One-step 9-stage Hermite–Birkhoff–Taylor ODE solver of order 10

A one-step 9-stage Hermite–Birkhoff–Taylor method of order 10, denoted by HBT(10)9, is constructed for solving nonstiff systems of first-order differential equations of the form y′=f(x,y), y(x ₀)=y ₀. The method uses y′ and higher derivatives y ⁽²⁾ to y ⁽⁴⁾ as in Taylor methods and is combined with a 9-stage Runge–Kutta method. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to Taylor- and Runge–Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than Dormand–Prince DP(8,7)13M. The stepsize is controlled by means of y ⁽²⁾ and y ⁽⁴⁾. HBT(10)9 is superior to DP(8,7)13M and Taylor method of order 10 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. These numerical results show the benefits of adding high-order derivatives to Runge–Kutta methods.     

Hermite–Birkhoff–Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size

A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y^′=f(t,y)$y^{\prime }=f(t,y)$ with initial conditions y(_t0)=_y0$y\left(t_0\right)=y_0$. Its formula uses y^′$y^{\prime }$, y^″$y^{″}$ and y^?$y^{?}$ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.     

On variable step Hermite–Birkhoff solvers combining multistep and 4-stage DIRK methods for stiff ODEs

Variable-step (VS) 4-stage k-step Hermite–Birkhoff (HB) methods of order p = (k + 2), p = 9, 10, denoted by HB (p), are constructed as a combination of linear k-step methods of order (p ? 2) and a diagonally implicit one-step 4-stage Runge–Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge–Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop L(a)-stable methods of order up to 11 with a > 63°. Fast algorithms are developed for solving these systems in O (p²) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsizes of these methods are controlled by a local error estimator. HB(p) of order p = 9 and 10 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8) and Ebadi et al. hybrid backward differentiation formulae of order 10 and 12, HBDF(10-12) in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.     

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge–Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples.     

Higher order recurrences and row sequences of Hermite–Padé approximation

ABSTRACT

We obtain extensions of the Poincaré and Perron theorems for higher order recurrence relations and apply them to obtain an inverse-type theorem for row sequences of the (type II) Hermite–Padé approximation of a vector of formal power series.     

On the order of a non-abelian representation group of a slim dense near hexagon

In this paper we study the possible orders of a non-abelian representation group of a slim dense near hexagon. We prove that if the representation group R of a slim dense near hexagon S is non-abelian, then R is a 2-group of exponent 4 and |R|=2 ^β , 1+NPdim(S)≤β≤1+dimV(S), where NPdim(S) is the near polygon embedding dimension of S and dimV(S) is the dimension of the universal representation module V(S) of S. Further, if β=1+NPdim(S), then R is necessarily an extraspecial 2-group. In that case, we determine the type of the extraspecial 2-group in each case. We also deduce that the universal representation group of S is a central product of an extraspecial 2-group and an abelian 2-group of exponent at most 4. This work was partially done when B.K. Sahoo was a Research Fellow at the Indian Statistical Institute, Bangalore Center with NBHM fellowship, DAE Grant 39/3/2000-R&D-II, Govt. of India.     

How to obtain higher-order multivariate Hermite expansion of Maxwell–Boltzmann distribution by using Taylor expansion?

We obtain the higher-order multivariate Hermite expansion of the Maxwell–Boltzmann distribution by using a new, compact tensorial notation and present a method to obtain the nth order multivariate Taylor expansion, which is identical to the nth order multivariate Hermite expansion of the Maxwell–Boltzmann distribution. This study enables us to find higher-order models of discrete kinetic theories such as the lattice Boltzmann theory.     

Construction of Interpolating Scaling Vectors with Hermite Interpolation Property

In this paper, we construct a new family of Hermite-type interpolating scaling vectors with compact support, of which the Hermite interpolation property generalizes the existing results of interpolating scaling vectors and Hermite interpolants. In terms of the Hermite interpolatory mask, we characterize the Hermite interpolation property, approximation property and symmetry property in detail. To illustrate these results, several examples with compact support and high smoothness are exhibited at the end of this paper.     

Rayleigh–Taylor instability of superposed barotropic fluids

Effects of compressibility on Rayleigh?CTaylor instability of superposed fluids are considered. The density is allowed to vary with pressure under the barotropy assumption. The small-compressibility limit is considered first in order to facilitate an analytical calculation. For the case with equal speeds of sound in the two superposed fluids, a non-trivial analytical compressibility correction to the Rayleigh?CTaylor growth rate becomes feasible if we perturbatively calculate the compressibility correction to O (g ²/k ² a ⁴). To this order, compressibility effects are found to reduce the growth rate. This trend is validated for arbitrary compressibility cases as well via an exact evaluation of the dispersion relation.     

A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes

In this paper, we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes based on the work (Computers & Fluids, 34: 642–663 (2005)) by Qiu and Shu, with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation laws. The key idea of HWENO is to evolve both with the solution and its derivative, which allows for using Hermite interpolation in the reconstruction phase, resulting in a more compact stencil at the expense of the additional work. The main difference between this work and the formal one is the procedure to reconstruct the derivative terms. Comparing with the original HWENO schemes of Qiu and Shu, one major advantage of new HWENO schemes is its robust in computation of problem with strong shocks. Extensive numerical experiments are performed to illustrate the capability of the method. Corresponding author This work was partially supported by the National Natural Science Foundation of China (Grant No. 10671097), the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simulations, Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and the Natural Science Foundation of Jiangsu Province (Grant No. BK2006511)     

Koksma–Hlawka type inequalities of fractional order

The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order is in . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.      

Construction of Runge–Kutta methods of Crouch–Grossman type of high order

An approach is described to the numerical solution of order conditions for Runge–Kutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg–Marquardt algorithm. Methods of order four and five are constructed and numerical experiments are presented which confirm that the derived methods have the expected order of accuracy.     

A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.     

Poincaré–Birkhoff–Witt Deformation of Koszul Calabi–Yau Algebras

The Calabi–Yau property of the Poincaré–Birkhoff–Witt deformation of a Koszul Calabi–Yau algebra is characterized. Berger and Taillefer (J Noncommut Geom 1:241–270, 2007, Theorem 3.6) proved that the Poincaré–Birkhoff–Witt deformation of a Calabi–Yau algebra of dimension 3 is Calabi–Yau under some conditions. The main result in this paper generalizes their result to higher dimensional Koszul Calabi–Yau algebras. As corollaries, the necessary and sufficient condition obtained by He et al. (J Algebra 324:1921–1939, 2010) for the universal enveloping algebra, respectively, Sridharan enveloping algebra, of a finite-dimensional Lie algebra to be Calabi–Yau, is derived.     

Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

A holomorphic function on a planar domain \(\Omega \) is said to possess a universal Taylor series about a point \(\zeta \) of \(\Omega \) if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in \(\mathbb {C}\backslash \Omega \) that have connected complement. In the case where \(\Omega \) is simply connected, such functions are known to be unbounded and to form a collection that is independent of the choice of \(\zeta \) . This paper uses tools from potential theory to show that, even for domains \(\Omega \) of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case, a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of \(\zeta \) . This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.     

Asymptotic bubble evolutions of the Rayleigh–Taylor instability

We examine the multiple harmonic model for the single-mode Rayleigh–Taylor instability, and present a new class of the asymptotic solution for the bubble evolution. Previously reported solutions for the bubble curvature and velocity from the model were quantitatively different from other theoretical models and numerical results, for small density jumps. The discrepancy between the theoretical models is resolved by our new approach to the model. Our solution agrees with the Layzer–Goncharov model, and gives the independence of the bubble curvature on the density ratio.     

Semi-symmetric Lorentzian metrics and three-dimensional curvature homogeneity of order one

We completely classify three-dimensional semi-symmetric Lorentzian manifolds which are curvature homogeneous up to order one. Curvature restrictions for semi-symmetry turn out to be the same ones which ensure the existence on these manifolds of a degenerate parallel null line field (Chaichi et al., J. Phys. A, Math. Gen. 38, 841–850, 2005). Supported by funds of MURST, GNSAGA and University of Salento.     

Pseudo almost automorphic solutions of fractional order neutral differential equation

In this paper we discuss the pseudo almost automorphic solution of a fractional order neutral differential equation in a Banach space X. The results are established using the Krasnoselskii’s fixed point theorem.