Stanly Steinberg

Mimetic Finite Difference Methods for Diffusion Equations

This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous anisotropic diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.

Testing Stability by Quantifier Elimination

For initial and initial-boundary value problems described by differential equations, stability requires the solutions to behave well for large times. For linear constant-coefficient problems, Fourier and Laplace transforms are used to convert stability problems to questions about roots of polynomials. Many of these questions can be viewed, in a natural way, as quantifier-elimination problems. The Tarski?Seidenberg theorem shows that quantifier-elimination problems are solvable in a finite number of steps. However, the complexity of this algorithm makes it impractical for even the simplest problems. The newer Quantifier Elimination by Partial Algebraic Decomposition (QEPCAD) algorithm is far more practical, allowing the solution of some non-trivial problems. In this paper, we show how to write all common stability problems as quantifier-elimination problems, and develop a set of computer-algebra tools that allows us to find analytic solutions to simple stability problems in a few seconds, and to solve some interesting problems in from a few minutes to a few hours.

Mapping ErbB receptors on breast cancer cell membranes during signal transduction.

Distributions of ErbB receptors on membranes of SKBR3 breast cancer cells were mapped by immunoelectron microscopy. The most abundant receptor, ErbB2, is phosphorylated, clustered and active. Kinase inhibitors ablate ErbB2 phosphorylation without dispersing clusters. Modest co-clustering of ErbB2 and EGFR, even after EGF treatment, suggests that both are predominantly involved in homointeractions. Heregulin leads to dramatic clusters of ErbB3 that contain some ErbB2 and EGFR and abundant PI 3-kinase. Other docking proteins, such as Shc and STAT5, respond differently to receptor activation. Levels of Shc at the membrane increase two- to five-fold with EGF, whereas pre-associated STAT5 becomes strongly phosphorylated. These data suggest that the distinct topography of receptors and their docking partners modulates signaling activities.

Generalization of symmetric α-stable Lévy distributions for q>1

The alpha-stable distributions introduced by Lévy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study sequences of long-range dependent random variables whose distributions have asymptotic power-law decay, and which are called (q,alpha)-stable distributions. These sequences are generalizations of independent and identically distributed alpha-stable distributions and have not been previously studied. Long-range dependent (q,alpha)-stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter q controls dependence. If q=1 then they are classical independent and identically distributed with alpha-stable Lévy distributions. In the present paper we establish basic properties of (q,alpha)-stable distributions and generalize the result of Umarov et al. [Milan J. Math. 76, 307 (2008)], where the particular case alpha=2,q[1,3) was considered, to the whole range of stability and nonextensivity parameters alpha(0,2] and q[1,3), respectively. We also discuss possible further extensions of the results that we obtain and formulate some conjectures.

Lie series, Lie transformations, and their applications

This paper is an exposition of the basic properties of Lie series and Lie transformations, which are now finding widespread applications. The applications are of two types: expanding solutions of Hamiltons equations and reducing (simplifying) Hamiltonians to normal form. The expansions are not power series but rather factored product expansions. These expansions have the advantage that the approximating systems are also hamiltonian. The normal form procedure has the advantage that it is canonical and explicit. In both cases the methods used are chosen so that they are easy to implement in a general purpose computer symbol manipulator.

The sensitivity and accuracy of fourth order finite-difference schemes on nonuniform grids in one dimension

Abstract We construct local fourth-order finite difference approximations of first and second derivatives, on nonuniform grids, in one dimension. The approximations are required to satisfy symmetry relationships that come from the analogous higher-dimensional fundamental operators: the divergence, the gradient, and the Laplacian. For example, we require that the discrete divergence and gradient be negative adjoint of each other, DIV∗ = − GRAD , and the discrete Laplacian is defined as LAP = DIVGRAD . The adjointness requirement on the divergence and gradient guarantees that the Laplacian is a symmetric negative operator. The discrete approximations we derive are fourth-order on smooth grids, but the approach can be extended to create approximations of arbitrarily high order. We analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids.

Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient

We derive conservative fourth- and sixth-order finite difference approximations for the divergence and gradient operators and a compatible inner product on staggered 1D uniform grids in a bounded domain. The methods combine standard centered difference formulas in the interior with new one-sided finite difference approximations near the boundaries. We derive compatible inner products for these difference methods that are high-order approximations of the continuum inner product. We also investigate defining compatible high-order divergence and gradient finite difference operators that satisfy a discrete integration by parts identity.

Applications of the lie algebraic formulas of Baker, Campbell, Hausdorff, and Zassenhaus to the calculation of explicit solutions of partial differential equations

Abstract We apply Lie algebraic methods of the type developed by Baker, Campbell, Hausdorff, and Zassenhaus to the initial value and eigenvalue problems for certain special classes of partial differential operators which have many important applications in the physical sciences. We obtain detailed information about these operators including explicit formulas for the solutions of the problems of interest. We have also produced a computer program to do most of the intermediate algebraic computations.

Variable Order Differential Equations with Piecewise Constant Order-Function and Diffusion with Changing Modes

In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-differential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Processes that have distinctive regimes of different types of diffusion depending on time are ubiquitous in the nature. Examples include diffusion in a heterogeneous media and protein movement in cell biology.

A Discrete Vector Calculus in Tensor Grids

Abstract Mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and differential forms identities for their derivation and analysis. Fully mimetic discretizations satisfy discrete analogs of the continuum theory results used to derive energy inequalities. Consequently, continuum arguments carry over and can be used to show that discrete problems are well-posed and discrete solutions converge. A fully mimetic discrete vector calculus on three dimensional tensor product grids is derived and its key properties proven. Opinions regarding the future of the field are stated.