) was modeled by using the ARX–polynomial cascade connection. By this means, an opportunity is obtained that the inverse polynomial is utilized to be able to control the ARX subsystem by a classical fuzzy PI controller. And therefore, the fuzzy PI controller performance in controlling nonlinear processes was enhanced. The success of the new approach was confirmed by the results obtained from the control operation carried out in different pH regions for a chemical neutralization process. And also, the results show that the proposed control strategy is as successful as the multiregion fuzzy PI controller in the highly nonlinear process control.

Development and analysis of fifth stage inverse polynomial scheme for the solution of stiff linear and nonlinear ordinary differential equations

This paper presents the development and analysis of fifth stage inverse polynomial scheme (ISM5) for the solution of Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) of stiff types. The properties of the scheme were investigated and analyzed. The new scheme is tested on three numerical examples and the results were compared with the Runge-Kutta method of order five (RK5) in the context of the analytical solution. The results generated via ISM5 agreed with the analytical solution and outperformed RK5. Hence, ISM5 is found to be accurate, efficient and a powerful method for obtaining the numerical solution of stiff ODEs.

PRELIMINARY STUDY ON WAVELET DENOISING METHOD FOR INVERSION OF SNOW DEPTH IN GNSS-MR

Abstract. GNSS-MR technology inverts snow depth by using low-altitude SNR data. This information contains rich surface information. Due to the complex surface environment, the signal received by the receiver contains noise，during snow depth inversion, it is not possible to extract relatively "pure" snow information. Therefore, the wavelet denoising method is used to compare with the traditional polynomial. The results show that the snow depth RMSE of the polynomial inversion is 8cm and the error is 7cm. Wavelet denoising inversion The snow depth RMSE is 5cm and the error is 3cm. The experimental verification wavelet denoising method can better eliminate the systematic error and improve the inversion precision.

On Inverses of Permutation Polynomials of Small Degree Over Finite Fields

Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all PPs of degree $\leq 6$ over finite fields $\mathbb{F}_{q}$ for all $q$ and the inverses of all PPs of degree $7$ over $\mathbb{F}_{2^n}$. The explicit inverse of a class of fifth degree PPs is the main result, which is obtained by using Lucas' theorem, some congruences of binomial coefficients, and a known formula for the inverses of PPs of finite fields.

Development and analysis of fifth stage inverse polynomial scheme for the solution of stiff linear and nonlinear ordinary differential equations

This paper presents the development and analysis of fifth stage inverse polynomial scheme (ISM5) for the solution of Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) of stiff types. The properties of the scheme were investigated and analyzed. The new scheme is tested on three numerical examples and the results were compared with the Runge-Kutta method of order five (RK5) in the context of the analytical solution. The results generated via ISM5 agreed with the analytical solution and outperformed RK5. Hence, ISM5 is found to be accurate, efficient and a powerful method for obtaining the numerical solution of stiff ODEs.

On the Inverse Degree Polynomial

Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials.

Open-boundary spectral and flux-balance Vlasov simulation

Simulations of one-dimensional Vlasov–Maxwell solutions with non-periodic boundary conditions are discussed. Results obtained using a recently developed filtered flux-balance simulation system are compared to those obtained using a filtered, Fourier–Fourier transformed system. Excellent agreement is confirmed except for the appearance of the Gibbs phenomenon on the discontinuous simulated solutions of the transformed system. Recovery of the flux-balance results from the Fourier transformed results using the inverse polynomial reconstruction method is demonstrated.

Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes

The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of non-Gaussian processes obtained by regular non-linear transformations of Gaussian processes. We provide the increasing-domain asymptotic properties of the (Gaussian) maximum likelihood and cross validation estimators of the covariance parameters of a non-Gaussian process of this class. We show that these estimators are consistent and asymptotically normal, although they are defined as if the process was Gaussian. They do not need to model or estimate the non-linear transformation. Our results can thus be interpreted as a robustness of (Gaussian) maximum likelihood and cross validation towards non-Gaussianity. Our proofs rely on two technical results that are of independent interest for the increasing-domain asymptotic literature of spatial processes. First, we show that, under mild assumptions, coefficients of inverses of large covariance matrices decay at an inverse polynomial rate as a function of the corresponding observation location distances. Second, we provide a general central limit theorem for quadratic forms obtained from transformed Gaussian processes. Finally, our asymptotic results are illustrated by numerical simulations.

Large data decay of Yang–Mills–Higgs fields on Minkowski and de Sitter spacetimes

In this article, we extend Eardley and Moncrief’s L∞ estimates [Commun. Math. Phys. 83(2), 193–212 (1982)] for the conformally invariant Yang–Mills–Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localize their estimates and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global L∞ estimates on the cylinder and extend Choquet-Bruhat and Christodoulou’s [Ann. Sci. Éc. Norm. Supér. 14(4), 481–506 (1981)] small data well-posedness result to large data. Finally, by employing an inverse conformal transformation, we deduce exponential decay rates for Yang–Mills–Higgs fields on de Sitter space and inverse polynomial decay rates on Minkowski space.

Use of the D-Shuttle dosimeter for radiation protection of members of the public: Characterization and feasibility study

The objective of this study was to characterize the latest version of the electronic D-Shuttle dosimeter (Chiyoda Technol Corp., Tokyo, Japan) for potential use in dosimetry of members of the public. The investigations comprise: (a) reproducibility, batch homogeneity, dose linearity, dose rate and angle dependence, and radiation hardness using irradiations at the secondary standard dosimetry laboratory of the Paul Scherrer Institute (PSI); (b) estimation of the environmental dose rate and its variability; and (c) feasibility tests for guests at PSI (short-term researchers). The performed verification measurements showed overall a good reproducibility, maximum 1.4% standard deviation (sd) per dosimeter, and maximum 1.0% sd for 5 dosimeters, in agreement with the manufacturer's specifications. The deviation from linearity was <10% and the deviation with energy was <30% in the range [60 keV, 1.25 MeV], except for 164 keV photons (N-200 radiation quality), where a deviation of almost 50% was observed with respect to the Cs-137 radiation quality. The angle dependence was within specifications (±10%) for Cs-137, but up to +30% for 83 keV photons (N-100 radiation quality). The drop in response for dose rates >0.5–1 mSv/h was well described by an inverse polynomial function, and no influence of a high dose irradiation (16.37 Sv) in subsequent low dose measurements was observed. Environmental background (BG) measurements demonstrated that the dosimeters are stable, showing a mean background indication of (70 ± 30) nSv/h, where the standard deviation is compatible with Poisson statistics assuming a sensitivity of 1 count per 10 nSv. This means that hourly indications above 220 nSv/h (BG + 5 sd) can be clearly distinguished from the background. The monitoring of the guests activities showed that the high sensitivity and hourly information can be useful to investigate out-of-the-ordinary exposures, representing a valuable tool for radiation protection of members of the public.

Further results on permutation polynomials via linear translators

In this paper, by using linear translators, we characterize a new class of permutation polynomials of the form [Formula: see text], which has a more general form than [Formula: see text]. Then, we present the compositional inverses of such permutation polynomials. Furthermore, by specifying the functions [Formula: see text] and [Formula: see text], we can get some new permutation polynomials of the forms [Formula: see text] and [Formula: see text] where [Formula: see text] is the trace function.

Modeling of lactation curves for prediction of standard lactation milk yield in Marathwadi buffaloes

The present study was carried out on 538 monthly milk yield records of Marathwadi buffaloes maintained at Cattle Breeding Farm, College of Veterinary and Animal Sciences, Udgir Dist-Latur Maharashtra spread over a period of 9 years. The monthly milk yield was recorded at 6th, 36th, 66th and so on till standard lactation with 30 days interval. The goodness of fit of 3 lactation curve models, viz. parabolic exponential function, inverse polynomial function and mixed log function were used on monthly milk yields in Marathwadi buffaloes. The highest coefficient of determination for fitting of lactation curve models was with mixed log function (97.63%), followed by parabolic exponential function (95%) and inverse polynomial function (82.70%). Further, the root mean square error was found least for mixed log function (1.70), followed by parabolic exponential function (3.55) and inverse polynomial function (4.61). It is therefore inferred that mixed log function fitted best for prediction and modeling of lactation curves based upon monthly milk yields.

Matrix biorthogonal polynomials on the real line: Geronimus transformations

In this paper, Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasi-definite matrix of bivariate generalized functions with a well-defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel-type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss–Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation in which a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses are performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel–Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular, a comparison of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given.

Two new classes of compactly supported radial basis functions for approximation of discrete and continuous data

Radial basis functions (RBFs), first proposed for interpolation of scattered data, have gotten the scientific community interest in the past two decades, with applications ranging from interpolation with the dual reciprocity approach of the boundary element method to mesh‐free finite element applications. The use of compactly supported RBFs (CSRBFs) has spread due to their localization properties. However, the mathematical derivation of continuous polynomial functions for different continuity requirements can be quite cumbersome, and although a few classes of functions have already been proposed, there is still room for nonpolynomial trial functions. In this paper, two new classes of CSRBFs are presented for which the continuity requirements are guaranteed. To justify the novelty claim, a view of RBF literature is conducted. The first function class proposed consists of a combination of different inverse polynomial functions, similar to inverse quadric functions, and is thus called inverse class. The second class is called the rational class, in which the functions are obtained as a ratio of two polynomial functions. The proposed functions are used on an approximation software, which takes advantage of their simplicity. The results demonstrate the accuracy and convergence of the proposed functions when compared to some referenced CSRBFs.

Cloaking of structures with convex or concave geometry using higher order parameterised function verified using ray tracing

The accomplishment of design for devices and applications such as cloaking has been feasible using transformation optics. This feasibility has been extended to the design of cloaks for concave bodies. Even though the process of cloaking in concave shaped bodies has been addressed in literature, the problem of poor accuracy and poor quality of cloaking remains. In accordance with the aim to outmanoeuvre the problem, the surface of interest which could be of convex or concave nature is modelled using a piecewise higher order parametric function. This function is expressed using Lagrange's interpolation. The calculation of piecewise model involves the inversion of the Lagrangian polynomial. But the inverse is numerically unstable because it is a Vandermonde matrix. Here, a method which is more stable than one that involves the direct inversion of the Vandermonde matrix, has been suggested using a recurrence relation. As the order of the function increases, it is observed that the efficiency of the cloaking improves further. It can be inferred that usage of the piecewise higher order parametric function along with a recurrence relation leads to an improvement in the quality and accuracy of cloaking.

Finite-Size Scaling of Out-of-Time-Ordered Correlators at Late Times.

Chaotic dynamics in quantum many-body systems scrambles local information so that at late times it can no longer be accessed locally. This is reflected quantitatively in the out-of-time-ordered correlator of local operators, which is expected to decay to 0 with time. However, for systems of finite size, out-of-time-ordered correlators do not decay exactly to 0 and in this paper we show that the residual value can provide useful insights into the chaotic dynamics. When energy is conserved, the late-time saturation value of the out-of-time-ordered correlator of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We provide both analytical arguments and numerical simulations to support this conclusion.

The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities

We consider the wave equation, [Formula: see text], in fixed flat Friedmann–Lemaître–Robertson–Walker and Kasner spacetimes with topology [Formula: see text]. We obtain generic blow up results for solutions to the wave equation toward the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to the solutions that blow up in an open set of the Big Bang hypersurface [Formula: see text]. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate [Formula: see text]-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the [Formula: see text] norms of the solutions blow up toward the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates, respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

The Spin $$\varvec{\pm }$$ ± 1 Teukolsky Equations and the Maxwell System on Schwarzschild

In this note, we prove decay for the spin ± 1 Teukolsky equations on the Schwarzschild spacetime. These equations are those satisfied by the extreme components ( $$\alpha $$ and $${\underline{\alpha }}$$ ) of the Maxwell field, when expressed with respect to a null frame. The subject has already been addressed in the literature, and the interest in the present approach lies in the connection with the recent work by Dafermos, Holzegel and Rodnianski on linearized gravity (Dafermos et al. in The linear stability of the Schwarzschild solution to gravitational perturbations, 2016. arXiv:1601.06467 ). In analogy with the spin $$\pm 2$$ case, it seems difficult to directly prove Morawetz estimates for solutions to the spin $$\pm 1$$ Teukolsky equations. By performing a differential transformation on the extreme components $$\alpha $$ and $${\underline{\alpha }}$$ , we obtain quantities which satisfy a Fackerell–Ipser Equation, which does admit a straightforward Morawetz estimate and is the key to the decay estimates. This approach is exactly analogous to the strategy appearing in the aforementioned work on linearized gravity. We achieve inverse polynomial decay estimates by a streamlined version of the physical space $$r^p$$ method of Dafermos and Rodnianski. Furthermore, we are also able to prove decay for all the components of the Maxwell system. The transformation that we use is a physical space version of a fixed-frequency transformation which appeared in the work of Chandrasekhar (Proc R Soc Lond Ser A 348(1652):39–55, 1976). The present note is a version of the author’s master thesis and also serves the “pedagogical” purpose to be as complete as possible in the presentation.

Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data II: The Exterior of the Black Hole Region

This is the second and last paper of a two-part series in which we prove the $$C^2$$ -formulation of the strong cosmic censorship conjecture for the Einstein–Maxwell–(real)–scalar–field system in spherical symmetry for two-ended asymptotically flat data. In the first paper, we showed that the maximal globally hyperbolic future development of an admissible asymptotially flat Cauchy initial data set is $$C^2$$ -future-inextendible provided that an $$L^2$$ -averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show that this lower bound is indeed satisfied for solutions arising from a generic set of Cauchy initial data. Roughly speaking, the generic set is open with respect to a (weighted) $$C^1$$ topology and is dense with respect to a (weighted) $$C^\infty $$ topology. The proof of the theorem is based on extensions of the ideas in our previous work on the linear instability of Reissner–Nordstrom Cauchy horizon, as well as a new large data asymptotic stability result which gives good decay estimates for the difference of the radiation fields for small perturbations of an arbitrary solution.