Polynomial Weight Research Articles

On the Widom factors for [formula omitted] extremal polynomials

We continue our study of the Widom factors for Lp(μ) extremal polynomials initiated in (Alpan and Zinchenko, 2020). In this work we characterize sets for which the lower bounds obtained in (Alpan and Zinchenko, 2020) are saturated, establish continuity of the Widom factors with respect to the measure μ, and show that despite the lower bound [W2,n(μK)]2≥2S(μK) for the equilibrium measure μK on a compact set K⊂R the general lower bound [Wp,n(μ)]p≥S(μ) is optimal even for measures dμ=wdμK with polynomial weights w on K⊂R. We also study pull-back measures under polynomial pre-images introduced in (Geronimo and Van Assche,1988), (Peherstorfer and Steinbauer, 2001) and obtain invariance of the Widom factors for such measures. Lastly, we study in detail the Widom factors for orthogonal polynomials with respect to the equilibrium measure on a circular arc and, in particular, find their limit, infimum, and supremum and show that they are strictly monotone increasing with the degree and strictly monotone decreasing with the length of the arc.

Power bounded operators and the mean ergodic theorem for subsequences

Let T be a power bounded Hilbert space operator without unimodular eigenvalues. We show that the subsequential ergodic averages N−1∑n=1NTan converge in the strong operator topology for a wide range of sequences (an), including the integer part of most of subpolynomial Hardy functions. Moreover, we show that the weighted averages N−1∑n=1Ne2πig(n)Tan also converge for many reasonable functions g. In particular, we generalize the polynomial mean ergodic theorem for power bounded operators due to ter Elst and the second author [16] to real polynomials and polynomial weights.

A second-order discretization for degenerate systems of stochastic differential equations

Abstract The paper proposes a new second-order weak approximation scheme for hypoelliptic diffusions or degenerate systems of stochastic differential equations satisfying a certain Hörmander condition. The scheme is constructed by a Gaussian process and a stochastic polynomial weight through a technique based on Malliavin calculus, and is implemented by a Monte Carlo method and a quasi-Monte Carlo method. A variance analysis for the Monte Carlo method is discussed, and further control variate methods are introduced to reduce the variance. The effectiveness of the proposed scheme is illustrated through numerical experiments for some hypoelliptic diffusions.

On Optimality of CSS Codes for Transversal T

In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal $T$ and $T^{-1}$ gates preserve the codespace. Our Heisenberg perspective reduces this to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries: (a) For any non-degenerate $[[ n,k,d ]]$ stabilizer code supporting a physical transversal $T$, there exists an $[[ n,k,d ]]$ CSS code with the same property; (b) Triorthogonal codes are the most general CSS codes that realize logical transversal $T$ via physical transversal $T$; (c) Triorthogonality is necessary for physical transversal $T$ on a CSS code to realize the logical identity. The main tool we use is a recent efficient characterization of certain diagonal gates in the Clifford hierarchy (arXiv:1902.04022). We refer to these gates as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing code constructions that realize logical gates via transversal $T$. We provide several examples and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal $\pi/2^{\ell}$ $Z$-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal $\pi/2^{\ell}$ $Z$-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal $1$- or $2$-local diagonal gate.

Optimal Exponential Decay for the Linearized Ellipsoidal BGK Model in Weighted Sobolev Spaces

This paper deals with the asymptotic behavior of solution to the linearized ellipsoidal BGK model in torus. We prove that the solution converges exponentially to the equilibrium in the weighted Sobolev spaces with polynomial weight. Our exponential decay rate $$e^{-\lambda t}$$ is optimal in the sense that $$\lambda >0$$ equals to the spectral gap of the linearized operator in the standard Hilbert space. Our strategy is taking advantage of the quantitative spectral gap estimates in a smaller reference Hilbert space, the factorization method, and the enlargement of the functional space for the associated semigroup.

Generalization of Sz\xe1sz\u2013Mirakjan\u2013Kantorovich operators using multiple Appell polynomials

The purpose of the present paper is to introduce and study a sequence of positive linear operators defined on suitable spaces of measurable functions on [0,infty ) and continuous function spaces with polynomial weights. These operators are Kantorovich type generalization of Jakimovski–Leviatan operators based on multiple Appell polynomials. Using these operators, we approximate suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,infty ) that do not constitute a subdivision of it. We also discuss the rate of convergence of these operators using moduli of smoothness.

Weighted Hurwitz Numbers and Topological Recursion

The KP and 2D Toda \(\tau \)-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel–Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations in the case of polynomial weight functions.

Regularization Estimates and Cauchy Theory for Inhomogeneous Boltzmann Equation for Hard Potentials Without Cut-Off

In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant frameworkfor this equation. Moreover, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.

General results on the divisibility of low weight polynomials in [formula omitted

In [6], Munemasa presented results concerning when a trinomial divides a trinomial in F2[x] and in [7], Panario, Sosnovski, Stevens and Wang presented a similar result for F3[x] along with a result for when a trinomial divides a binomial in the same polynomial ring. In this paper, by using a different technique from the one used in those papers, we generalize and extend these results to Fq[x] for any prime power q. We also provide results and conjectures resulting from a computer search for small q.

On random polynomials generated by a symmetric three-term recurrence relation

ABSTRACTWe investigate the sequence of random polynomials generated by the three-term recurrence relation , , with initial conditions , , assuming that is a sequence of positive i.i.d. random variables. is a sequence of orthogonal polynomials on the real line, and is the characteristic polynomial of a Jacobi matrix . We investigate the relation between the common distribution of the recurrence coefficients and two other distributions obtained as weak limits of the averaged empirical and spectral measures of . Our main result is a description of combinatorial relations between the moments of the aforementioned distributions in terms of certain classes of coloured planar trees. Our approach is combinatorial, and the starting point of the analysis is a formula of P. Flajolet for weight polynomials associated with labelled Dyck paths.

Quantitative estimations of bivariate summation‐integral–type operators

In this paper, we study the approximation properties of bivariate summation‐integral–type operators with two parameters . The present work deals within the polynomial weight space. The rate of convergence is obtained while the function belonging to the set of all continuous and bounded function defined on ([0],∞)(×[0],∞) and function belonging to the polynomial weight space with two parameters, also convergence properties, are studied. To know the asymptotic behavior of the proposed bivariate operators, we prove the Voronovskaya type theorem and show the graphical representation for the convergence of the bivariate operators, which is illustrated by graphics using Mathematica. Also with the help of Mathematica, we discuss the comparison by means of the convergence of the proposed bivariate summation‐integral–type operators and Szász‐Mirakjan‐Kantorovich operators for function of two variables with two parameters to the function. In the same direction, we compute the absolute numerical error for the bivariate operators by using Mathematica and is illustrated by tables and also the comparison takes place of the proposed bivariate operators with the bivariate Szász‐Mirakjan operators in the sense of absolute error, which is represented by table. At last, we study the simultaneous approximation for the first‐order partial derivative of the function.

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

In this note we investigate the multi-parameter Potential Theory on the weighted d-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

Exact and Approximate Weighted Model Integration with Probability Density Functions Using Knowledge Compilation

Weighted model counting has recently been extended to weighted model integration, which can be used to solve hybrid probabilistic reasoning problems. Such problems involve both discrete and continuous probability distributions. We show how standard knowledge compilation techniques (to SDDs and d-DNNFs) apply to weighted model integration, and use it in two novel solvers, one exact and one approximate solver. Furthermore, we extend the class of employable weight functions to actual probability density functions instead of mere polynomial weight functions.

A third-order weak approximation of multidimensional Itô stochastic differential equations

Abstract This paper proposes a new third-order discretization algorithm for multidimensional Itô stochastic differential equations driven by Brownian motions. The scheme is constructed by the Euler–Maruyama scheme with a stochastic weight given by polynomials of Brownian motions, which is simply implemented by a Monte Carlo method. The method of Watanabe distributions on Wiener space is effectively applied in the computation of the polynomial weight of Brownian motions. Numerical examples are shown to confirm the accuracy of the scheme.

Mountain railway alignment optimization using stepwise & hybrid particle swarm optimization incorporating genetic operators

Optimizing railway alignments is a quite complex and time-consuming engineering problem. The huge continuous search space, complex constraints, implicit objective function and infinite potential alternatives of this problem pose many challenges. Especially in mountainous regions, finding a near-optimal alignment for extremely complex terrain and constraints is a most arduous task, which cannot be solved satisfactorily with most existing methods. In this study, a stepwise & hybrid particle swarm-genetic algorithm is developed for railway alignment optimization in mountainous regions. It is a continuous search method suitable for railway alignment design. A stepwise horizontal–vertical–integral approach which defines the horizontal and vertical alignments as two kinds of particles, is proposed to solve the three-dimensional railway alignment optimization problem. To enhance the initial diversity and momentum, butterfly-shaped areas are preset on a path generated with a bidirectional distance transform for initializing horizontal particles. For the solution method, specific genetic operators, including roulette wheel selection, four crossovers and two mutations are integrated into the stepwise particle swarm method to address parameter-dependent performance and avoid premature convergence. In addition, a cubic polynomial weight update strategy is employed for thoroughly searching the problem space. This synthesis method has been applied to a real-world case in a very mountainous region. The detailed data analyses demonstrate that it can offer more promising solutions compared with alternatives designed by experienced designers and those generated with a genetic algorithm or non-stepwise particle swarm algorithm.

Emergence of exponentially weighted Lp-norms and Sobolev regularity for the Boltzmann equation

We consider the homogeneous Boltzmann equation for Maxwell and hard potentials, without cutoff, and study the appearance and propagation of Lp-norms, including polynomial and exponential weights. Propagation of Sobolev regularity with such weights is also considered. Classical and novel ideas are combined to elaborate an elementary argument that proves the result in the full range of integrability and singularity . For the case , we use an adaptation of the classical level set method by De Giorgi. All arguments are performed using the weak formulation which simplifies considerably the technicalities.

Trend to the equilibrium for the Fokker-Planck system with an external magnetic field

We consider the Fokker-Planck equation with an external magnetic field. Global-in-time solutions are built near the Maxwellian, the global equilibrium state for the system. Moreover, we prove the convergence to equilibrium at exponential rate. The results are first obtained on spaces with an exponential weight. Then they are extended to larger functional spaces, like certain Lebesgue spaces with polynomial weights and modified weighted Sobolev spaces, by the method of factorization and enlargement of the functional space developed in [Gualdani, Mischler, Mouhot, 2017].

From Boltzmann to incompressible Navier–Stokes in Sobolev spaces with polynomial weight

We study the Boltzmann equation on the [Formula: see text]-dimensional torus in a perturbative setting around a global equilibrium under the Navier–Stokes linearization. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a [Formula: see text]-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently of the Knudsen number. Finally, we prove well-posedness of the Cauchy problem for the nonlinear Boltzmann equation in perturbative setting and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal. Furthermore, this result only requires derivatives in the space variable and allows to connect solutions to the incompressible Navier–Stokes equations in these spaces.

Developing an Optimal Class of Generic Sixteenth-Order Simple-Root Finders and Investigating Their Dynamics

Developed here are sixteenth-order simple-root-finding optimal methods with generic weight functions. Their numerical and dynamical aspects are investigated with the establishment of a main theorem describing the desired optimal convergence. Special cases with polynomial and rational weight functions have been extensively studied for applications to real-world problems. A number of computational experiments clearly support the underlying theory on the local convergence of the proposed methods. In addition, to investigate the relevant global convergence, we focus on the dynamics of the developed methods, as well as other known methods through the visual description of attraction basins. Finally, we summarized the results, discussion, conclusion, and future work.

Relative profiles and extended weight polynomials of almost affine codes

In this paper we study various aspects concerning almost affine codes, a class including, and strictly larger than, that of linear codes. We use the combinatorial tool demi-matroids to show how one can define relative length/dimension and dimension/length profiles of flags (chains) of almost affine codes. In addition we show two specific results. One such result is how one can express the relative length/dimension profiles (also called relative generalized Hamming weights) of a pair of codes in terms of intersection properties between the smallest of these codes and subcodes of the largest code. The other result tells how one can find the extended weight polynomials, expressing the number of codewords of each possible weight, for each code in an infinite hierarchy of extensions of a code over a given alphabet.