Polynomial Space Research Articles

Tchebycheffian B-splines in isogeometric Galerkin methods

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.

A C0 interior penalty method for mth-Laplace equation

In this paper, we propose a C0 interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝd, where m and d can be any positive integers. The standard H1-conforming piecewise r-th order polynomial space is used to approximate the exact solution u, where r can be any integer greater than or equal to m. Unlike the interior penalty method in Gudi and Neilan [IMA J. Numer. Anal. 31 (2011) 1734–1753], we avoid computing Dm of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete Hm-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete Hm-norm. The error estimate under the low regularity assumption of the exact solution is also obtained. Numerical experiments validate our theoretical estimate.

Deciding path size of nondeterministic (and input-driven) pushdown automata

The degree of ambiguity (respectively, the path size) of a nondeterministic automaton, on a given input, measures the number of accepting computations (respectively, the number of all computations). It is known that deciding the finiteness of the degree of ambiguity of a nondeterministic pushdown automaton is undecidable. Also, it is undecidable for a given k≥3 to decide whether the path size of a nondeterministic pushdown automaton is bounded by k.As the main result, we show that deciding the finiteness of the path size of a nondeterministic pushdown automaton can be done in polynomial time. Also, we show that the k-path problem for nondeterministic input-driven pushdown automata (respectively, for nondeterministic finite automata) is complete for exponential time (respectively, complete for polynomial space).

Proximity Search for Maximal Subgraph Enumeration

This paper proposes a new general technique for maximal subgraph enumeration which we call proximity search, whose aim is to design efficient enumeration algorithms for problems that could not be solved by existing frameworks. To support this claim and illustrate the technique we include output-polynomial algorithms for several problems for which output-polynomial algorithms were not known, including the enumeration of maximal bipartite subgraphs, maximal -degenerate subgraphs (for bounded ), maximal induced chordal subgraphs, and maximal induced trees. Using known techniques, such as reverse search, the space of all maximal solutions induces an implicit directed graph called “solution graph” or “supergraph,” and solutions are enumerated by traversing it; however, nodes in this graph can have exponential out-degree, thus requiring exponential time to be spent on each solution. The novelty of proximity search is a formalization that allows us to define a better solution graph, and a technique, which we call canonical reconstruction, by which we can exploit the properties of given problems to build such graphs. This results in solution graphs whose nodes have significantly smaller (i.e., polynomial) out-degree with respect to existing approaches, but that remain strongly connected, so that all solutions can be enumerated in polynomial delay by a traversal. A drawback of this approach is the space required to keep track of visited solutions, which can be exponential; we further propose a technique to induce a parent-child relationship among solutions and achieve polynomial space when suitable conditions are met.

Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians

We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\langle \alpha_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $\alpha_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\check{S}^{\dagger}_{W}$ of the quotient difference operator $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\widehat{S}$ is a linear difference operator with constant coefficients depending on $\alpha_{i}$ and $\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which is annihilated by $\check{S}^{\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\in\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.

Polynomial null solutions to bosonic Laplacians, bosonic Bergman and Hardy spaces

AbstractA bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

Faster Graph Coloring in Polynomial Space

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.

A decision framework with nonlinear preferences and unknown weight information for cloud vendor selection

Cloud vendor selection (CVS) is a complex decision-making problem, which actively adheres to human behavior/cognition. The complex nature of the problem is due to personal biases/hesitation, trade-offs among attributes, uncertainty in rating, and the nonlinear relationship among cloud vendors and associated attributes. In recent times, researchers started paying more attention to user/expert behavior, which led to non-linear decision-making. Most of the extant decision models for CVS considered the linear form of decision-making, which is not realistic due to expert opinions' complexity and dynamism. Motivated by the claim, in this paper, a non-linear decision approach is put forward for CVS. Likert scale rating is adopted for rating cloud vendors based on some attributes, which are transformed to polynomial space from the linear fuzzy space. After this, weights of attributes are determined by using CRITIC in the non-linear space. Following this, cloud vendors are ranked in a personalized fashion using the proposed algorithm that encompasses the WASPAS procedure and rank fusion schemes. Finally, a case study is exemplified to validate the usefulness of the decision approach. Comparison and sensitivity analysis showcases the efficacy and robustness of the developed approach.

On the SAV‐DG method for a class of fourth order gradient flows

AbstractFor a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty‐free discontinuous Galerkin (DG) spatial discretization leads to SAV‐DG schemes. These schemes are linear and shown unconditionally energy stable. However, the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre‐evaluate the auxiliary variable in the piecewise polynomial space. As a result the computational complexity of reduces to when exploiting the conjugate gradient (CG) solver. This hybrid SAV‐DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV‐DG schemes.

Parameter identification for Hammerstein nonlinear system with polynomial and state space model

This study investigates a two-stage parameter identification algorithm for the Hammerstein nonlinear system based on special test signals. The studied Hammerstein nonlinear system has a static nonlinear subsystem represented by polynomial basis function and a dynamic linear subsystem described by canonical observable state space model, and special test signals composed of binary signals and random signals are applied to parameter identification separation of the nonlinear subsystem and linear subsystem. The detailed identification procedures consist of two main steps. Firstly, using the characteristics that binary signals do not excite the static nonlinear subsystem, the dynamic linear subsystem parameters are identified through recursive least squares algorithm based on input-output data of binary signals. Secondly, unmeasurable state variables of the identified system are replaced with estimated values, thus the nonlinear subsystem parameters are obtained using recursive least squares algorithm with the help of input-output data of random signals. The efficiency and accuracy of proposed identification scheme are confirmed on experiment results of a numerical simulation and a practical nonlinear process, and experimental simulation results show that the developed two-stage identification algorithm has excellent predictive performance for identifying the Hammerstein nonlinear state space systems.

VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV

AbstractWe prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$ . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$ -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus

Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. Harmonic forms are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular cohomology class is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed piecewise polynomial differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed polynomial forms, that is, arbitrary degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.

Linear Pseudospectral Method with Chebyshev Collocation for Optimal Control Problems with Unspecified Terminal Time

In this paper, a linear Chebyshev pseudospectral method (LCPM) is proposed to solve the nonlinear optimal control problems (OCPs) with hard terminal constraints and unspecified final time, which uses Chebyshev collocation scheme and quasi-linearization. First, Taylor expansion around the nonlinear differential equations of the system is used to obtain a set of linear perturbation equations. Second, the first-order necessary conditions for OCPs with these linear equations and unspecified terminal time are derived, which provide the successive correction formulas of control and terminal time. Traditionally, these formulas are linear time varying and cannot be solved in an analytical manner. Third, Lagrange interpolation, whose supporting points are orthogonal Chebyshev–Gauss–Lobatto (CGL), is employed to discretize the resulting problem. Therefore, a series of analytical correction formulas are successfully derived in approximating polynomial space. It should be noted that Chebyshev approximation is close to the best polynomial approximation, and CGL points can be solved in closed form. Finally, LCPM is applied to the air-to-ground missile guidance problem. The simulation results show that it has high computational efficiency and convergence rate. A comparison with the other typical OCP solvers is provided to verify the optimality of the proposed algorithm. In addition, the results of Monte Carlo simulations are presented, which show that the proposed algorithm has strong robustness and stability. Therefore, the proposed method has potential to be onboard application.

Eichler-Shimura isomorphism and vector-valued Hecke eigenpolynomials

The Eichler-Shimura isomophism gives a relation between holomorphic cusp forms and period polynomials. In [21], Paşol and Popa introduced vector-valued period polynomials and studied various properties of such polynomials. In this paper, we find a basis for the space of vector-valued period polynomials consisting of Hecke eigenpolynomials, and relate the basis to a Miller-like basis of the space of weakly holomorphic cusp forms by calculation of the matrix representation of the Hecke operator T˜l. Furthermore, we find the relation between our Hecke eigenpolynomials and the even and odd parts of the period polynomials induced from holomorphic Hecke eigenforms, and examine algebracity of the Hecke eigenpolynomials.

Encoder-X: Solving Unknown Coefficients Automatically in Polynomial Fitting by Using an Autoencoder

Modeling, prediction, and recognition tasks depend on the proper representation of the objective curves and surfaces. Polynomial functions have been proved to be a powerful tool for representing curves and surfaces. Until now, various methods have been used for polynomial fitting. With a recent boom in neural networks, researchers have attempted to solve polynomial fitting by using this end-to-end model, which has a powerful fitting ability. However, the current neural network-based methods are poor in stability and slow in convergence speed. In this article, we develop a novel neural network-based method, called Encoder-X, for polynomial fitting, which can solve not only the explicit polynomial fitting but also the implicit polynomial fitting. The method regards polynomial coefficients as the feature value of raw data in a polynomial space expression and therefore polynomial fitting can be achieved by a special autoencoder. The entire model consists of an encoder defined by a neural network and a decoder defined by a polynomial mathematical expression. We input sampling points into an encoder to obtain polynomial coefficients and then input them into a decoder to output the predicted function value. The error between the predicted function value and the true function value can update parameters in the encoder. The results prove that this method is better than the compared methods in terms of stability, convergence, and accuracy. In addition, Encoder-X can be used for solving other mathematical modeling tasks.

A family of H-div-div mixed triangular finite elements for the biharmonic equation

A family of Pk+22×2-PkH(divdiv) mixed finite elements on triangular grids is constructed for the biharmonic equation. For the H(divdiv) mixed finite element functions, both the stress tensor and its divergence have continuous normal components, σh⋅n and divσh⋅n, on each edge. The double divergence of the H(divdiv) mixed finite element space is exact the full discontinuous Pk polynomial space on a triangular grid. The stability of the mixed method is established. The stress solution converges at the optimal order in L2 and H(divdiv) norms. Four-order of superconvergence is proved in L2 norm for the displacement solution. The Pk discrete solution is processed on each triangle to obtain an optimal order Pk+4 solution there. Numerical results are presented verifying the theory. Additional computation shows that all existing H(divdiv) mixed finite elements fail to solve jump-coefficient biharmonic equations while the new mixed element converges independently of the interface-jump. This is because the H(divdiv) element here is the full H(divdiv)−Pk+2 space while the existing mixed elements are subspaces of the H(divdiv)−Pk+2 space, on a triangular grid.

Algebraic Number Starscapes

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by the resulting images, which we have called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases. The geometry describes and explains the notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. Meanwhile, the images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. In particular, the paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural action. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries of and its unit tangent bundle, respectively. Utilizing this geometry, we determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We reconsider the fundamental questions of the Diophantine approximation of complex numbers by algebraic numbers of bounded degree, from the geometric perspective developed. In the quadratic case (approximation by quadratic irrationals), we consider approximation in terms of hyperbolic distance between roots in the complex plane and the discriminant as a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponent k for which an algebraic target α has infinitely many approximations β whose hyperbolic distance from α does not exceed . It turns out to fall into two cases, depending on whether α lies on the image of a plane of rational slope in coefficient space (a rational geodesic). The result comes as an application of Schmidt’s subspace theorem. Our results recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered. Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.

Numerical simulation of thermal flows and entropy generation of magnetized hybrid nanomaterials filled in a hexagonal cavity

The current study addresses the features of entropy generation and thermal flows regarding magnetized hybrid nanofluid in the presence of a cylinder in a closed hexagonal domain. The hexagonal cavity comprises two heated horizontal walls, and two are insulated while the other walls are cold. The whole system has been modeled as coupled non-linear partial differential equations. Also, normalized the governing coupled equation by utilizing a proper pair of variables and are computed with a finite element approach. For the approximation of velocity profiles, a finite element space involving the quadratic polynomial (P2) is selected whereas the pressure and temperature estimation is accomplished through a space of linear polynomial (P1). An analogy is handed with published findings at confining instance. The degree of freedom and grid convergence test is considered for the kinetic energy (KE) and Bejan number (Be). The study reveals a major role in enhancing the heat transfer rate due to hybrid nano-particles. The results show that the increase of magnetic field effect considers the reduction of the heat transfer because the conduction motion occupies the motion of the fluid flow. When the Hartmann number is increased, the magnetic entropy is raised, too. For intensified the Hartmann numbers, the highest ratio of heat transfer occurs for the case of the hybrid nanoparticles, and then MgO-water, followed by the Ag-water. The nature of thermal flow parameters has been scrutinized.

Inclusion method of optimal constant with quadratic convergence for [formula omitted]-projection error estimates and its applications

We present an interval inclusion method for optimal constants of second-order error estimates of H01-projections to finite-degree polynomial spaces. These constants can be applied to error estimates of the Lagrange-type finite element method. Moreover, the proposed a priori error estimates are applicable to residual iteration techniques for the verification of solutions to nonlinear elliptic equations. Some numerical examples by the finite element method will be shown for comparison with other approaches, which confirm us the actual usefulness of the results in this paper for the numerical verification method for PDEs.