Sum Of Polynomials Research Articles

On Complete Rational Arithmetic Sums of Polynomial Values

New estimates are obtained for complete arithmetic sums of polynomial values (exponential sums, sums of Dirichlet characters, and sums of Bernoulli polynomials) in the case where the derivative of the polynomial in the argument of the sum has no multiple roots modulo primes dividing the period of these arithmetic sums.

Efficient uncertainty quantification in fully-integrated surface and subsurface hydrologic simulations

Abstract Although high performance computers and advanced numerical methods have made the application of fully-integrated surface and subsurface flow and transport models such as HydroGeoSphere common place, run times for large complex basin models can still be on the order of days to weeks, thus, limiting the usefulness of traditional workhorse algorithms for uncertainty quantification (UQ) such as Latin Hypercube simulation (LHS) or Monte Carlo simulation (MCS), which generally require thousands of simulations to achieve an acceptable level of accuracy. In this paper we investigate non-intrusive polynomial chaos for uncertainty quantification, which in contrast to random sampling methods (e.g., LHS and MCS), represents a model response of interest as a weighted sum of polynomials over the random inputs. Once a chaos expansion has been constructed, approximating the mean, covariance, probability density function, cumulative distribution function, and other common statistics as well as local and global sensitivity measures is straightforward and computationally inexpensive, thus making PCE an attractive UQ method for hydrologic models with long run times. Our polynomial chaos implementation was validated through comparison with analytical solutions as well as solutions obtained via LHS for simple numerical problems. It was then used to quantify parametric uncertainty in a series of numerical problems with increasing complexity, including a two-dimensional fully-saturated, steady flow and transient transport problem with six uncertain parameters and one quantity of interest; a one-dimensional variably-saturated column test involving transient flow and transport, four uncertain parameters, and two quantities of interest at 101 spatial locations and five different times each (1010 total); and a three-dimensional fully-integrated surface and subsurface flow and transport problem for a small test catchment involving seven uncertain parameters and three quantities of interest at 241 different times each. Numerical experiments show that polynomial chaos is an effective and robust method for quantifying uncertainty in fully-integrated hydrologic simulations, which provides a rich set of features and is computationally efficient. Our approach has the potential for significant speedup over existing sampling based methods when the number of uncertain model parameters is modest ( ≤ 20). To our knowledge, this is the first implementation of the algorithm in a comprehensive, fully-integrated, physically-based three-dimensional hydrosystem model.

Observer‐based non‐linear H ∞ attitude control for a flexible satellite

This study focuses on the observer-based non-linear H ∞ (OBNH) attitude control for a flexible satellite with external disturbances and local state-region constraints. By considering the structural features of the attitude system and applying Lyapunov stability theory, the solvable conditions of the OBNH attitude control problem are provided. An advantage of this study is that the state-feedback controller and the reduced-order state observer can be designed independently, thus significantly reducing the complexity of the attitude control algorithm. Furthermore, based on the polynomial sum of squares (SOSs) and the generalised S-procedure techniques, the above problem can be transformed into a convex optimisation problem with SOS constraints. This conversion can effectively overcome the computational difficulties widely present in the control problem of non-linear systems. Finally, a numerical example is given to illustrate the effectiveness and feasibility of the proposed approach.

Free‐surface tracking in 2D with the harmonic polynomial cell method: Two alternative strategies

AbstractSeveral cases of nonlinear wave propagation are studied numerically in two dimensions within the framework of potential flow. The Laplace equation is solved with the harmonic polynomial cell (HPC) method, which is a field method with high‐order accuracy. In the HPC method, the computational domain is divided into overlapping cells. Within each cell, the velocity potential is represented by a sum of harmonic polynomials. Two different methods denoted as immersed boundary (IB) and multigrid (MG) are used to track the free surface. The former treats the free surface as an IB in a fixed Cartesian background grid, while the latter uses a free‐surface fitted grid that overlaps with a Cartesian background grid. The simulated cases include several nonlinear wave mechanisms, such as high steepness and shallow‐water effects. For one of the cases, a numerical scheme to suppress local wave breaking is introduced. Such scheme can serve as a practical mean to ensure numerical stability in simulations where local breaking is not significant for the result. For all the considered cases, both the IB and MG method generally give satisfactory agreement with known reference results. Although the two free‐surface tracking methods mostly have similar performance, some differences between them are pointed out. These include aspects related to modeling of particular physical problems as well as their computational efficiency when combined with the HPC method.

On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers

In this article we analyze the global diffeomorphism property of polynomial maps $$F:\mathbb {R}^n\rightarrow \mathbb {R}^n$$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials $$\Vert F\Vert _2^2$$ . This allows us to identify a class of polynomial maps F for which their global diffeomorphism property on $$\mathbb {R}^n$$ is equivalent to their Jacobian determinant $$\det JF$$ vanishing nowhere on $$\mathbb {R}^n$$ . In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.

Deformation rates in northern Cascadia consistent with slow updip propagation of deep interseismic creep

Interpretations of interseismic slip deficit on the northern Cascadia megathrust are complicated by an enigmatic ‘gap’ between the downdip limit of the locked region, inferred from kinematic inversions of deformation rates, and the top of the episodic tremor and slip (ETS) zone. Recent inversions of global positioning system (GPS) and tide gauge/leveling data for shear stress rates acting on the megathrust found a ∼21 km locking depth with a steep slip-rate gradient at its base is required to fit the data. Previous studies have assumed the depth distribution of interseismic slip rate to be time invariant; however, steep slip-rate gradients could also result from the updip propagation of slip into the locked region. This study explores models where interseismic slip penetrates up into the locked zone. We consider the creeping region, corresponding to the gap and the ETS zone, as a quasi-static crack driven by the plate velocity at its downdip end. We derive a simple model that allows for crack propagation over time, and provides analytical expressions for stress drop within the crack, slip and slip rate on the fault. It is convenient to expand the non-singular slip-rate distribution in a sum of Chebyshev polynomials. Estimation of the polynomial coefficients is underdetermined, yet provides a useful way of testing particular solutions and provides bounds on the updip propagation rate. When applied to the deformation rates in northern Cascadia, best-fitting models reveal that a very slow updip propagation, between 30 and 120 m yr−1 along the fault, could explain the steep slip-rate profile, needed to fit the data. This work provides a new tool for estimating interseismic slip rates, between purely kinematic inversions and full physics-based modeling, allowing for the possibility for updip expansion of the creeping zone.

On restarting the tensor infinite Arnoldi method

An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov–Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by applying them to solve large scale NEPs that arise from a delay differential equation and a wave propagation problem. The advantages in comparison to other restart methods are also illustrated.

New LMI conditions for admissibility analysis of time-varying descriptor systems

This paper presents new admissibility analysis conditions for discrete-time descriptor systems. Firstly, it proposes an extension, expressed as necessary and sufficient admissibility conditions, of admissibility criteria developed in the literature for linear time-invariant systems in order to incorporate the time-varying case. This extension exploits time-varying slack variables and uses the linear matrix inequality (LMI) formulation. Then, this extension is applied to the case of polytopic descriptor systems with bounded time-varying parameters. The use of sum of squares polynomials allows to eliminate the criteria parametric dependence and to provide new admissibility analysis conditions expressed as a LMIs feasibility problem. This new admissibility analysis result uses parameter-dependent slack variables and may be less conservative than the results already proposed in the literature as sustained by our numerical evaluation.

Extension of Satellite Altimetry Jason-2 Sea Level Anomalies Towards the Red Sea Coast Using Polynomial Harmonic Techniques

ABSTRACTSatellite altimetry data are facing big challenges near the coasts. These challenges arise due to the fundamental difficulties of correction and land contamination in the foot print, which result in rejection of these data near the coast. Several studies have been carried out to extend these data towards the coast. Over the Red Sea, altimetry data consist of gaps, which extend to about 30–50 km from the coast. Two methods are used for processing and extending Jason-2 satellite altimetry sea level anomalies (SLAs) towards the Red Sea coast; Fourier Series Model (FSM), and the polynomial sum of sine model (SSM). FSM model technique uses Fourier series and statistical analysis reflects strong relationship with both the observation and AVISO data, with strong and positive correlation. The second prediction technique, SSM model, depends on the polynomial sum of sine, and does not reflect any relationship with the observations and AVISO data close to the coast and the correlation coefficient (CC) is weak and negative. The FSM model output results in SLA data significantly better and more accurate than the SSM model output.

Efficient Ehrlich–Aberth iteration for finding intersections of interpolating polynomials and rational functions

We analyze the problem of carrying out an efficient iteration to approximate the eigenvalues of some rank structured pencils obtained as linearization of sums of polynomials and rational functions expressed in (possibly different) interpolation bases. The class of linearizations that we consider has been introduced by Robol, Vandebril and Van Dooren in [17]. We show that a traditional QZ iteration on the pencil is both asymptotically slow (since it is a cubic algorithm in the size of the matrices) and sometimes not accurate (since in some cases the deflation of artificially introduced infinite eigenvalues is numerically difficult). To solve these issues we propose to use a specifically designed Ehrlich–Aberth iteration that can approximate the eigenvalues in O(kn2) flops, where k is the average number of iterations per eigenvalue, and n the degree of the linearized polynomial. We suggest possible strategies for the choice of the initial starting points that make k asymptotically smaller than O(n), thus making this method less expensive than the QZ iteration. Moreover, we show in the numerical experiments that this approach does not suffer of numerical issues, and accurate results are obtained.

Bifurcation of Limit Cycles from the Center of a Quintic System via the Averaging Method

This paper deals with the bifurcation of limit cycles for a quintic system with one center. Using the averaging method, we explain how limit cycles can bifurcate from the periodic annulus around the center of the considered system by adding perturbed terms which are the sum of homogeneous polynomials of degree [Formula: see text] for [Formula: see text]. We show that up to first-order averaging, at most five limit cycles can bifurcate from the period annulus of the unperturbed system for [Formula: see text], at most [Formula: see text] limit cycles can bifurcate from the periodic annulus of the unperturbed system for any [Formula: see text], and the upper bound is sharp for [Formula: see text] and for [Formula: see text].

Period Monotonicity for Weight-Homogeneous Differential Systems

In this article, integrability, center, and monotonicity of associated period function for -quasi-homogeneous vector fields are investigated. We are concerned with family of vector field given by sum, finite or infinite number of quasi-homogeneous polynomials not necessarily to be sharing the same weights. The investigation is done by utilizing method of computing focal values. As an application of the result, a particular family of (p, q)-quasi-homogeneous vector field is studied to find conditions for center, monotonicity and consequently an explicit form for the associated period function.

On an elementary version of I.M. Vinogradov’s method

We prove estimates for complete rational arithmetic sums of Bernoulli polynomials whose arguments are formed by the fractional parts of values of a polynomial with rational coefficients. The results are applied to the problem of finding the convergence exponent for the mean values of the sums under consideration.

A Framework for Structured Linearizations of Matrix Polynomials in Various Bases

We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in nonmonomial bases and allows us to represent polynomials expressed in product families, that is, as a linear combination of elements of the form $\phi_i(\lambda) \psi_j(\lambda)$, where $\{ \phi_i(\lambda) \}$ and $\{ \psi_j(\lambda) \}$ can either be polynomial bases or polynomial families which satisfy some mild assumptions. We show that this general construction can be used for many different purposes. Among them, we show how to linearize sums of polynomials and rational functions expressed in different bases. As an example, this allows us to look for intersections of functions interpolated on different nodes without converting them to the same basis. We then provide some constructions ...

Some polynomial invariants of virtual links via a parity

We introduce the even index polynomial of virtual link diagrams and some even state sum polynomials which are the even Jones–Kauffman, even Miyazawa and even arrow polynomials. By simple computation, we can distinguish virtual links with these even polynomial invariants. Also, we give examples so that these invariants are different from the original ones.

Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane foam Viscoelastic Response under Quasi-Static Compressive Tests

The flexible polyurethane foam exhibits a highly nonlinear and viscoelastic behavior under quasi-static and uni-axial compressive tests. The dependence of the displacement rate during the loading phase is significantly higher than that of the unloading phase. A new identification of the Force-Displacement curve based on experimental observations is proposed. The total foam response is modeled as a sum of a nonlinear elastic component and a viscoelastic component. The elastic force is modeled as a sum of orthogonal polynomials in displacement, while the viscoelastic force is modeled according to the hereditary approach in the loading half-cycle and the fractional derivative approach in the unloading half-cycle. The objective of this paper is to develop a model able to make predictions of the total foam force as well as simulations of its components. A parameter calibration procedure is established according to the difference force method between two unloading responses corresponding to two different displacement rates of the same foam sample. The validity of the model results is discussed by addressing three efficiency requirements: the accuracy of simulations to experimental measurements, the repeatability of results for both tests, and the accordance of predicted components of the total response with the phenomenological identification of the Force-Displacement curve. The proposed optimization methodology is found to simulate reasonably well the responses of three different types of soft foam. Some morphologic characteristics of these foams have clear influence on viscoelastic damping and residual effects.

Circular free spectrahedra

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by eit: up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials.

Polynomial interpolation and sums of powers of integers

ABSTRACTIn this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, Pk(n) and Qk(n), such that Pk(n) = Qk(n) = fk(n) for n = 1, 2,… , k, where fk(1), fk(2),… , fk(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of the first n positive integers Sk(n) = 1k + 2k + ⋅⋅⋅ + nk, and show that Sk(n) admits the polynomial representations Sk(n) = Pk(n) and Sk(n) = Qk(n) for all n = 1, 2,… , and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for Sk(n) alternative to the well-known formula of Bernoulli.

Tails of Polynomials of Random Variables and Stable Limits for Nonconventional Sums

We obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails and derive stable limit theorems for nonconventional sums of such polynomials

Invariance of polynomial inequalities under polynomial maps

We prove the invariance under simple polynomial maps of a few useful (multivariate) polynomial inequalities. The proofs are elementary and based on an algebraic decomposition of any polynomial as a sum of polynomials, which we call q-coordinates, composed with the polynomial map q, balanced with monomials of bounded degree.