Finite-order Polynomials Research Articles

Hilbert's tenth problem for lacunary entire functions of finite order

AbstractIn the context of Hilbert's tenth problem, an outstanding open case is that of complex entire functions in one variable. A negative solution is known for polynomials (by Denef) and for exponential polynomials of finite order (by Chompitaki, Garcia‐Fritz, Pasten, Pheidas, and Vidaux), but no other case is known for rings of complex entire functions in one variable. We prove a negative solution to the analogue of Hilbert's tenth problem for rings of complex entire functions of finite order having lacunary power series expansion at the origin.

A coherent integration method of an active sonar for maneuvering turning target detection.

The detection probability of underwater weak targets using active sonar is low, and inter-pulse coherent integration can improve the signal-to-noise ratio of echoes. When a target executes a maneuvering turn, complex range and Doppler frequency migrations occur during the coherent integration time that decrease the coherent integration gain. Most existing integration methods simplify the target motion to a finite-order polynomial model but fail to integrate a maneuvering turning target (MTT) due to model mismatch. Hence, this study proposes an underwater MTT integration method based on the modified Radon-Fourier transform. The proposed method constructs a theoretically accurate motion model for the MTT and a phase compensation function to compensate for the Doppler frequency migration. Furthermore, it yields a well-focused integration peak in the range-velocity and offset angle-turn rate dimensions and accurately estimates the target motion parameters. Moreover, the proposed method is suitable for targets with radial and oblique uniform motions. The effectiveness of the proposed method is demonstrated through simulations and a lake test. The proposed method demonstrates good integration performance, with an integration gain approximately 4-7 dB higher than that of traditional methods when using 30 integration pulses.

Self-Directed Mobile Robot Navigation Based on Functional Firefly Algorithm (FFA)

This paper proposes an optimized mobile robot navigation strategy using a functional firefly algorithm (FFA) and choice function. This approach has two key advantages: first, the linear objective function performs efficiently with the single degree and finite-order polynomial time operation, and second, the cartesian constraint performs compactly with the chosen degree of freedom on the finite interval. This functional approach optimizes the size of operational parameters in context with key size, operation time, and a finite range of verification. The choice function achieves parameter order (size) reduction. The attraction characteristic of fireflies is represented by the choice function for optimizing the choice between low and high intensities of fireflies. In 2D and 3D environments, the proposed robot navigation performs well in an uncertain environment with static and dynamic obstacles. This efficiency includes the robot’s speed as determined by the choice function’s minimum path lengths. The collision-free path is achieved by the non-void family of non-void sets. The obtained results are optimal in terms of path length and navigational time. The proposed controller is also compared with the other existing controllers, and it is observed that the FFA gives the shortest path in less time for the same environmental condition.

Collaborative Filtering Model of Graph Neural Network Based on Random Walk

This paper proposes a novel graph neural network recommendation method to alleviate the user cold-start problem caused by too few relevant items in personalized recommendation collaborative filtering. A deep feedforward neural network is constructed to transform the bipartite graph of user–item interactions into the spectral domain, using a random wandering method to discover potential correlation information between users and items. Then, a finite-order polynomial is used to optimize the convolution process and accelerate the convergence of the convolutional network, so that deep connections between users and items in the spectral domain can be discovered quickly. We conducted experiments on the classic dataset MovieLens-1M. The recall and precision were improved, and the results show that the method can improve the accuracy of recommendation results, tap the association information between users and items more effectively, and significantly alleviate the user cold-start problem.

Complex migration correction and coherent integration algorithm for radar target with arbitrary moving direction

Long-time coherent integration is conducive to improving the radar detection performance for weak moving targets, whereas range migration (RM) and Doppler frequency migration (DFM) might occur when the target has high speed and high maneuverability. Although a variety of methods have been proposed to correct RM and DFM, they are widely based on the target's radial motion model, which is expressed as a finite order polynomial with respect to slow time. However, when an offset exists between the target's moving direction and radar sight line, model mismatch will occur and deteriorate the integration performance. In this paper, the accurate echo model for the moving target with arbitrary moving direction is established, then the modified Radon Fourier transform (MRFT) algorithm is proposed to realize coherent integration via removing the complex RM and DFM. Moreover, the generalized form of MRFT, i.e., modified generalized RFT (MGRFT), is introduced to focus the target with high maneuverability. Both simulation and real data processed results have been provided to validate the effectiveness of the proposed method. According to the coherent integration results for the target with oblique maneuvering motion, the signal-to-noise ratio (SNR) gain of the proposed method is about 4.5 dB higher than that of the existing GRFT method with the same computational complexity.

Accurate modeling and coherent integration method for weak maneuvering target with oblique movement

Prolonging the integration time is a widely used method to enhance the detection performance for weak targets, whereas range migration (RM) and Doppler frequency migration (DFM) might occur when the target moves with high speed and high maneuverability. Although various algorithms have been proposed to solve the migration problem and realize coherent integration, they are based on the assumption that the target is moving radially relative to radar, i.e., the instantaneous slant range from radar to target is expressed as a finite order polynomial with respect to slow time. However, the target in the practical scene is probably moving obliquely relative to radar, thus modeling error will exist between the polynomial range expression and the target’s actual instantaneous slant range from radar. We focus on the accurate modeling and coherent integration problems for the weak maneuvering target with oblique movement, where the oblique movement model (OMM) of target echo is established. Subsequently, the OMM-generalized Radon–Fourier transform method is proposed to coherently accumulate the target’s energy via correcting the complex RM and DFM. Both simulation and real data processed results have been provided to demonstrate the effectiveness of the proposed method.

High-order point-value enhanced finite volume method for two-dimensional hyperbolic equations on unstructured meshes

Presently, an approximate delta function (ADF) is defined in multi-dimensional space in the form of a finite-order polynomial that holds identical integral properties to the Dirac delta function when used in conjunction with a finite-order polynomial integrand spanning a finite domain. By using this ADF over a two-dimensional unstructured mesh, a compact high-order point-value enhanced finite volume method (PFV) is constructed. The corresponding scheme is capable of storing and updating cell-averaged values inside each element as well as the unknown quantities and their derivatives (when needed) on vertex points. The sharing of vertex information with surrounding elements reduces the number of degrees of freedom relative to other compact and high-order methods of comparable convergence rate. To ensure conservation, cell-averaged values are updated using an identical approach to the one traditionally followed in the finite volume method. To verify the performance of the PFV scheme at successive orders, its accuracy and stability are vetted using benchmark problems associated with the two-dimensional wave and Euler equations encompassing a total of six standard test cases.

Accelerated translational motion compensation with contrast maximisation optimisation algorithm for inverse synthetic aperture radar imaging

Range alignment of traditional translational motion compensation for inverse synthetic aperture radar imaging generally cannot be implemented accurately under low signal-to-noise ratio, resulting in the following phase adjustment invalid. In this study, a novel accelerated translational motion compensation with contrast maximisation optimisation algorithm is proposed. Translational motion is first modelled as a parametric finite order polynomial. The translational motion property can be compactly expressed by a polynomial coefficient vector. Meanwhile, the image contrast is utilised to estimate the polynomial coefficient vector based on the maximum contrast optimisation, implemented by Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. BFGS is an effective quasi-Newton algorithm, yielding fast convergence and small computational complexity. Moreover, a method called pseudo Akaike information criterion is also proposed to determine the polynomial order adaptively. Both simulated and real data experiments are provided for a clear demonstration of the proposed algorithm.

Nonpolynomial Higgs interactions and vacuum stability

The possible violation of the conventional lower Higgs mass stability bound by the discovered Higgs boson has far reaching consequences within particle physics and cosmology. We discuss the possibility that nonpolynomial bare interactions seeded at some high-momentum scale can considerably diminish the lower Higgs mass bound without introducing a metastability in the Higgs effective potential. For this, we classify various deformations of the usual quartic bare potential regarding their impact on stable IR physics. We perform the analysis in a large N_{mathrm {f}} expansion, addressing the convergence of the obtained results by taking 1/N_{mathrm {f}} corrections into account as well. In addition, we investigate the renormalization group flow of the scalar potential on a nonperturbative level. Within these approximations, we are able to identify bare potentials that lead to Higgs masses below stability mass bounds obtained from finite-order polynomial bare interactions without introducing a metastability in the effective potential.

On Wick Polynomials of Boson Fields in Locally Covariant Algebraic QFT

This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over globally hyperbolic spacetimes and quantized through the known functorial machinery in terms of local $^*$-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles, in addition to the Lorentzian metric. The mass and the coupling constants are in particular viewed as background fields. Wick powers of the quantized vector field are axiomatically defined imposing in particular local covariance, scaling properties and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is specialized to the case of general tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. The notion of local smooth dependence on polynomial scalar invariants is made precise in the text.

On the adequacy of the polynomial approximation to the exponential growth curve model

Exponential growth curves are usually approximated with a finite order polynomial curve in the study of trending curves. This is done because the most popular way of removing the trend component is by differencing. This paper shows that the trend curve cannot be removed by differencing when the trend curve is the exponential growth curve. Exponential curves are transcendental functions which can be reduced to a finite order polynomial by Maclaurin series expansion. The objective of this paper is to examine the adequacy of the polynomial approximation of the exponential growth curve with respect to its growth rate and sample size. The coefficients of the associated polynomial curve were obtained theoretically by the use of Maclaurin series expansion method. Next, exponential growth curves with varying growth rates and sample sizes were simulated. Adequate polynomials were fitted to the simulated exponential growth curves and the coefficients obtained were compared with the theoretical coefficients using absolute error and paired tests. Results obtained show that adequacy depend on both growth rate and sample size. For the purpose of statistical analysis, the highest sample size of 28 is not useful, especially in times series analysis where the demand of samples of 60 or more is made.

A point-value enhanced finite volume method based on approximate delta functions

We revisit the concept of an approximate delta function (ADF), introduced by Huynh (2011) [1], in the form of a finite-order polynomial that holds identical integral properties to the Dirac delta function when used in conjunction with a finite-order polynomial integrand over a finite domain. We show that the use of generic ADF polynomials can be effective at recovering and generalizing several high-order methods, including Taylor-based and nodal-based Discontinuous Galerkin methods, as well as the Correction Procedure via Reconstruction. Based on the ADF concept, we then proceed to formulate a Point-value enhanced Finite Volume (PFV) method, which stores and updates the cell-averaged values inside each element as well as the unknown quantities and, if needed, their derivatives on nodal points. The sharing of nodal information with surrounding elements saves the number of degrees of freedom compared to other compact methods at the same order. To ensure conservation, cell-averaged values are updated using an identical approach to that adopted in the finite volume method. Here, the updating of nodal values and their derivatives is achieved through an ADF concept that leverages all of the elements within the domain of integration that share the same nodal point. The resulting scheme is shown to be very stable at successively increasing orders. Both accuracy and stability of the PFV method are verified using a Fourier analysis and through applications to the linear wave and nonlinear Burgers' equations in one-dimensional space.

Classical Dynamics of Fullerenes

The classical mechanics of large molecules and fullerenes is studied. The approach is based on the model of collective motion of these objects. The mixed Lagrangian (material) and Eulerian (space) description of motion is used. In particular, the Green and Cauchy deformation tensors are geometrically defined. The important issue is the group-theoretical approach to describing the affine deformations of the body. The Hamiltonian description of motion based on the Poisson brackets methodology is used. The Lagrange and Hamilton approaches allow us to formulate the mechanics in the canonical form. The method of discretization in analytical continuum theory and in classical dynamics of large molecules and fullerenes enable us to formulate their dynamics in terms of the polynomial expansions of configurations. Another approach is based on the theory of analytical functions and on their approximations by finite-order polynomials. We concentrate on the extremely simplified model of affine deformations or on their higher-order polynomial perturbations.

Multipartite Fully Entangled Fraction

Fully entangled fraction is a definition for bipartite states, which is tightly related to bipartite maximally entangled states, and has clear experimental and theoretical significance. In this work, we generalize it to multipartite case, we call the generalized version multipartite fully entangled fraction (MFEF). MFEF measures the closeness of a state to GHZ states. The analytical expressions of MFEF are very difficult to obtain except for very special states, however, we show that, the MFEF of any state is determined by a system of finite-order polynomial equations. Therefore, the MFEF can be efficiently numerically computed.

Swarm Motion as Particles of a Continuum With Communication Delays

In this paper, we give an upper bound for the communication delay in a multi-agent system (MAS) that evolves under a recently developed continuum paradigm for formation control. The MAS is treated as particles of a continuum that transforms under special homeomorphic mapping, called a homogeneous map. Evolution of an MAS in ℝn is achieved under a special communication topology proposed by Rastgoftar and Jayasuriya (2014, “Evolution of Multi Agent Systems as Continua,” ASME J. Dyn. Syst. Meas. Control, 136(4), p. 041014) and (2014, “An Alignment Strategy for Evolution of Multi Agent Systems,” ASME J. Dyn. Syst. Meas. Control, 137(2), p. 021009), employing a homogeneous map specified by the trajectories of n+1 leader agents at the vertices of a polytope in ℝn, called the leading polytope. The followers that are positioned in the convex hull of the leading polytope learn the prescribed homogeneous mapping through local communication with neighboring agents using a set of communication weights prescribed by the initial positions of the agents. However, due to inevitable time-delay in getting positions and velocities of the adjacent agents through local communication, the position of each follower may not converge to the desired state given by the homogeneous map leaving the possibility that MAS evolution may get destabilized. Therefore, ascertaining the stability under time-delay is important. Stability analysis of an MAS consisting of a large number of agents, leading to higher-order dynamics, using conventional methods such as cluster treatment of characteristic roots (CTCR) or Lyapunov–Krasovskii are difficult. Instead we estimate the maximum allowable communication delay for the followers using one of the eigenvalues of the communication matrix that places MAS evolution at the margin of instability. The proposed method is advantageous because the transcendental delay terms are directly used and the characteristic equation of MAS evolution is not approximated by a finite-order polynomial. Finally, the developed framework is used to validate the effect of time-delays in our previous work.

Control point based exact description of trigonometric/hyperbolic curves, surfaces and volumes

Using the normalized B-bases of vector spaces of trigonometric and hyperbolic polynomials of finite order, we specify control point configurations for the exact description of the zeroth and higher order (mixed partial) derivatives of integral curves and (hybrid) multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variables. Based on homogeneous coordinates and central projection, we also propose algorithms for the control point and weight based exact description of the zeroth order (partial) derivative of the rational counterpart of these integral curves and surfaces. The core of the proposed modeling methods relies on basis transformation matrices with entries that can be efficiently obtained by order elevation.

Stein–Malliavin approximations for nonlinear functionals of random eigenfunctions on [formula omitted

We investigate Stein–Malliavin approximations for nonlinear functionals of geometric interest for random eigenfunctions on the unit d-dimensional sphere Sd, d≥2. All our results are established in the high energy limit, i.e. as the corresponding eigenvalues diverge. In particular, we prove a quantitative Central Limit Theorem for the excursion volume of Gaussian eigenfunctions; this goal is achieved by means of several results of independent interest, concerning the asymptotic analysis for the variance of moments of Gaussian eigenfunctions, the rates of convergence in various probability metrics for Hermite subordinated processes, and quantitative Central Limit Theorems for arbitrary polynomials of finite order or general, square-integrable, nonlinear transforms. Some related issues were already considered in the literature for the 2-dimensional case S2; our results are new or improve the existing bounds even in these special circumstances. Proofs are based on the asymptotic analysis of moments of all order for Gegenbauer polynomials, and make extensive use of the recent literature on so-called fourth-moment theorems by Nourdin and Peccati.

Exceptional and regular spectra of a generalized Rabi model

We study the spectrum of the generalized Rabi model in which co- and counter-rotating terms have different coupling strengths. It is also equivalent to the model of a two-dimensional electron gas in a magnetic field with Rashba and Dresselhaus spin-orbit couplings. Like in case of the Rabi model, the spectrum of the generalized Rabi model consists of the regular and the exceptional parts. The latter is represented by the energy levels which cross at certain parameters' values which we determine explicitly. The wave functions of these exceptional states are given by finite order polynomials in the Bargmann representation. The roots of these polynomials satisfy a Bethe ansatz equation of the Gaudin type. At the exceptional points the model is therefore quasi-exactly solvable. An analytical approximation is derived for the regular part of the spectrum in the weak- and strong-coupling limits. In particular, in the strong-coupling limit the spectrum consists of two quasi-degenerate equidistant ladders.

Coherent states formulation of polymer field theory

We introduce a stable and efficient complex Langevin (CL) scheme to enable the first direct numerical simulations of the coherent-states (CS) formulation of polymer field theory. In contrast with Edwards' well-known auxiliary-field (AF) framework, the CS formulation does not contain an embedded nonlinear, non-local, implicit functional of the auxiliary fields, and the action of the field theory has a fully explicit, semi-local, and finite-order polynomial character. In the context of a polymer solution model, we demonstrate that the new CS-CL dynamical scheme for sampling fluctuations in the space of coherent states yields results in good agreement with now-standard AF-CL simulations. The formalism is potentially applicable to a broad range of polymer architectures and may facilitate systematic generation of trial actions for use in coarse-graining and numerical renormalization-group studies.

ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

AbstractIn this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.