Constructive Polynomial Research Articles

Multiway dynamic nonlinear global neighborhood preserving embedding method for monitoring batch process

Aiming at the dynamic and nonlinear characteristics of batch process, a multiway dynamic nonlinear global neighborhood preserving embedding algorithm is proposed. For the nonlinear batch process monitoring, kernel mapping is widely used to eliminate nonlinearity by projecting the data into high-dimensional space, but the nonlinear relationships between batch process variables are limited by many physical constraints, and the infinite-order mapping is inefficient and redundant. Compared with the basic kernel mapping method which provides an infinite-order nonlinear mapping, the proposed method considers the dynamic and nonlinear characteristics with many physical constraints and preserves the global and local structures concurrently. First, the time-lagged window is used to remove the auto-correlation in time series of process variables. Second, a nonlinear method named constructive polynomial mapping is used to avoid unnecessary redundancy and reduce computational complexity. Third, the global neighborhood preserving embedding method is used to extract structures fully after the dynamic and nonlinear characteristics are processed. Finally, the effects of the proposed algorithm are demonstrated by a mathematical model and the penicillin fermentation process.

Constructive Polynomial Partitioning for Algebraic Curves in $\mathbb{R}^3$ with Applications

In 2015, Guth [Math. Proc. Cambridge Philos. Soc., 159 (2015), pp. 459--469] proved that for any set of $k$-dimensional bounded complexity varieties in ${\mathbb R}^d$ and for any positive integer ...

A new general model for quantum image histogram (QIH)

An image histogram diagram illustrates the frequency of intensity values occurring in an image. A histogram figure has two dimensions including the x-axis and the y-axis. The color values are demonstrated by the former, and the latter indicates the number of pixels having the corresponding color values. Because quantum image processing (QImP) has been developed rapidly in recent years, in this paper, a new general approach to construct quantum image histogram (QIH) for quantum grayscale and RGB images is proposed, which is based on two kinds of quantum images named novel enhanced quantum representation of digital images (NEQR) and novel colored quantum images (NCQI). QIH manipulates two entangled qubit sequences with the aim of storing and preparing image histogram information, constructing a quantum model for illustrating image histogram. This model requires $$(q+2n+1)$$ qubits based on NEQR image and $$3\times (q+2n+1)$$ qubits based on RGB image to construct the image histogram, when size of image is $$2^{n}\times 2^{n}$$ with gray (color) range $$2^{q}$$ . Moreover, our constructive polynomial preparation proves that the time complexity of QIH is not more than $$O(nq2^{2n+q})$$ . To the best of our knowledge, this is the first general approach for encoding histogram information based on quantum RGB and grayscale images, and it may open a new window to further statistical analysis procedures related to QImP. For instance, a practical use of QIH as a strict criterion to evaluate quantum data hiding algorithms is also discussed in this paper.

Improved latent variable models for nonlinear and dynamic process monitoring

The kernel mapping is a popular method for designing nonlinear process monitoring techniques. In most cases, kernel mapping is achieved by using the radial basis kernel function which, in theory, is able to provide an infinite order nonlinear mapping. However, such an infinite order mapping might be redundant and inefficient as the order of nonlinear relationship between process variables of industrial process system is often limited by many physical constraints. To address this issue, an efficient nonlinear mapping method called the constructive polynomial mapping (CPM) is utilized in lieu of the radial basis kernel function to avoid the excessive modeling redundancy. In addition, the CPM is integrated with the dynamic principal component analysis and the linear Gaussian state space model to build improved latent variable models for nonlinear and dynamic process monitoring. The promising performance of the proposed models has been demonstrated through two case studies.

Minimization and maximization versions of the quadratic travelling salesman problem

The travelling salesman problem (TSP) asks for a shortest tour through all vertices of a graph with respect to the weights of the edges. The symmetric quadratic travelling salesman problem (SQTSP) associates a weight with every three vertices traversed in succession. If these weights correspond to the turning angles of the tour, we speak of the angular-metric travelling salesman problem (Angle TSP). In this paper, we first consider the SQTSP from a computational point of view. In particular, we apply a rather basic algorithmic idea and perform the separation of the classical subtour elimination constraints on integral solutions only. Surprisingly, it turns out that this approach is faster than the standard fractional separation procedure known from the literature. We also test the combination with strengthened subtour elimination constraints for both variants, but these turn out to slow down the computation. Secondly, we provide a completely different, mathematically interesting MILP linearization for the Angle TSP that needs only a linear number of additional variables while the standard linearization requires a cubic one. For medium-sized instances of a variant of the Angle TSP, this formulation yields reduced running times. However, for larger instances or pure Angle TSP instances, the new formulation takes more time to solve than the known standard model. Finally, we introduce MaxSQTSP, the maximization version of the quadratic travelling salesman problem. Here, it turns out that using some of the stronger subtour elimination constraints helps. For the special case of the MaxAngle TSP, we can observe an interesting geometric property if the number of vertices is odd. We show that the sum of inner turning angles in an optimal solution always equals . This implies that the problem can be solved by the standard ILP model without producing any integral subtours. Moreover, we give a simple constructive polynomial time algorithm to find such an optimal solution. If the number of vertices is even, the optimal value lies between 0 and and these two bounds are tight, which can be shown by an analytic solution for a regular n-gon.

Graphical characterization of the set of all flat outputs for structured linear discrete-time systems

This paper deals with the characterization of flat outputs for structured linear discrete-time systems. More precisely, considering a structured discrete-time linear system, we provide a complete set of the flat outputs using constructive polynomial complexity order algorithms. The proposed method is simple to implement. It is based on usual algorithms dedicated to the computation of successors and predecessors of vertex subsets and to the computation and the ordering of strongly connected components in a digraph.

Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincare inequality, a cancellation property and a simple induction argument. We also construct a quasi-interpolation operator, built on local averages over stars, which is well defined for functions in $$L^1$$L1. We derive optimal error estimates for any polynomial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have comparable size, which yields optimal anisotropic error estimates over $$n$$n-rectangular domains. The interpolation theory extends to cases when the error and function regularity require different weights. We conclude with three applications: nonuniform elliptic boundary value problems, elliptic problems with singular sources, and fractional powers of elliptic operators.

On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

AbstractA proper edge coloring of a graph with colors 1, 2, 3, … is called an interval coloring if the colors on the edges incident to each vertex form an interval of integers. A bipartite graph is ‐biregular if every vertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. We prove that all (3, 6)‐biregular graphs have interval colorings and that all (3, 9)‐biregular graphs having a cubic subgraph covering all vertices of degree 9 admit interval colorings. Moreover, we prove that slightly weaker versions of the conjecture hold for (3, 5)‐biregular, (4, 6)‐biregular, and (4, 8)‐biregular graphs. All our proofs are constructive and yield polynomial time algorithms for constructing the required colorings.

Formula omitted] optimization and fixed poles of sampled-data systems under colored noise

The paper considers the sampled-data control problem for multivariable continuous LTI processes when colored stationary stochastic disturbances act on their inputs. The design problem is to find a causal stabilizing controller, which ensures the minimal value of the mean output variance. Based on the parametric transfer matrix (PTM) concept, a constructive polynomial solution of that design problem is provided. A number of general properties of the optimal system is derived, which are important for practical applications. In particular, the existence of fixed poles is shown. A consequence of this fact is a principal performance limitation of the H2 optimal control loop.

Filtered hyperinterpolation: a constructive polynomial approximation on the sphere

This paper considers a fully discrete filtered polynomial approximation on the unit sphere \({\mathbb{S}^{d}.}\) For \({f \in C(\mathbb{S}^{d}),V_{L,N}^{(a)} \, f}\) is a polynomial approximation which is exact for all spherical polynomials of degree at most L, so it inherits good convergence properties in the uniform norm for sufficiently smooth functions. The oscillations often associated with polynomial approximation of less smooth functions are localised by using a filter with support [0, a] for some a > 1, and with the value 1 on [0, 1]. The allowed choice of filters includes a recently introduced filter with minimal smoothness, and other smoother filters. The approximation uses a cubature rule with N points which is exact for all polynomials of degree \({t = \left\lceil{a L}\right\rceil+L-1.}\) The main theoretical result is that the uniform norm \({\|V_{L,N}^{(a)} \|}\) of the filtered hyperinterpolation operator is bounded independently of L, providing both good convergence and stability properties. Numerical experiments on \({\mathbb{S}^{2}}\) with a variety of filters, support intervals and cubature rules illustrate the uniform boundedness of the operator norm and the convergence of the filtered hyperinterpolation approximation for both an arbitrarily smooth function and a function with derivative discontinuities.

A flexible representation of quantum images for polynomial preparation, image compression, and processing operations

A Flexible Representation of Quantum Images (FRQI) is proposed to provide a representation for images on quantum computers in the form of a normalized state which captures information about colors and their corresponding positions in the images. A constructive polynomial preparation for the FRQI state from an initial state, an algorithm for quantum image compression (QIC), and processing operations for quantum images are combined to build the whole process for quantum image processing on FRQI. The simulation experiments on FRQI include storing, retrieving of images and a detection of a line in binary images by applying quantum Fourier transform as a processing operation. The compression ratios of QIC between groups of same color positions range from 68.75 to 90.63% on single digit images and 6.67---31.62% on the Lena image. The FRQI provides a foundation not only to express images but also to explore theoretical and practical aspects of image processing on quantum computers.

The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

AbstractElliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

Contributions to constructive polynomial ideal theory XXIII

In a 1941 paper (which is condensed here), N. M. Gjunter refers to some of his unknown papers from 1910, 1913 and 1925, which are partially written in context with the 1941 paper. Thus Gjunter had already pursued a constructive theory of polynomial ideals in 1913. Among other results, he proves the inequality attributed to Macaulay/Sperner in 1913 (from 1927 and 1930, respectively). Also discussed are results analogous to more recent work. The author hereby finishes his sequence of articles.

Learning polynomial feedforward neural networks by genetic programming and backpropagation

This paper presents an approach to learning polynomial feedforward neural networks (PFNNs). The approach suggests, first, finding the polynomial network structure by means of a population-based search technique relying on the genetic programming paradigm, and second, further adjustment of the best discovered network weights by an especially derived backpropagation algorithm for higher order networks with polynomial activation functions. These two stages of the PFNN learning process enable us to identify networks with good training as well as generalization performance. Empirical results show that this approach finds PFNN which outperform considerably some previous constructive polynomial network algorithms on processing benchmark time series.

Constructive Polynomial Approximation on the Sphere

This paper considers the problem of constructive approximation of a continuous function on the unit sphere Sr−1⊆Rr by a spherical polynomial from the space Pn of all spherical polynomials of degree ⩽n. In particular, for r=3 it is shown that the hyperinterpolation approximation Lnf (in which the Fourier coefficients in the exact L2 orthogonal projection Pnf are approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ⩽2n) has the exact order ‖Ln‖≍n1/2 for its uniform norm, provided the underlying quadrature rule satisfies an additional “quadrature regularity” assumption. For r=3, this rate of growth is the same as that of ‖Pn‖, and is known to be optimal among all linear projections on Pn. For r⩾3 an upper bound on ‖Ln‖ of non-optimal asymptotic order O(n(r−1)/2) also holds, without any special assumption on the quadrature rule.

Constructive polynomial approximation with a priori error bounds for nonlinear initial value differential problems

In this paper, we consider an initial value problem y′ = f( x, y), y(0) = y 0, where f is a continuous function satisfying a Lipschitz condition. First, the function f( x, y) is approximated by a Bernstein polynomial in two variables, B n ( f; x, y), of an appropriate degree according to a prescribed accuracy. Second, by application of the Frobenius method to the initial value problem y′ = B n ( f; x, y), y(0) = y 0, an exact series solution of the latter problem is constructed. Finally, the infinite series solution is truncated in order to obtain an explicit polynomial whose error with respect to the exact solution of the original problem is less than the prescribed accuracy, in an existence and uniqueness domain which is determined in terms of the initial data.

Single machine hierarchical scheduling with customer orders and multiple job classes

In this paper, we consider single machine bi-criteria scheduling problems where jobs are from multiple job classes and there are customer orders such that each customer order consists of at least one job from each of the job classes. A set-up time is needed whenever the machine is changed over from a job in one class to a job in another class. One objective is to minimize the makespan which is equivalent to minimizing the total set-up time. The other objective is to minimize total carrying costs of the customer orders. The carrying cost of a customer order is measured by the length of the time interval between the completion times of the first job and the last job in the customer order. We propose constructive polynomial time algorithms for the two hierarchical scheduling problems with these two objectives, i.e. the two scheduling problems in which one objective is to be optimized while holding the other objective fixed at its optimal value.

Constructive Polynomial Approximation on Spheres and Projective Spaces

Constructive Polynomial Approximation on Spheres and Projective Spaces

Constructive polynomial approximation on spheres and projective spaces.

This paper contains constructive generalizations to functions defined on spheres and projective spaces of the Jackson theorems on polynomial approximation. These results, (3.3) and (4.6), give explicit methods of constructing uniform approximations to smooth functions on these spaces by polynomials, together with error estimates based on the smoothness of the function and the degree of the polynomial. The general method used exploits the fact that each space considered is the orbit of some compact subgroup, G, of an orthogonal group acting on a Euclidean space. For such homogeneous spaces a general result (2.1) is proved which shows that a G-invariant linear method of polynomial approximation to continuous functions can be modified to yield a linear method which produces better approximations to k-times differentiable functions. Jackson type theorems (3.4) are also proved for functions on the unit ball (which is not homogeneous) in a Euclidean space.