Self-Organized Task Allocation for Service Tasks in Computing Systems with Reconfigurable Components

A self-organized scheme for the allocation service tasks in adaptive or organic computing systems is proposed. Such computing systems are highly self-organized and the components ideally adapt to the needs of users or the environment. Typically, the components of such systems need some service from time to time in order perform their work efficiently. Since the type of service tasks will often change in this systems it is attractive to use reconfigurable hardware to perform the service tasks. The studied system consists of normal worker components and the helper components which have reconfigurable hardware and can perform different service tasks. The speed with which a service task is executed by a helper depends on its actual configuration. Different strategies for the helpers to decide about service task acceptance and reconfiguration are proposed. These task acceptance strategies are inspired by stimulus-threshold models that are used to explain task allocation in social insects. Analytical results for a system with two reconfigurable helpers are presented together with simulation results for larger systems. This work was supported by the German Research Foundation (DFG) through the project Organisation and Control of Self-Organising Systems in Technical Compounds within SPP 1183.     

A generalization of carries processes and Eulerian numbers

We study a generalization of Holteʼs amazing matrix, the transition probability matrix of the Markov chains of the ‘carries’ in a non-standard numeration system. The stationary distributions are explicitly described by the numbers which can be regarded as a generalization of the Eulerian numbers and the MacMahon numbers. We also show that similar properties hold even for the numeration systems with the negative bases.     

Expansion of vectors by powers of a matrix

In this paper, we investigate the problem of expansion of any d-dimensional vector in powers of a dilation matrix M, where a dilation matrix is an integer matrix of size d × d with all modules of its eigenvalues more than one. We consider this expansion as a multidimensional system of numeration, where we take the matrix as the base of the system of numeration and a special set of vectors as the set of digits. We give a constructive method of expansion of integer vectors in powers of a dilation matrix and prove the existence of expansion for any real vector. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 199–218.     

Numeration and enumeration

In this paper, numeration systems defined by recurrent sequences are considered. We present a class of recurrences yielding numeration systems for which the words corresponding to greedy expressions for natural numbers are easily described. Those sequences, in turn, enumerate classes of words with forbidden substrings.     

Random variables determined by the distributions of their digits in a numeration system with complex base

We study the distributions of complex-valued random variables determined by the distributions of their digits in a numeration system with complex base. We establish sufficient conditions for the singularity of such random variables, in particular, in the cases where their spectrum has Lebesgue measure zero (C-type singular distribution) or is a rectangle (S-type singular distribution). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1715–1720, December, 1998.     

Student Outcomes in a Local Systemic Change Project

The Local Systemic Change initiative of the National Science Foundation supports projects focusing primarily on teacher enhancement through extensive professional development and the use of standards‐based curriculum materials. The underlying rationale is that the effective use of such materials will ultimately result in enhanced student learning. However, the research base regarding the impact of these efforts on student learning is rather lean. This paper describes the results of a curriculum‐aligned assessment comprising selected items from the Third International Mathematics and Science Study. The assessment was administered to fifth graders involved in a Local Systemic Change project to address the following questions: How does the performance of students involved in the project compare nationally and internationally? and Does length of involvement in the project make a difference in student performance? Additional evidence to relate student outcomes to the project's systemic change efforts are provided.     

Smooth finite-dimensional approximations of distributed optimization problems via control discretization

Approximating finite-dimensional mathematical programming problems are studied that arise from piecewise constant discretization of controls in the optimization of distributed systems of a fairly broad class. The smoothness of the approximating problems is established. Gradient formulas are derived that make use of the analytical solution of the original control system and its adjoint, thus providing an opportunity for algorithmic separation of numerical optimization and the task of solving a controlled initial-boundary value problem. The approximating problems are proved to converge to the original optimization problem with respect to the functional as the discretization is refined. The application of the approach to optimization problems is illustrated by solving the semilinear wave equation controlled by applying an integral criterion. The results of numerical experiments are analyzed.     

Reviewers for Volume 12

When taken literally, mathematical texts are concerned with objects of a specific kind and learning mathematics, among other things, requires students to make sense of those mathematical objects. Understanding mathematical objects is commonly described as a cognitive construction. It is proposed and substantiated that for those objects to emerge for the individual, the construction processes have to be supplemented by a deliberate decision to view, treat, use, and investigate a structure or a collection of items as a unified object. This decision strongly depends on the mediation by symbols, diagrams, and notational systems.     

Formation of Mathematical Objects as Decision Making

When taken literally, mathematical texts are concerned with objects of a specific kind and learning mathematics, among other things, requires students to make sense of those mathematical objects. Understanding mathematical objects is commonly described as a cognitive construction. It is proposed and substantiated that for those objects to emerge for the individual, the construction processes have to be supplemented by a deliberate decision to view, treat, use, and investigate a structure or a collection of items as a unified object. This decision strongly depends on the mediation by symbols, diagrams, and notational systems.     

Combinatorial lemma and structure of a class of binary codes

A class of complete binary code systems is presented in which one of the symbols of the alphabet is used in any code combination not more than once, in terms of prefix codes of this class and of the operations of inversion and substitution of codes.Translated from Matematicheskie Zametki, Vol. 7, No. 3, pp. 325–331, March, 1970.In conclusion the author expresses his gratitude to G. Pollak who proposed to characterize the factorizations N by a system of calculus with a variable base, and to L. Redei who discussed the problems related to the factorization of finite sets.     

Robust design of rule-based fuzzy controllers

This paper deals with the design of stable and robust rule-based fuzzy control systems. New expressions to compute indices which provide a measure of the stability and robustness of fuzzy control systems are presented. The relation between the modification of the rules and the stability is studied through the so-called sensitivity indices. The paper presents procedures that make use of these indices to improve the design of fuzzy control systems, including the modification of the rules to obtain the global stability of an unstable system with multiple attractors, and to improve the dynamic behavior or the robustness of a non-linear plant. An example with a fuzzy controller for a system with non-linear damping and saturation in the actuation is presented to illustrate the design procedure.     

Students’ understanding of algebraic notation: A semiotic systems perspective

The ideas of equivalence and variable are two of the most fundamental concepts in algebra. Most studies of students’ understanding of these concepts have posited a gap between the students’ conceptions and the institutional meanings for the symbols. In contrast, this study develops a theoretical framework for describing the ways undergraduate students use personal meanings for symbols as they appropriate institutional meanings. To do this, we introduce the idea of semiotic systems as a framework for understanding the ways students use collections of signs to engage in mathematical activity and how the students use these signs in meaningful ways. The analysis of students’ work during task-based interviews suggests that this framework allows us to identify the ways in which seemingly idiosyncratic uses of the symbols are evidence of meaning-making and, in many cases, how the symbol use enables the student to engage productively in the mathematical activity.     

A diagrammatic view of the equals sign: arithmetical equivalence as a means,not an end

It is recommended in the mathematics education literature that pupils be presented with equality statements that can be assessed for numerical balance by attending to notational structure rather than computation. I describe an alternative, diagrammatic approach in which pupils do not assess statements but instead use them to make substitutions of notation. I report on two trials of a computer-based task conducted with pairs of pupils and highlight two findings. First, the pupils found it useful to articulate the distinct substitutive effects of commutative (‘swap’, ‘switch’) and partitional (‘split’, ‘separate’) statements when working on the task. Secondly, the pupils did not notice that some of the statements presented were in fact false, which suggests their substituting activities were independent of numerical equivalence conceptions. This demonstrates that making substitutions offers task designers a mathematical utility for equality statements that is distinct from, but complementary to, assessing numerical balance.     

An Ito–Taylor weak 3.0 method for stochastic dynamics of nonlinear systems

A new framework for development of order 3.0 weak Taylor scheme towards stochastic modeling and dynamics of coupled nonlinear systems is presented. The proposed method is derived by including third order multiple stochastic integral terms of Ito–Taylor expansion and developing them for a wide class of stochastic nonlinear systems. For computing the system responses of linear and a wide class of nonlinear structural systems, the use of lower order integration schemes is sufficient. But for highly non-linear stochastically driven systems like base isolated hysteretic systems and degrading stochastic systems the evaluation of higher order terms is necessary. Additionally, the use of higher order integration schemes for stochastic dynamics of higher dimensional nonlinear systems remains a challenge due to the arising mathematical complexities with the increase in the number of DOFs (degrees-of-freedom) which really necessitates the development of the proposed algorithm. The proposed algorithm is verified using a representative class of coupled nonlinear system in presence and absence of nonlinear degradation and hysteretic oscillators. The efficiency of the proposed numerical scheme over classical integration schemes is demonstrated through a practical engineering problem. Finally, an automated extension of the proposed algorithm is presented by generalizing it for a system of N-DOFs.     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

Lag and anticipatory synchronization based parameter estimation scheme in modulated time-delayed systems

In this article, we have shown that using delay dynamical systems as base, one can use the modulation of the various parameters, to transmit multiple signals through a single chaotic time series. It is shown that under certain conditions the original signals and parameters of the driving signals can be recovered with the help of an adaptive demodulator. An important aspect of the present method is that the communication is possible between two time-delayed systems with parameter mismatch using lag and anticipatory synchronization. The whole idea is presented on the basis of Mackey–Glass system, with variable time delay.     

The sum-of-digits function of canonical number systems: Distribution in residue classes

In the present paper we investigate the sum-of-digits function for canonical number systems. We are interested in its re-partition in arithmetic progressions and similar results for sum sets. The original problem goes back to Gelfond, who proved the independence of the distribution of the digits and their sum of digits. The present paper extends results by Thuswaldner and Mauduit and Sárközy to this kind of numeration systems.     

A simulation tool for voltage control studies in power systems

An electrical power system is a large scale system composed of complicated and sophisticated combination of multiple electronic and electromechanical components. In general, these components are nonlinear. Power system is also characterized by a wide range of normal operating conditions which continuously vary. To help the designer study the voltage control problems in power systems, a simulation tool is presented in this paper. It is based on decomposition of the overall system simulation task into three subtasks so as to attain both computational efficiency and flexibility. The use of the proposed simulation tool in a voltage control problem is also presented.     

Students' understanding of the notion of function in dynamic geometry environments

To make optimal use of computational environments, one must understand how students interact with the environments and how students' mathematical thinking is reflected and affected by their use of the environments. Similarly, to make sense of research on students' thinking and learning, one must understand how the environments and contexts used in the research may affect the conclusions one derives.The research on students' learning of functions has approached the topic in terms of symbols and graphs (see, for example, Leinhardt et al. (1990) for a review of work up to that point; Harel and Dubinsky (1992) for a collection of research; and Dugdale et. al. (1995), for some recent thinking about implications for curriculum reform using technology). Dynamic geometry environments (DGEs) like Cabri Geometry or Geometer's Sketchpad, offer us an opportunity to get a new perspective on these old and important issues. DGEs let students build geometrical constructions and then drag certain objects around the screen in a continuous manner while observing how the entire construction responds dynamically. In this way DGEs model functional relationships that are not specified by symbols or represented by graphs.Based on interviews with undergraduate mathematics majors, this paper presents preliminary observations that confirm some old results and raise some new questions about students' notions of function.     

Model Predictive Control for Heating Ventilating and Air Conditioning Systems

In heating ventilating and air conditioning (hvac) systems, model predictive control (mpc) is rarely used up to now. However, the following properties of hvac systems make mpc a well suited control methodology: the plant is a multiple input, multiple output system and its inputs are constrained – both in their value and in their rate of change. Moreover, several disturbances acting on the plant like varying outdoor air temperature or outdoor humidity can be measured. Furthermore, time constants are relatively large which makes it easy to perform the required optimization of the mpc strategy in time. This contribution presents the application of mpc to a real world hvac system. The considered hvac system consists of standard industrial components. The core components of the system are water-to-air heat exchangers, both for heating and for cooling purposes as well as a steam humidifier. Hence, air temperature and air humidity can be varied with the help of the investigated hvac system. The derivation of the plant model based on thermodynamic relations is presented as well as the application of the mpc strategy to the real world system. The plant contains some nonlinear elements which have to be dealt with when applying the mpc strategy. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)