A finite element method for a single phase quasilinear free boundary problem in one space dimension

In this paper, we apply finite element Galerkin method to a singlephase quasilinear Stefan problem with a forcing term. To construct the fully discrete approximation we apply the extrapolated Crank-Nicolson method and we derive the optimal order of convergence 2 in the temporal direction inL ²,H ¹ normed spaces.     

Option pricing under a normal mixture distribution derived from the Markov tree model

We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ⁺, and σ⁻ required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expires. Comparing against the Black–Scholes model, we find that the MT model’s prices are closer to market prices.     

A geometric approach to error estimates for conservation laws posed on a spacetime

We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov’s method, we derive an L¹ error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.     

Covering groups of almost simple groups as Galois groups over ℚab(t)

In this paper we construct Galois extensions with the rigidity method and apply a criterion [15] for solving central embedding problems over ?^ab(t) to realize regularly the covering groups of most of the classical groups and the sporadic groups as Galois groups over ?^ab(t).     

On the determination of the basin of attraction of a periodic orbit in two-dimensional systems

We consider the general nonlinear differential equation with x∈R² and develop a method to determine the basin of attraction of a periodic orbit. Borg's criterion provides a method to prove existence, uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction. In order to use the criterion one has to find a function W∈C¹(R²,R) such that LW(x)=W^′(x)+L(x) is negative for all x∈K, where K is a positively invariant set. Here, L(x) is a given function and W^′(x) denotes the orbital derivative of W. In this paper we prove the existence and smoothness of a function W such that LW(x)=−μ‖f(x)‖. We approximate the function W, which satisfies the linear partial differential equation W^′(x)=〈∇W(x),f(x)〉=−μ‖f(x)‖−L(x), using radial basis functions and obtain an approximation w such that Lw(x)<0. Using radial basis functions again, we determine a positively invariant set K so that we can apply Borg's criterion. As an example we apply the method to the Van-der-Pol equation.     

Relaxation limit for a symmetrically hyperbolic system

In this paper, we apply the invariant region theory to get an a prioriL^∞ estimate of the relaxation approximated solutions to the Cauchy problem of a symmetrically hyperbolic system with stiff relaxation and dominant diffusion, and then obtain that the relaxation approximated solutions converge almost everywhere to the equilibrium state of the symmetrical system with the aid of the compactness framework about the scalar equation.     

The cross ratio and Clifford algebras

Motivated by recent quaternionic approach of Bobenko and Pinkall to the complex cross ratio we presenta simple method to eva]uate the cross ratio in the Euclidean space? ⁿ identifying the space with vectors generating the Clifford algebraC(n). We apply the Clifford cross ratio to describe discrete analogues of orthogonal nets in? ⁿ .     

Trigonometric wavelet method for some elliptic boundary value problems

In this paper, we apply trigonometric wavelet method to investigate the numerical solution of elliptic boundary value problem in an exterior unit disk. The simple computation formulae of the entries in the stiffness matrix are obtained. It shows that we only need to compute 2(j²+1) elements of one ²j+2ײj+2 stiffness matrix. Moreover, the error estimates of the approximation solutions are given and some test examples are presented.     

One-dimensional Lagrangians generated by a quadratic form

In this paper we apply the classical variational calculus to the Lagrangians of particular type. Namely, we establish formulas for the 1^st integrals and propose a technique for obtaining invariant 1^st integrals. We also deduce the differential homogeneity conditions for Lagrangians with respect to multiplication of a path x(t) by a function c(t) and with respect to the change of the parameter t = t(s).     

ADI preconditioned Krylov methods for large Lyapunov matrix equations

In the present paper, we propose preconditioned Krylov methods for solving large Lyapunov matrix equations AX+XA^T+BB^T=0. Such problems appear in control theory, model reduction, circuit simulation and others. Using the Alternating Direction Implicit (ADI) iteration method, we transform the original Lyapunov equation to an equivalent symmetric Stein equation depending on some ADI parameters. We then define the Smith and the low rank ADI preconditioners. To solve the obtained Stein matrix equation, we apply the global Arnoldi method and get low rank approximate solutions. We give some theoretical results and report numerical tests to show the effectiveness of the proposed approaches.     

A strong law of large numbers for vector gaussian martingales and a statistical application in linear regression

For a d-dimensional gaussian martingae M with tensor increasingprocess we prove that <M>⁺_tM_t converges in R^d with probability 1 as t → ∞ and the limit is zero a.s. iff tr <M>⁺_t tends to zero. We apply this result to study the strong consistency of estimates in a linear regression model.     

Convergence analysis of the spectral collocation methods for two-dimensional nonlinear weakly singular Volterra integral equations*

We apply Jacobi spectral collocation approximation to a two-dimensional nonlinear weakly singular Volterra integral equation with smooth solutions. Under reasonable assumptions on the nonlinearity, we carry out complete convergence analysis of the numerical approximation in the L^∞-norm and weighted L²-norm. The provided numerical examples show that the proposed spectral method enjoys spectral accuracy.     

A combined mixed finite element method and local discontinuous Galerkin method for miscible displacement problem in porous media

A combined method consisting of the mixed finite element method for flow and the local discontinuous Galerkin method for transport is introduced for the one-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in L∞(0,T;L2) for concentration c,in L2(0,T;L2)for cxand L∞(0,T;L2) for velocity u are derived. The main technical difficulties in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method,the nonlinearity,and the coupling of the models. Numerical experiments are performed to verify the theoretical results. Finally,we apply this method to the one-dimensional compressible miscible displacement problem and give the numerical experiments to confirm the efficiency of the scheme.     

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

In this work we study the period function T of solutions to the conservative equation x^″(t)+f(x(t))=0. We present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H(x,y)=F(x)+n^-1|y|ⁿ. Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation Δu=γu^γ-1 in two dimensions.     

L 1 C 1 polynomial spline approximation algorithms for large data sets

In this article, we address the problem of approximating data points by C ¹-smooth polynomial spline curves or surfaces using L ₁-norm. The use of this norm helps to preserve the data shape and it reduces extraneous oscillations. In our approach, we introduce a new functional which enables to control directly the distance between the data points and the resulting spline solution. The computational complexity of the minimization algorithm is nonlinear. A local minimization method using sliding windows allows to compute approximation splines within a linear complexity. This strategy seems to be more robust than a global method when applied on large data sets. When the data are noisy, we iteratively apply this method to globally smooth the solution while preserving the data shape. This method is applied to image denoising.     

The quasilinearization method for boundary value problems on time scales

In this paper, we apply the method of quasilinearization to a family of boundary value problems for second order dynamic equations −y^Δ∇+q(t)y=H(t,y) on time scales. The results include a variety of possible cases when H is either convex or a splitting of convex and concave parts and whether lower and upper solutions are of natural form or of natural coupled form.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.     

A note on strong matrix summability via ideals

Following the line of recent works of Das et al. (2011) [15] and Savas and Das (2011) [19] we apply the notion of ideals to strong matrix summability, and in this note, we introduce the notion of strong A^I-summability with respect to an Orlicz function. We make certain investigations on the classes of sequences arising out of this new summability method.     

Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry

We give two sufficient conditions for a branch consisting of non-trivial solutions of an abstract equation in a Banach space not to have a (secondary) bifurcation point when the equation has a certain symmetry. When the nonlinearity f is of Allen-Cahn type (for instance f(u)=u−u³), we apply these results to an unbounded branch consisting of non-radially symmetric solutions of the Neumann problem on a disk D⊂R²