Scott D. Thomson is deputy executive director, NASSP, Reston, Va. Parents often complain about counseling services in their schools. Teachers frequently criticize &dquo;that counseling department&dquo; as well. Even students-the recipients of counseling-don’t appear to appreciate the service. Typically students rely more upon parents or peers for advice on education or personal matters than upon counselors. What has gone wrong? Counselors are hard-working, dedicated persons with good intentions. Trained and subsidized during the heyday of NDEA, they now form in secondary schools a corps of experienced professionals. Should they not be the most effective and appreciated of school personnel? The villian in this scenario is ambiguity, a lack of preciseness in determining the role and function of counselors. Each group has its own expectations. Whether student or parent or principal or counselor-each has a private definition of what it is counselors are supposed to do. For instance, parents want advice for students on course selection and assistance with college entrance; students want class schedule changes and discussion on advanced education; teachers want behavior

An Introduction: New Opportunities for Counseling Services

Padé-type approximation and general orthogonal polynomials, by Claude Brezinski. Pp 250. 62 Swiss francs. 1980. ISBN 3-7643-1100-2 (Birkhäuser)

This book explains recent results in the theory of moving frames that concern the symbolic manipulation of invariants of Lie group actions. In particular, theorems concerning the calculation of generators of algebras of differential invariants, and the relations they satisfy, are discussed in detail. The author demonstrates how new ideas lead to significant progress in two main applications: the solution of invariant ordinary differential equations and the structure of Euler-Lagrange equations and conservation laws of variational problems. The expository language used here is primarily that of undergraduate calculus rather than differential geometry, making the topic more accessible to a student audience. More sophisticated ideas from differential topology and Lie theory are explained from scratch using illustrative examples and exercises. This book is ideal for graduate students and researchers working in differential equations, symbolic computation, applications of Lie groups and, to a lesser extent, differential geometry.

/pdf/a-practical-guide-to-the-invariant-calculus-169ee5onxg.pdf

A Practical Guide to the Invariant Calculus

Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.

Difference Equations by Differential Equation Methods

In this paper we discuss the structure of recursion operators. We show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries. This enables us to compute recursion operators more systematically. Under certain conditions (which hold for all examples known to us) Nijenhuis operators are well defined, i.e., they give rise to hierarchies of infinitely many commuting symmetries of the operator. Moreover, the nonlocal part of a Nijenhuis operator contains the candidates of roots and coroots.

/pdf/integrable-systems-and-their-recursion-operators-16ty7kkvb9.pdf

Integrable systems and their recursion operators

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed. The method assumes that the PΔEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PΔEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PΔEs classified by Adler, Bobenko, and Suris and systems of PΔEs including the integrable two-component potential Korteweg---de Vries lattice system, as well as nonlinear Schrodinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for PΔEs recently derived by Hietarinta (J. Phys. A, Math. Theor. 44:165204, 2011). The method is algorithmic and is being implemented in Mathematica.

/pdf/symbolic-computation-of-lax-pairs-of-partial-difference-1y4jvafwme.pdf

Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube

We show, in full generality, that the staircase method (Papageorgiou et al 1990 Phys. Lett. A 147 106?14, Quispel et al 1991 Physica A 173 243?66) provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg?De Vries equation, the five-point Bruschi?Calogero?Droghei equation, the quotient-difference (QD)-algorithm and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with , then one can introduce q ? 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.

/pdf/the-staircase-method-integrals-for-periodic-reductions-of-293uv2840j.pdf

The staircase method: integrals for periodic reductions of integrable lattice equations

We describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils. In finitely many directions, we arrive at discrete Goursat problems and in the remaining directions we find Cauchy problems. Next, we consider (s1, s2)-periodic initial value problems. In the Goursat directions, the periodic solutions are generated by correspondences. In the Cauchy directions, assuming the equation to be multi-linear, the periodic solution can be obtained uniquely by iteration of a particularly simple mapping, whose dimension is a piecewise linear function of s1, s2.

Initial value problems for lattice equations

We show, in full generality, that the staircase method provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg-De Vries equation, the five-point Bruschi-Calogero-Droghei equation, the QD-algorithm, and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r<n, then one can introduce q<= 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular 2D lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P\Delta Es) is reviewed. The method assumes that the P\Delta Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P\Delta Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable P\Delta Es classified by Adler, Bobenko, and Suris and systems of P\Delta Es including the integrable 2-component potential Korteweg-de Vries lattice system, as well as nonlinear Schroedinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for P\Delta Es recently derived by Hietarinta (J. Phys. A: Math. Theor., 44, 2011, Art. No. 165204). The method is algorithmic and is being implemented in Mathematica.

Peter H. van der Kamp

Papers

Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube

The staircase method: integrals for periodic reductions of integrable lattice equations

Initial value problems for lattice equations

The staircase method: integrals for periodic reductions of integrable lattice equations

Symbolic Computation of Lax Pairs of Partial Difference Equations Using Consistency Around the Cube