Sum Of Polynomials Research Articles

DUAL RECURSIVE FORMULAS FOR THE SUMS OF POWERS OF INTEGERS

In this note, we introduce the concept of a dual pair of recursive formulas for the sums of powers of integers Central to this concept is the symmetry property exhibited by the power sum polynomial . We illustrate the concept by some examples taken from the literature, and derive our own pair of dual recursive formulas for .

Application of optimized polynomial controllers based on Bayesian optimization and weight matrix adjustment

AbstractThis paper uses a polynomial controller with non‐linear performance that is a summation of polynomials in non‐linear states. The optimal performance of this controller depends to a large extent on the weighting matrices elements whose determination is not easy via analytical methods for high dimension control systems. This paper focuses on optimizing a polynomial controller applied to a 20‐story non‐linear benchmark building using the Bayesian optimization (BO). BO is a powerful strategy for optimizing functions that are expensive to evaluate. In setting up the optimization problem, design variables are the parameters of weighting matrices. The objective function is a ratio of the controlled and uncontrolled response for the maximum absolute acceleration of the building, and the maximum inter‐story drift is taken as the constraint. The efficiency of the proposed control scheme is tested on a 20‐story non‐linear benchmark building equipped with magnetorheological dampers. The considered evaluation criteria are compared with other control methods. The results show that the considered algorithm is desirable in reducing the maximum seismic response under different earthquakes.

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

Exact and Approximate Moment Derivation for Probabilistic Loops With Non-Polynomial Assignments

Many stochastic continuous-state dynamical systems can be modeled as probabilistic programs with nonlinear non-polynomial updates in non-nested loops. We present two methods, one approximate and one exact, to automatically compute, without sampling, moment-based invariants for such probabilistic programs as closed-form solutions parameterized by the loop iteration. The exact method applies to probabilistic programs with trigonometric and exponential updates and is embedded in the Polar tool. The approximate method for moment computation applies to any nonlinear random function as it exploits the theory of polynomial chaos expansion to approximate non-polynomial updates as the sum of orthogonal polynomials. This translates the dynamical system to a non-nested loop with polynomial updates, and thus renders it conformable with the Polar tool that computes the moments of any order of the state variables. We evaluate our methods on an extensive number of examples ranging from modeling monetary policy to several physical motion systems in uncertain environments. The experimental results demonstrate the advantages of our approach with respect to the current state-of-the-art.

Correction to the paper “The polynomial sieve and equal sums of like polynomials” (IMRN, Vol. 2015, No. 7, 1987–2019)

Abstract This paper corrects an error in an earlier work of the author.

Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

Complex valued measures of finite total variation are a powerful signal model in many applications. Restricting to the d-dimensional torus, finitely supported measures can be exactly recovered from their trigonometric moments up to some order if this order is large enough. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the 1-Wasserstein distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the indicator function on the support of the measure and to converge to zero outside.

Sums of random polynomials with differing degrees

Let μ \mu and ν \nu be probability measures in the complex plane, and let p p and q q be independent random polynomials of degree n n , whose roots are chosen independently from μ \mu and ν \nu , respectively. Under assumptions on the measures μ \mu and ν \nu , the limiting distribution for the zeros of the sum p + q p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n → ∞ n \to \infty . In this paper, we generalize and extend this result to the case where p p and q q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of μ \mu and ν \nu , scaled by the limiting ratio of the degrees of p p and q q . Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q p + q for any pair of measures μ \mu and ν \nu , with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.

Monochromatic quotients, products and polynomial sums in the rationals

Let k,a∈N and let p1,⋯,pk∈Q[n] with zero constant term. We show that for any finite coloring of Q, there are non-zero x,y∈Q such that there exists a color which contains the set{x,xya,x+p1(y),⋯,x+pk(y)} and there are non-zero v,u∈Q such that there exists a color which contains the set{v,v⋅ua,v+p1(u),⋯,v+pk(u)}.

A new invariant of planar knotoids and finite-type invariants

In this paper, we introduce a new polynomial invariant of planar (but not spherical) knotoids, which we call the winding signed sum polynomial. This Laurent polynomial invariant of planar knotoids denoted by [Formula: see text] is a type-one Vassiliev invariant. This invariant might tell whether a planar knotoid is a zero-height or nonzero-height planar knotoid. It also might distinguish between a planar knotoid and its inverse, while the affine index polynomial defined by Gügümcü and Kauffman in [New invariants of knotoids, Eur. J. Combin. 65 (2017) 186–229] (that is also a type-one Vassiliev invariant) cannot distinguish between a planar knotoid and its inverse. We also define some geometric invariants of planar knotoids, and give lower bounds for these invariants using the winding signed sum polynomial, which helps in computing these geometric invariants that are easy to define, but hard to calculate.

Shape-Constrained Regression Using Sum of Squares Polynomials

Shape-Constrained Regression Using Algebraic Techniques How can one fit a multivariate polynomial to data points in such a way that this polynomial is guaranteed to have certain shape constraints, such as convexity or monotonicity in some of its variables? In “Shape-constrained regression using sum-of-squares polynomials,” M. Curmei and G. Hall propose a hierarchy of semidefinite programs to address this problem. They show that polynomial functions that are optimal to any fixed level of our hierarchy form a consistent estimator of the underlying shape-constrained function, which generates the data points. As a by-product of the proof, they establish that sum-of-squares-convex polynomials are dense in the set of polynomials that are convex over an arbitrary box. They further demonstrate the performance of the regressor for the problem of computing optimal transport maps in a color transfer task and that of estimating the optimal value function of a conic program. A real-time application of the latter problem to inventory management contract negotiation is presented.

A fuzzy based intelligent scheme for enhancing the performance of the optimal controllers by online weighting matrix selection in seismically excited nonlinear buildings

In the present work, an effective strategy is proposed to improve the efficiency of the optimal controllers that are based on minimizing a performance index by using a fuzzy controller. In the used fuzzy system, the weighting matrices of the optimal controller are tuned online according to the response of the structural system. In order to assess the effectiveness of the proposed control scheme, its function is investigated in two phases on a 3-story nonlinear benchmark building equipped with MR dampers. In the first phase, a linear quadratic Gaussian (LQG) controller is combined with a fuzzy logic system (Fuzzy-LQG) and the efficiency of the used technique is compared with three different cases of the conventional LQG controllers. The results show that the Fuzzy-LQG, in addition to using an optimal level of demand energy, has a better performance than the conventional LQG methods in controlling the maximum responses of the structure. In the second phase, the performance of a polynomial optimal controller that is a summation of polynomials in nonlinear states in combination with the fuzzy system is investigated. At this stage, the effectiveness of the designed control system (Fuzzy Polynomial) in terms of seismic performance criteria with the results of a conventional LQG active control system, an optimal adaptive neuro-fuzzy inference system (ANFIS) and a neural network predictive control (NNPC) system is evaluated. The results show that this method is successful in improving the structural performance of a nonlinear building subjected to earthquakes.

A polynomial invariant of Kauffman type for knotoids

We construct a three-variable polynomial invariant for unoriented kd([Formula: see text])-linkoid diagrams including knotoid ones as a certain weighted sum of polynomials on oriented kd([Formula: see text])-linkoid diagrams associated with a given kd([Formula: see text])-linkoid diagram.

Total Orderization Invariant Maps on Distributive Lattices

Given any finite subset A of order n of a distributive lattice and kin {1,ldots ,n}, there is a natural extension of the median operation to n variables which generalizes the notion of the kth smallest element of A. By applying each of these operations to A, a totally ordered set to(A) is obtained. We refer to to(A) as the total orderization of A. After developing a brief theory of total orderization invariant maps on distributive lattices, it is shown in this paper how these functions generalize and provide new characterizations for symmetric continuous positively homogeneous functions, bounded orthosymmetric multilinear maps, and certain power sum polynomials on vector lattices. These theorems generalize several results by Bernau, Huijsmans, Kusraev, Azouzi, Boulabiar, Buskes, Boyd, Ryan, and Snigireva and in turn reveal novel properties of the various maps studied in this paper.

Unitary matrix integrals, symmetric polynomials, and long-range random walks

Unitary matrix integrals over symmetric polynomials play an important role in a wide variety of applications, including random matrix theory, gauge theory, number theory, and enumerative combinatorics. We derive novel results on such integrals and apply these and other identities to correlation functions of long-range random walks (LRRW) consisting of hard-core bosons. We generalize an identity due to Diaconis and Shahshahani which computes unitary matrix integrals over products of power sum polynomials. This allows us to derive two expressions for unitary matrix integrals over Schur polynomials, which can be directly applied to LRRW correlation functions. We then demonstrate a duality between distinct LRRW models, which we refer to as quasi-local particle-hole duality. We note a relation between the multiplication properties of power sum polynomials of degree n and fermionic particles hopping by n sites. This allows us to compute LRRW correlation functions in terms of auxiliary fermionic rather than hard-core bosonic systems. Inverting this reasoning leads to various results on long-range fermionic models as well. In principle, all results derived in this work can be implemented in experimental setups such as trapped ion systems, where LRRW models appear as an effective description. We further suggest specific correlation functions which may be applied to the benchmarking of such experimental setups.

Adelic amplitudes and intricacies of infinite products

For every prime number p it is possible to define a p-adic version of the Veneziano amplitude and its higher-point generalizations. Multiplying together the real amplitude with all its p-adic counterparts yields the adelic amplitude. At four points it has been argued that the adelic amplitude, after regulating the product that defines it, equals one. For the adelic 5-point amplitude, there exist kinematic regimes where no regularization is needed. This paper demonstrates that in special cases within this regime, the adelic product can be explicitly evaluated in terms of ratios of the Riemann zeta function, and observes that the 5-point adelic amplitude is not given by a single analytic function. Motivated by this fact to study new regularization procedures for the 4-point amplitude, an alternative formalism is presented, resulting in non-constant amplitudes that are piecewise analytic in the three scattering channels, including one non-constant adelic amplitude previously suggested in the literature. Decomposing the residues of these amplitudes into weighted sums of Gegenbauer polynomials, numerical evidence indicates that in special ranges of spacetime dimensions all the coefficients are positive, as required by unitarity.

Fast Verification of Control Barrier Functions via Linear Programming

Control barrier functions are a popular method of ensuring system safety, and these functions can be used to enforce invariance of a set under the dynamics of a system. A control barrier function must have certain properties, and one must both formulate a candidate control barrier function and verify that it does indeed satisfy the required properties. Targeting the latter problem, this paper presents a method of verifying any finite number of candidate control barrier functions with linear programming. We first apply techniques from real algebraic geometry to formulate verification problem statements that are solvable numerically. Typically, semidefinite programming is used to verify candidate control barrier functions, but this does not always scale well. Therefore, we apply a method of inner-approximating the set of sums of squares polynomials that significantly reduces the computational complexity of these verification problems by transcribing them to linear programs. We give explicit forms for the resulting linear programs, and simulation results for a satellite inspection problem show that the computation time needed for verification can be reduced by more than 95%.

Determination of asymptotic normalization coefficients for the channel $$^{16}$$O$$\rightarrow \alpha +^{12}$$C: excited state $$^{16}$$O($$0^+; 6.05$$ MeV)

Asymptotic normalization coefficients (ANC) determine the overall normalization of cross sections of peripheral radiative capture reactions. In the present paper, we treat the ANC $C$ for the virtual decay $^{16}$O$(0^+; 6.05$ MeV)$\to \alpha+^{12}$C(g.s.), the known values of which are characterized by a large spread $(0.29-1.65)\times 10^3$ fm$^{-1/2}$. The ANC $C$ is found by analytic continuation in the energy of the $\alpha^{12}$C $s$-wave scattering amplitude, known from the phase-shift analysis of experimental data, to the pole corresponding to the $^{16}$O bound state and lying in the unphysical region of negative energies. To determine $C$, two different methods of analytic continuation are used. In the first method, the scattering data are approximated by the sum of polynomials in energy in the physical region and then extrapolated to the pole. The best way of extrapolation is chosen on the basis of the exactly solvable model. Within the second approach, the ANC $C$ is found by solving the Schr\"odinger equation for the two-body $\alpha^{12}$C potential, the parameters of which are selected from the requirement of the best description of the phase-shift analysis data at a fixed experimental binding energy of $^{16}$O$(0^+; 6.05$ MeV) in the $\alpha+^{12}$C channel. The values of the ANC $C$ obtained within these two methods lie in the interval (886--1139) fm$^{-1/2}$.

Power sum polynomials and the ghosts behind them

The power sum polynomial associated to a multi-subset of the projective plane text {PG}(2,q) is the sum of the (q-1)-th powers of the Rédei factors of the points in the multi-subset. The classification of multi-subsets having the same power sum polynomial passes through the determination of those multi-subsets associated to the zero polynomial, called ghosts. In this paper we provide new classes of ghosts and compute the dimension of the ghost subspace by exploiting the linear code generated by the lines of text {PG}(2,q) and its dual. Moreover, we explicitly enumerate and classify ghosts for planes of order 2, 3, 4.

Sums of Separable and Quadratic Polynomials

We study separable plus quadratic (SPQ) polynomials, that is, polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials but negative already for bivariate quartic SPQ polynomials. We use our decomposition result for convex SPQ polynomials to show that convex SPQ polynomial optimization problems can be solved by “small” semidefinite programs. For question (ii), we provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. We end by presenting applications of SPQ polynomials to upper bounding sparsity of solutions to linear programs, polynomial regression problems in statistics, and a generalization of Newton’s method that incorporates separable higher order derivative information. Funding: This work was partially supported by an AFOSR MURI award, the Defense Advanced Research Projects Agency Young Faculty Award, the Princeton School of Engineering and Applied Sciences Innovation Award, the National Science Foundation Faculty Early Career Development Program Award, the Google Faculty Award, and the Sloan Fellowship.

Unfolding-synthesis technique for digital pulse processing, Part 2: Synthesis

Pulse shaping is an important step in the unfolding-synthesis technique. In this paper we present efficient digital pulse-shaping algorithms that utilize repeated sum polynomials. These algorithms address the most common constraint in pulse shape synthesis — the finite duration of the pulses. The presented digital methods for efficient real-time synthesis use only basic digital signal processing functions (addition, constant multiplication, and shift), thereby minimizing required signal processing resources. A differentiation technique to decompose pulse shapes defined by polynomials is presented and used to synthesize arbitrary trapezoidal/triangular pulse shapes. The synthesis of rational, exponential, trigonometric and other non-polynomial defined pulse shapes can be approximated in real time. A methodology to approximate non-polynomial defined pulse shapes is described, and Gaussian and sinusoidal pulse shapes are synthesized via polynomial approximation and linear interpolation. The pulse shape synthesis algorithms are presented in recursive form and are suitable for efficient implementation by using integer only arithmetic.