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This is a conceptual paper based on existing literature aiming to provide practical information on designing and implementing activities to promote children's computational thinking. Computational thinking is a relatively new term in early childhood education that refers to a specific problem-solving thinking process involving various logical and analytical thinking skills. Four foundational skills have been identified as core thinking skills of computational thinking: decomposition, abstraction, pattern recognition, and algorithm. We explain these four skills in this paper and their practical applications to teaching and learning in early childhood education. Early computational thinking skills are found in common early childhood activities. This paper identifies activities teachers can use in the classroom to explicitly promote children's computational thinking and provides a new perspective on how to adapt classroom activities to integrate computational thinking. In particular, we emphasize the need to vary the demands of the content in the activity and incorporate computational thinking based on children's needs and development to ensure that children progress through the thinking process. [ABSTRACT FROM AUTHOR]

Abstract: The long‐standing debate into the potential benefit of developing mathematical thinking skills through learning to program has been reignited with the widespread introduction of programming in schools across many countries, including England where it is a statutory requirement for all pupils to be taught programming from 5 years old. Algorithm is introduced early in the English computing curriculum, yet there is limited knowledge of how young pupils view this concept. This paper explores pupils' (aged 10–11) understandings of algorithm following their engagement with 1 year of ScratchMaths, a curriculum designed to develop computational and mathematical thinking skills through learning to program. A total of 181 pupils from 6 schools undertook a set of written tasks to assess their interpretations and evaluations of different algorithms that solve the same problem, with a subset of these pupils subsequently interviewed to probe their understandings in greater depth. We discuss the different approaches identified, the evaluation criteria they used, and the aspects of the concept that pupils found intuitive or challenging, such as simplification and abstraction. The paper ends with some reflections on the implications of the research, concluding with a set of recommendations for pedagogy in developing primary pupils' algorithmic thinking. [ABSTRACT FROM AUTHOR]

Integrating mathematics in architectural design and construction helped in generating various design solutions. Thus, aesthetical performance had developed with the involving of advanced mathematics fields and computational software. The main intention of this paper is to study the importance of mathematics in aesthetical architecture performance. Consequently, the research methodology carries out an analysis of different case studies that offered a better perception about the degree of mathematics effectiveness on the aesthetical performance in architecture. The research will investigate to what extent mathematics forms in architecture had actually a great effect on aesthetical architecture performance. [ABSTRACT FROM AUTHOR]

A (combinatorial) channel consists of pairs of words representing all possible input-output channel situations. In a past paper, we formalized the intuitive concept of 'largest amount of errors' detectable by a given language L, by defining the maximal error-detecting capabilities of L with respect to a given class of channels, and we showed how to compute all maximal error-detecting capabilities (channels) of a given regular language with respect to the class of rational channels and a class of channels involving only the substitution-error type. In this paper we resolve the problem for channels involving any combination of the basic error types: substitution, insertion, deletion. Moreover, we consider the problem of finding the inverses of these channels, in view of the fact that L is error-detecting for γ if and only if it is error-detecting for the inverse of γ. We also discuss a natural method of reducing the problem of computing (inner) distances of a given regular language L to the problem of computing maximal error-detecting capabilities of L. [ABSTRACT FROM AUTHOR]

Logic regression, as developed by Ruczinski, Kooperberg, and LeBlanc [1], is a multivariable regression methodology that constructs logical relationships among Boolean predictor variables that best predicts a Boolean dependent variable. More specifically, it finds a regression model of the form g[E∣Y]=b0+b1 L1+b2L2+⋯bmLm where both the coefficients b0, b1,…,bm and the logical expressions Lj, j=1,…, m are determined, whereby a logical expression one means logical relationships among the predictor variables, such as "X1, X2 are true but not X5 " , or "X3, X5, X7 are true but not X1 or X2 " In paper [1] the authors investigate the use a simulated annealing algorithm. In this paper. the Group Method of Data Handling (GMDH) is used. [ABSTRACT FROM AUTHOR]

Hardware/software (HW/SW) partitioning is one of the key challenges in HW/SW codesign. This paper presents efficient algorithms for the HW/SW partitioning problem, which has been proved to be NP-hard. We reduce the HW/SW partitioning problem to a variation of knapsack problem that is approximately solved by searching 1D solution space, instead of searching 2D solution space in the latest work cited in this paper, to reduce time complexity. Three heuristic algorithms are proposed to determine suitable partitions to satisfy HW/SW partitioning constraints. We have shown that the time complexity for partitioning a graph with n nodes and m edges is significantly reduced from O(dx ⋅ dy ⋅ n³) to O(n log n + d ⋅ (n + m)), where d and dx ⋅ dy are the number of the fragments of the searched 1D solution space and the searched 2D solution space, respectively. The lower bound on the solution quality is also proposed based on the new computing model to show that it is comparable to that reported in the literature. Moreover, empirical results show that the proposed algorithms produce comparable and often better solutions when compared to the latest algorithm while reducing the time complexity significantly. [ABSTRACT FROM AUTHOR]

The aim of our paper is to explain a mathematical model as a special case of symbolic knowledge map. Each knowledge mapping is a visualization of knowledge for the purpose of eliciting, sharing and expanding. Tools of such visualization can be of various types. But in reality many types of so-called knowledge maps are only data flow or information flow diagrams. Our paper will define the most important features which every knowledge map must satisfy, for instance it must include chronological, hierarchical, associative, causal and evaluative relationships, it must improve the quality of knowledge etc. In our paper we will prove that a mathematical model satisfies all requirements to be called a knowledge map. Neither definition nor categorization and taxonomy of knowledge mapping are unified in the literature so the authors try to start with working on this field. Knowledge map is a visual interception of knowledge with the aim of its storage, sharing and development. Weak descriptive knowledge maps may be used for explaining the ideas and concepts connected with OR models, as well as for explaining the new knowledge gained with the models, in a well-structured form. Strong descriptive knowledge maps can serve to describe real relations between the objects of the models or real elements in relation to their positioning. In this case the object placing does not describe only its physical position but also, for instance, its economical indexes. Like the normative OR models, the normative knowledge maps show the normative solution, or help to find the best, desirable or advisable solution. After suggestions of how to categorize knowledge maps (above) mathematical models of various types with all features and properties are presented as a knowledge map. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider completely generalized nonlinear quasi-variational-like inclusions in Banach spaces and propose an Ishikawa type iterative algorithm for computing their approximate solutions by applying the new notion of -proximal mapping given in [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308]. We prove that the approximate solutions obtained by the proposed algorithm converge to the exact solution of our quasi-variational-like inclusions. The results presented in this paper extend and improve the corresponding results of [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308; X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; N.J. Huang, Generalized nonlinear variational inclusions with non-compact valued mappings, Appl. Math. Lett. 9(3) (1996) 25–29; A. Hassouni, A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185(3) (1994) 706–712]. Some special cases are also discussed. [Copyright &y& Elsevier]

The aim of this paper is to exploit the algorithms of paper Experimental Math. 8 (1999) in order to produce a new algebraic method for computing efficiently absolute Lagrange resolvent, a fundamental tool in constructive algebraic Galois theory. This article is composed of two parts.The main idea of the first part is to break up the calculation of absolute resolvent into smaller computations. Since a multi-resolvent is a factor of a resolvent, the whole resolvent may be computed by means of several multi-resolvents.The idea of the second part is that an irreducible polynomial over Z might be reducible over Z /pZ for certain integer p. So the first part can be applied and then, the Chinese remainder theorem allows to lift up the resolvent over Z. [Copyright &y& Elsevier]

Isometric subgraphs of hypercubes, or partial cubes as they are also called, are a rich class of graphs that include median graphs, subdivision graphs of complete graphs, and classes of graphs arising in mathematical chemistry and biology. In general, one can recognize whether a graph on n vertices and m edges is a partial cube in O(mn) steps, faster recognition algorithms are only known for median graphs. This paper exhibits classes of partial cubes that are not median graphs but can be recognized in O(m log n) steps. On the way relevant decomposition theorems for partial cubes are derived, one of them correcting an error in a previous paper (Eur. J. Combin. 19 (1998) 677). [Copyright &y& Elsevier]

In this paper, we introduce and study fuzzy variational-like inclusion, fuzzy resolvent equation and H(.,.)-ø-η-accretive operator in real uniformly smooth Banach spaces. It is established that fuzzy variational-like inclusion is equivalent to a fixed point problem as well as to a fuzzy resolvent equation. This equivalence is used to define an iterative algorithm for solving fuzzy resolvent equation. Some examples are given. [ABSTRACT FROM AUTHOR]

This paper presents a new method and algorithms for dividing modular numbers on the basis of the use of dividend and divisor values relative with respect to the full range of a residue number system. The proposed modular division algorithm uses elementary operations of register shift and addition, which makes the algorithm very simple and fastest to date. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes an algorithm which produces all ideals compatible with a given surjective Frobenius near-splitting. [Copyright &y& Elsevier]

Let G be a connected graph. A spanning tree T of G is a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. If their distances in T and G differ by at most t, then T is an additive tree t-spanner of G. In this paper, we show that any permutation graph has an additive tree 2-spanner, and it can be found in O(n) time sequentially or in O(log n) time with O(n/log n) processors on the EREW PRAM computational model by using a previously published algorithm for finding a tree 3-spanner of a permutation graph. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new algorithm that computes the local algebras of the roots of a zero-dimensional polynomial equation system, with a number of operations in the coefficient field that is polynomial in the number of variables, in the evaluation cost of the equations and in a Bézout number. [Copyright &y& Elsevier]

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zeroth Betti number, and the Euler–Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zeroth one. As a consequence we also obtain algorithms for computing semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in singly exponential time, which improves on the complexity of the previously published algorithms for this problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, by using the theories and methods of mathematical analysis and computer algebra, a reliable algorithm of finite difference method for solving singular perturbation problems was established, a new matlab procedure ehssp was implemented, too. Several linear and non-linear problems are presented to illustrate the implementation of the program. [Copyright &y& Elsevier]

Abstract: A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of in T induces a connected subgraph of G for every . The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively. [Copyright &y& Elsevier]

Abstract: In this paper, by using theories of stochastic process and computer algebra, a reliable algorithm for computing the high-order transition probability matrix of finite Markov chain is established, a new mechanical procedure markovproc is established, too. The procedure markovproc give not only the exact expression of the Markov chains, but also the limiting probability. Some examples are presented to illustrate the implementation of the algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, by using the theories and methods of mathematics analysis and computer algebra, a reliable algorithm for solving high-order nonlinear Volterra–Fredholm integro-differential equations was established, and a new Maple procedure voltfredproc was established too. The results of the examples indicated that the procedure voltfredproc of Taylor polynomial method is simple and effective, and could provide an accuracy approximate solution or exact solution of the high-order nonlinear Volterra–Fredholm integro-differential equation. This would be useful for solving integro-differential equation, integral equations and ordinary differential equation. [Copyright &y& Elsevier]

Abstract: Resultant is a classical algebraic tool for determining whether or not a polynomial systems have a common root without explicitly solving for the roots. But, the complexity of resultants can be very high in practice. In this paper, by using the theories and methods of computer algebra, two reliable algorithms for computing multi-resultant problems-Dixon resultant and Macaulay resultant were established, two new Maple procedures dixonres and macaulayres were established, too. Some examples are presented to illustrate the implementations of the programs. This would be useful for solving a system of polynomial equations. [Copyright &y& Elsevier]

In this paper it is shown that the Doubling Algorithm for solving the discrete time Riccati Equation also provides the steady state solution of a related dual Riccati Equation. [ABSTRACT FROM AUTHOR]

Abstract: One of the main problems dealing with iterative methods for solving polynomial systemsis the initialization of the iteration. This paper provides an algorithm to initialize the search of solutions of polynomial systems. [Copyright &y& Elsevier]

Abstract: In numerous applications involving finite fields, we often need high-order elements. Ideally we should be able to obtain a primitive element for any finite field in reasonable time. However, if the prime factorization of the group order is unknown, we do not know how to achieve the goal. We thus turn our attentions to a less ambitious problem: constructing an element of provably high order. In this paper, we survey various algorithms that find an element of high order for general or special finite fields. [Copyright &y& Elsevier]

Abstract: In this paper, the WZ algorithm for the eigenvalue problem of a complex matrix is given. By comparing the WZ method with the LR method, the necessary and sufficient condition of the convergence of the algorithm is proposed. The relation between LR factorization with WZ factorization is proved. The illustrative example demonstrated the high precision of the proposed method. [Copyright &y& Elsevier]

Abstract: One approach for computations with special functions in computer algebra is the systematic use of analytic functions whenever possible. This naturally leads to problems of how to answer questions about analytic functions in a fully effective way. Such questions comprise the determination of the radius of convergence or the evaluation of the analytic continuation of the function at the endpoint of a broken line path. In this paper, we propose a first definition for the notion of an effective analytic function and we show how to effectively solve several types of differential equations in this context. We will limit ourselves to functions in one variable. [Copyright &y& Elsevier]

In this paper, we propose a semi-numerical algorithm for computing absolute factorization of multivariate polynomials. It is based on some properties appearing after a generic change of coordinate. Using numerical computation, Galois group action and rational approximation, this method provides an efficient probabilistic algorithm for medium degrees. Two implementations are presented and compared to other algorithms. [Copyright &y& Elsevier]

In Ref. [J. Comput. Math. 20 (3) (2002) 301], a new superlinearly convergent algorithm of sequential systems of linear equations for nonlinear optimization problems with inequality constraints was proposed. Since the new algorithm only needs to solve four systems of linear equations having a same coefficient matrix per iteration, the computation amount of the algorithm is much less than that of the existing sequential quadratic programming algorithms per iteration. Under some mild assumptions, the new algorithm is globally convergent and its rate of convergence is one-step superlinearly. In this paper, it is shown that the new algorithm also can be used to deal with nonlinear optimization problems having nonlinearly equality and inequality constraints, by solving an auxiliary problem. Some numerical results are reported. [Copyright &y& Elsevier]

In his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis (“Omega Calculus”) as a computational method for solving problems in connection with linear diophantine inequalities and equations. The technique has recently been given a new life by G.E. Andrews and his coauthors, who had the idea of marrying it with the tools of to-day’s Computer Algebra. The theory consists of evaluating a certain type of rational function of the form A(λ)-1B(1/λ)-1 by the MacMahon Ω operator. So far, the case where the two polynomials A and B are factorized as products of polynomials with two terms has been studied in details. In this paper we study the case of arbitrary polynomials A and B. We obtain an algorithm for evaluating the Ω operator using the coefficients of those polynomials without knowing their roots. Since the program efficiency is a persisting problem in several-variable polynomial Calculus, we did our best to make the algorithm as fast as possible. As an application, we derive new combinatorial identities. [ABSTRACT FROM AUTHOR]

Checking regularity (or singularity) of interval matrices is a known NP-hard problem. In this paper a general algorithm for checking regularity/singularity is presented which is not a priori exponential. The algorithm is based on a theoretical result according to which regularity may be judged from any single component of the solution set of an associated system of linear interval equations. Numerical experiments (with interval matrices up to the size n = 50) confirm that this approach brings an essential decrease in the amount of operations needed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions for set-valued mappings. We construct some new iterative algorithms for this generalized nonlinear implicit quasivariational inclusions for set-valued mappings without compactness. We prove the existence of solutions for this generalized nonlinear implicit quasivariational inclusions for set-valued mappings without compactness and the convergence of iterative sequences generated by the algorithms. We also give an application to generalized nonlinear implicit variational inequalities. [ABSTRACT FROM AUTHOR]

Given, a graph G = (V, E), with two vertices s and t specified as the origin and destination, respectively; a positive integer K called the obstruction number; and (positive) weights l (·) on the edges; then, two players P1 and P2 play a game with the following rules. P1 starts from s, and tries to arrive at t by tracing an edge on each move. P2 cuts in his turn, P2 cuts some edges. The maximum total number of edges that P2 can cut is K during the game. It is assumed that both P1 and P2 can see the whole graph. P1 tries to minimize the sum of traveled edge weights (called the path length), and P2 tries to maximize the path length. This paper shows that the path length when the two players do their best can be determined in O(nk-l (m + n log n)) (m = |E| n = (|V|) time. It is also shown that the problem of deciding whether or not the path length when the two players do their best is not greater than a certain positive integer is P-SPACE-complete, even if the weight on each edge is 1. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(2): 13–23, 1998 [ABSTRACT FROM AUTHOR]

This paper discusses the problems of finding one of the largest similar substructures in tree Tb to tree Ta, where both Ta and Tb are rooted and ordered trees (RO-trees) or unrooted and cyclically ordered trees (CO-trees). A maximal closest common ancestor mapping and a largest similar substructure in Tb to Ta based on this mapping are defined and two algorithms are proposed for finding one of the largest similar substructures for RO-trees and that for CO-trees. The time and space complexities of the algorithm for RO-trees are OT(NaNb) and OS(NaNb), respectively; and those of the algorithm for CO-trees are OT(mambNaNb) and OS((ma + mb)NaNb), respectively, where ma(mb) and Na(Nb) are the largest degree of a vertex and the number of vertices of Ta(Tb), respectively. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(2): 92–104, 1997 [ABSTRACT FROM AUTHOR]

Combinatorics on words began more than a century ago with a demonstration that an infinitely long string with no repetitions could be constructed on an alphabet of only three letters. Computing all the repetitions (such as TTT…or …CGACGA…) in a given string x of length n is one of the oldest and most important problems of computational stringology, requiring time in the worst case. About a dozen years ago, it was discovered that repetitions can be computed as a by-product of the Θ(n)-time computation of all the maximal periodicities or runs in x. However, even though the computation is linear, it is also brute force: global data structures, such as the suffix array, the longest common prefix array and the Lempel-Ziv factorization, need to be computed in a preprocessing phase. Furthermore, all of this effort is required despite the fact that the expected number of runs in a string is generally a small fraction of the string length. In this paper, I explore the possibility that repetitions (perhaps also other regularities in strings) can be computed in a manner commensurate with the size of the output. [ABSTRACT FROM AUTHOR]