Non-embeddable simple relation algebras

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.? The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jónsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.? Andréka, Jónsson, Németi [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”. Received November 27, 1996; accepted in final form July 3, 1997.     

Integrability of multivariate subordinated Lévy processes in Hilbert space

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].     

Solution of Maxwell equation in axisymmetric geometry by Fourier series decompostion and by use of H (rot) conforming finite element

Summary. This study deals with the mathematical and numerical solution of time-harmonic Maxwell equation in axisymmetric geometry. Using Fourier decomposition, we define weighted Sobolev spaces of solution and we prove expected regularity results. A practical contribution of this paper is the construction of a class of finite element conforming with the H (rot) space equipped with the weighted measure rdrdz. It appears as an extension of the well-known cartesian mixed finite element of Raviart-Thomas-Nédélec [11]–[15]. These elements are built from classical lagrangian and mixed finite element, therefore no special approximations functions are needed. Finally, following works of Mercier and Raugel [10], we perform an interpolation error estimate for the simplest proposed element. Received March 15, 1996 / Revised version received November 30, 1998 / Published online December 6, 1999     

The effect of spatial quadrature on finite element galerkin approximations to hyperbolic integro-differential equations

The purpose of this paper is to study the effect of numerical quadrature on the finite element approximations to the solutions of hyperbolic intego-differential equations. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in L ^∞(H ¹)L ^∞(L ²) norms and quasi-optimal estimate in L ^∞(L ^∞) norm using energy arguments. Further, optimal L(L ²)-estimates are shown to hold with minimal smoothness assumptions on the initial functions. The analysis in the present paper not only improves upon the earlier results of Baker and Dougalis [SIAM J. Numer. Anal. 13 (1976), pp. 577-598] but also confirms the minimum smoothness assumptions of Rauch [SIAM J. Numer. Anal. 22 (1985), pp. 245-249] for purely second order hyperbolic equation with quadrature.     

Composite mixed finite elements on plane quadrilaterals

Mixed finite elements over a plane convex quadrilateral are obtained by assembling two Raviart-Thomas mixed finite elements over triangles. The macroelement is given by an eliminating procedure of the degrees of freedom related to the common edge to the two triangles. This procedure results in a finite element with a space of interpolating functions containing the polynomials of degree ? l, where l is the greater integer for which the same property is satisfied by the relevant Raviart-Thomas [Mathematical Aspects of Finite Element Methods, Roma 1975, I. Galligani and E. Magenes, Eds., Lecture Notes in Mathematics Vol. 606, Springer-Verlag, Berlin, 1975] mixed finite element. The interpolation error is estimated by means of the technique of almost equivalent affine element as given by Ciavaldini and Nédélec [Rev. Fr. Autom. Inf. Recher. Opérationnelle Ser. Rouge R2 , 29–45 (1974)]. © 1993 John Wiley & Sons, Inc.     

Mesh‐centered finite differences from unconventional mixed‐hybrid nodal finite elements

It is shown how mesh‐centered finite differences can be obtained from unconventional mixed‐hybrid nodal finite elements. The classical Raviart‐Thomas schemes of index k (RTk) are based on interpolation parameters that are cell and/or edge moments. For the unconventional form (URTk), they become point values at Gaussian points. In particular, the scheme URT1 is fully described. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Novel numerical techniques for the finite moment log stable computational model for European call option

Option pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black–Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second-order spatial derivative of the BS model is replaced by a fractional operator of order α which generates an α-stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank–Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi-conjugate gradient stabilized method (Bi-CGSTAB) and the fast Bi-CGSTAB (FBi-CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher-order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.     

Multilevel methods for mixed finite elements in three dimensions

In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof. Received November 4, 1996 / Revised version received February 2, 1998     

Theoretical tools to solve the axisymmetric Maxwell equations

In this paper, the mathematical tools, which are required to solve the axisymmetric Maxwell equations, are presented. An in‐depth study of the problems posed in the meridian half‐plane, numerical algorithms, as well as numerical experiments, based on the implementation of the theory described hereafter, shall be presented in forthcoming papers. In the present paper, the attention is focused on the (orthogonal) splitting of the electromagnetic field in a regular part and a singular part, the former being in the Sobolev space H¹ component‐wise. It is proven that the singular fields are related to singularities of Laplace‐like operators, and, as a consequence, that the space of singular fields is finite dimensional. This paper can be viewed as the continuation of References (J. Comput. Phys. 2000; 161 : 218–249, Modél. Math. Anal. Numér, 1998; 32 : 359–389) Copyright © 2002 John Wiley & Sons, Ltd.     

Finite element approximations with quadrature for second‐order hyperbolic equations

In this article, the effect of numerical quadrature on the finite element Galerkin approximations to the solution of hyperbolic equations has been studied. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in the L^∞(H¹), L^∞(L²) norms, whereas quasi‐optimal estimate is derived in the L^∞(L^∞) norm using energy methods. The analysis in the present paper improves upon the earlier results of Baker and Dougalis [SIAM J Numer Anal 13 (1976), pp 577–598] under the minimum smoothness assumptions of Rauch [SIAM J Numer Anal 22 (1985), pp 245–249] for a purely second‐order hyperbolic equation with quadrature. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 537–559, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10022     

Invariant Kekulé structures in fullerene graphs

Fullerene graphs are trivalent plane graphs with only hexagonal and pentagonal faces. They are often used to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric Kekulé structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene and molecules with totally symmetric Kekulé structures could have special physical and chemical properties, as suggested in [Austin, S.J, and J. Baker, P. W. Fowler, D. E. Manolopoulos, Bond-stretch Isomerism and the Fullerenes, J. Chem. Soc. Perkin Trans. 2 (1994), 2319–2323] and [Rogers, K.M., and P. W. Fowler, Leapfrog fullerenes, Huckel bond order and Kekulé structures, J. Chem. Soc. Perkin Trans. 2 (2001), 18–22]. All fullerenes with at least ten symmetries were studied in [Graver, J.E. The Structure of Fullerene Signature, DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64, AMS (2005), 137–166.] and a complete catalog was given in [Graver, J. E. Catalog of All Fullerene with Ten or More Symmetries DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64 AMS (2005), 167–188]. Starting from this catalog in [Bogaerts, M., and G. Mazzuoccolo, G.Rinaldi, Totally symmetric Kekulé structures in fullerene graphs with ten or more symmetries, MATCH Communications in Mathematical and in Computer Chemistry 69 (2013), 677–705] we established exactly which of them have at least one totally symmetric Kekulé structure.     

Applications of Rosenbrock-type methods to the p-version of finite elements

In quasistatic solid mechanics the spatial as well as the temporal domain need to be discetized. For the spatial discretization usually elements with linear shape functions are used even though it has been shown that generally the p-version of the finite elemente method yields more effective discretizations, see e.g. [1], [2]. For the temporal discretization diagonal-implicit, see e.g. [4], and especially linear-implicit Runge-Kutta schemes, see e.g. [5], [6], have for smooth problems proven to be superior to the frequently applied Backward-Euler scheme (BE). Thus an approach combining the p-version of the finite element method with linear-implicit Runge-Kutta schemes, so-called Rosenbrock-type methods, is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust fitted operator finite difference method for a two-parameter singular perturbation problem1

We consider singularly perturbed two-point BVPs with two small parameters ? and μ multiplying the derivatives. It is pointed out in P.A. Farrel (Sufficient conditions for uniform convergence of a class of difference schemes for singular perturbation problems, IMA J. Numer. Anal. 7 (1987), pp. 459–472), that ‘in general, the exponentially fitted finite difference methods (EFFDMs) are more effective inside the layers. However, though these methods are uniformly convergent, they do not give fairly good approximations in the whole interval of interest’. In this paper, we study that the non-standard finite difference method (NSFDM) that we develop overcomes this weakness. Like EFFDMs, the NSFDM is also a method of fitted operator type. Secondly, unlike several earlier works (see, for example Gracia et al., Appl. Numer. Math. 56 (2006), pp. 962–980) where the authors use a combination of approaches in various regions, the method presented in this paper consists of just one scheme throughout the domain of interest. This is very important because it increases the possibilities of extending the approach both for higher dimensional and higher order problems. Combination of schemes usually suffers from the drawback that their selection is mostly based on the relative values of ? and μ, otherwise they fail to provide monotonic solutions. We also investigate a number of issues associated with a variety of NSFDMs and finally provide some comparative numerical results.     

New variational-asymptotic formulations for 3-D stress analyses of laminated composite shells with a circular hole

A new approach for three-dimensional stress analyses in composite cylindrical shells is presented. The method of composite expansions along with Hellinger-Reissner variational formulation is employed to derive the interior and edge layer problems for high order approximations. Classical assumptions have been justified and new approximations have been established. These formulations are directed especially towards, new high integrity mixed-hybrid finite element schemes. The expository examples chosen are of cross-ply and angle-ply laminated shells. The circumferential location of the delamination failure initiation, for angle-ply laminates containing a circular hole, is within a sector located symmetrically around the perpendicular direction to the applied load.     

On the number of join-irreducibles in a congruence representation of a finite distributive lattice

For a finite lattice L, let $ \trianglelefteq_L $ denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form $ \trianglelefteq_L $, as follows: Theorem. Let $ \trianglelefteq $ be a quasi-ordering on a finite set P. Then the following conditions are equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L, let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L is the congruence lattice of L. It is well-known that the inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive lattice D, let us define the join- excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$ , for any finite distributive lattice D; the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota     

Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems

We design a wavelet optimized finite difference (WOFD) scheme for solving self-adjoint singularly perturbed boundary value problems. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. Small dissipation of the solution is captured significantly using an adaptive grid. The adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples have been solved and compared with non-standard finite difference schemes in [J.M.S. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238]. The proposed method outperforms the non-standard finite difference for studying singular perturbation problems for small dissipations (very small ) and effective grid generation. Therefore, the proposed method is better for studying the more challenging cases of singularly perturbed problems.     

Implicit complementarity problems on isotone projection cones

Isac and Németh [G. Isac and A. B. Németh, Projection methods, isotone projection cones and the complementarity problem, J. Math. Anal. Appl. 153 (1990), pp. 258–275] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this article an iterative algorithm is studied in connection with an implicit complementarity problem. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones, extending the results of Németh [S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350 (2009), pp. 340–370]. Some existing concepts from the latter paper are extended to solve the problem of finding nonzero solutions of the implicit complementarity problem.     

Nonlinear Dunkl-wave equations

In this article, we introduce a class of nonlinear wave equations associated with the Dunkl operators, we study local and global well-posedness. Next, we establish the linearization of bounded energy solutions in the spirit of Gérard [P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal. 141 (1996), pp. 60–98]. The proof uses Strichartz-type inequalities and the energy estimate.     

Influence oflow Mach numbers on thenumerical dissipation of Godunov-typeschemes: a comparison between Roeand HLLscheme

Upwind schemes for the Euler equations face three kinds of problems in the low Mach number regime: The stiffness due to the presence of fast acoustic and slow entropy and shear waves can be overcome – at least for steady problems – by preconditioning the physical equations, see for example [1, 2]. Secondly, the 𝒪(M²)-pressure variations get lost in the O(1)-global pressure due to finite precision arithmetics. This cancellation problem was dealt with in [3]. The third problem originates in the numerical viscosity of the upwind schemes and is – in the author's view of the matter – not fully understood to date. Asymptotic analyses of the upwind schemes such as [4] suggest that the pressure field will completely degenerate, producing variations of the wrong order of magnitude. Our numerical andanalytical results give a more precise picture of the problem and are – at least in parts – contradicting the established view. We will prove in this paper that complete Riemann solvers such as Roe's behave completely different to incomplete Riemann solvers such as HLL in the low Mach number regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)