Appropriate problems for learning and for performing—an issue for teacher training

Selecting, modifying or creating appropriate problems for mathematics class has become an activity of increaing importance in the professional development of German mathematics teachers. But rather than asking in general: “What is a good problem?” there should be a stronger emphasis on considering the specific goal of a problem, e.g.: “What are the ingredients that make a problem appropriate for initiating a learning process” or “What are the characteristics that make a problem appropriate for its use in a central test?” We propose a guiding scheme for teachers that turns out to be especially helpful, since the newly introduced orientation on outcome standards a) leads to a critical predominance of test items and b) expects teachers to design adequate problems for specific learning processes (e.g. problem solving, reasoning and modelling activities).     

Graphical Diagnostics for Markov Models for Categorical Data

Markov models are widely used as a method for describing categorical data that exhibit stationary and nonstationary autocorrelation. However, diagnostic methods are a largely overlooked topic for Markov models. We introduce two types of residuals for this purpose: one for assessing the length of runs between state changes, and the other for assessing the frequency with which the process moves from any given state to the other states. Methods for calculating the sampling distribution of both types of residuals are presented, enabling objective interpretation through graphical summaries. The graphical summaries are formed using a modification of the probability integral transformation that is applicable for discrete data. Residuals from simulated datasets are presented to demonstrate when the model is, and is not, adequate for the data. The two types of residuals are used to highlight inadequacies of a model posed for real data on seabed fauna from the marine environment.

Supplemental materials, including an R-package RMC with functions to perform the diagnostic measures on the class of models considered in this article, are at the journal’s website. The R-package is also available at CRAN.     

Existence results for Hughes' model for pedestrian flows

In this paper we prove two global existence results for Hughes' model for pedestrian flows under assumptions that ensure that the traces of the solutions along the turning curve are zero for all positive times.     

Algorithm for compiling optimal schedule for parallel-sequential networks

An algorithm is described for compiling an optimal schedule for a computational process simulated by a parallel-sequential network, and also for a homogeneous computational system.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 48, pp. 200–204, 1974.     

Identities for Anderson generating functions for Drinfeld modules

Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on special values of positive characteristic $L$ -series and in transcendence and algebraic independence problems. In the present paper we investigate techniques for expressing Anderson generating functions in terms of the defining polynomial of the Drinfeld module and determine new formulas for periods and quasi-periods.     

Call for Papers: Bayesian Inference for Stochastic Processes

Call for Papers

Call for Papers: Bayesian Inference for Stochastic Processes     

Equations for coefficients of tactical decomposition matrices for t-designs

Equations for coefficients of tactical decomposition matrices for 2-designs are well-known and they have been used for constructions of many examples of 2-designs. In this paper, we generalize these equations and propose an explicit equation system for coefficients of tactical decomposition matrices for \(t\text{-}(v,k,\lambda _t)\) designs, for any integer value of \(t.\)      

An inequality for Steklov eigenvalues for planar domains

Study of the zeta function associated to the Neumann operator on planar domains yields an inequality for Steklov eigenvalues for planar domains.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Polytope Lyapunov Functions for Stable and for Stabilizable LSS

We present a new approach for constructing polytope Lyapunov functions for continuous-time linear switching systems (LSS). This allows us to decide the stability of LSS and to compute the Lyapunov exponent with a good precision in relatively high dimensions. The same technique is also extended for stabilizability of positive systems by evaluating a polytope concave Lyapunov function (“antinorm”) in the cone. The method is based on a suitable discretization of the underlying continuous system and provides both a lower and an upper bound for the Lyapunov exponent. The absolute error in the Lyapunov exponent computation is estimated from above and proved to be linear in the dwell time. The practical efficiency of the new method is demonstrated in several examples and in the list of numerical experiments with randomly generated matrices of dimensions up to 10 (for general linear systems) and up to 100 (for positive systems). The development of the method is based on several theoretical results proved in the paper: the existence of monotone invariant norms and antinorms for positively irreducible systems, the equivalence of all contractive norms for stable systems and the linear convergence theorem.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

Algorithm for choosing conjunctions for logical recognition methods

A new family of recognition algorithms for problems with binary information is constructed. A natural method is proposed for ordering conjunctions used in pattern recognition problems in the standard formulation and with binary features.     

A sufficient condition for GPN-stability for delay differential equations

Summary In this paper the author considers a linear test delay differential equation with non-constant coefficients related to the definitions of PN and GPN-stability for numerical methods. He defines stability properties for an ordinary differential equation together with stability properties of interpolants for numerical methods and in this way he gives sufficient conditions for GPN-stability.This work was supported by the Italian M.P.I. (funds 40%) and by C.N.R.     

New methods for determining quasi-stationary distributions for markov chains

We shall be concerned with the problem of determining quasi-stationary distributions for Markovian models directly from their transition rates Q. We shall present simple conditions for a μ-invariant measure m for Q to be μ-invariant for the transition function, so that if m is finite, it can be normalized to produce a quasi-stationary distribution.     

Conditions for Optimal Scheduling for the Three-machine Problem

This note proposes a simpler formulation for the conditionssuggested by Eltman & Townsend for the three-machine problem,which if satisfied must result in one job preceding anotherin an optimal sequence. These are compared with Johnson's conditions.     

Polynomial cost for solving IVP for high-index DAE

We show that the cost of solving initial value problems for high-index differential algebraic equations is polynomial in the number of digits of accuracy requested. The algorithm analyzed is built on a Taylor series method developed by Pryce for solving a general class of differential algebraic equations. The problem may be fully implicit, of arbitrarily high fixed index and contain derivatives of any order. We give estimates of the residual which are needed to design practical error control algorithms for differential algebraic equations. We show that adaptive meshes are always more efficient than non-adaptive meshes. Finally, we construct sufficiently smooth interpolants of the discrete solution. AMS subject classification (2000) 34A09, 65L80, 68Q25     

Some Complexity Results for Zero Finding for Univariate Functions

This paper gives a survey of complexity results for zero finding. We focus on zero finding for univariate, at least continuous functions with a change of sign on some interval. One section, however, deals with zero finding for complex polynomials. We present asymptotic, worst case, and average case results.     

A framework for decision support systems for scheduling problems

Considering a decision support system as a tool where executive's judgment can be included along with the mathematical tool kit of the management scientist, this paper shows the need to include problem management as an integral component of the decision support system for scheduling problems. A methodology based on the resolution of conflicts among various constraints in scheduling problems is proposed to implement the problem management system in a decision support system for these problems. The paper concludes with some guidelines to create a workable framework for providing effective decision support to solve scheduling problems and the identification of some fruitful directions for future research.     

Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems

A new approach for constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners for this system are then constructed based on a triangulation of the domain into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory.     

Challenges for mathematical and statistical modelling for metrology

There are challenges in disseminating information concerning mathematical and statistical modelling for metrology. We specify these challenges and consider ways they are being met. In addition to journal papers, the value of supporting software and other aspects are indicated. Relevant work in the UK Software Support for Metrology programme is outlined. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)