Sylvester’s double sums: The general case

In 1853 Sylvester introduced a family of double-sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to characterize all members of the family, providing a uniform method.     

An elementary proof of Sylvester’s double sums for subresultants

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester’s formula was also recently proved by Lascoux and Pragacz using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.     

Sylvester double sums and subresultants

Sylvester double sums, introduced first by Sylvester (see [Sylvester, 1840] and [Sylvester, 1853]), are symmetric expressions of the roots of two polynomials, while subresultants are defined through the coefficients of these polynomials (see Apery and Jouanolou (2006) and Basu et al. (2003) for references on subresultants). As pointed out by Sylvester, the two notions are very closely related: Sylvester double sums and subresultants are equal up to a multiplicative non-zero constant in the ground field. Two proofs are already known: that of Lascoux and Pragacz (2003), using Schur functions, and that of d’Andrea et al. (2007), using manipulations of matrices. The purpose of this paper is to give a new simple proof using similar inductive properties of double sums and subresultants.     

Morley’s theorem revisited: Origami construction and automated proof

Morley’s theorem states that for any triangle, the intersections of its adjacent angle trisectors form an equilateral triangle. The construction of Morley’s triangle by the straightedge and compass method is impossible because of the well-known impossibility result for angle trisection. However, by origami, the construction of an angle trisector is possible, and hence that of Morley’s triangle. In this paper we present a computational origami construction of Morley’s triangle and an automated correctness proof of the generalized Morley’s theorem.During the computational origami construction, geometrical constraints in symbolic representation are generated and accumulated. Those constraints are then transformed into algebraic forms, i.e. a set of polynomials, which in turn are used to prove the correctness of the construction. The automated proof is based on the Gröbner bases method. The timings of the experiments of the Gröbner bases computations for our proofs are given. They vary greatly depending on the origami construction methods, the algorithms for the Gröbner bases computation, and variable orderings.     

An extension of Kedlaya’s algorithm for hyperelliptic curves

In this paper we describe a generalisation and adaptation of Kedlaya’s algorithm for computing the zeta-function of a hyperelliptic curve over a finite field of odd characteristic that the author used for the implementation of the algorithm in the Magma library. We generalise the algorithm to the case of an even degree model. We also analyse the adaptation of working with the xⁱdx/y³ rather than the xⁱdx/y differential basis. This basis has the computational advantage of always leading to an integral transformation matrix whereas the latter fails to in small genus cases. There are some theoretical subtleties that arise in the even degree case where the two differential bases actually lead to different redundant eigenvalues that must be discarded.     

The effect of nonhuman’s external regulation on young children’s self-regulation to regulate their own process of learning

The present study sought to examine the effect of nonhuman’s external regulation on children’s self-regulation to regulate their own process of learning and to what extent did children succeed in terms of application when they talk and think while act alone with nonhuman’s external regulator. The Aginian’s methodology (, , ,  and ) that used an isolated, computer-based learning system and acts as a standalone learning environment with special set of tasks was used by hundred healthy preschool children. The results showed that young children were able to regulate their own process of learning and engage with their full free-will without the need of their real teacher’s regulation. The conclusion provided evidence that the verbalization of thinking aloud should occur spontaneously by nature, the nonhuman’s external regulation has a positive effect on young children’s development when they act with their full free-will, and has a positive effect on their behavior either.     

The effect of nonhuman’s external regulation on children’s responses to detect children with developmental problems (DP) associated with the natural development of self-regulation during learning tasks

The present study sought to examine the effect of the nonhuman’s external regulation on children’s responses during learning tasks to detect children with developmental problems (DP) associated with the natural development process of self-regulation. The material was an isolated, computer-based learning system that acts as a standalone learning environment and used by 100 preschool children, which were randomly selected from ten preschools without revising their medical files. Participants were classified by the system itself during learning progression in three essential groups based on Aginian’s zone of children regulation (ZCR), which is “the equilibrium point in the self-regulation’s development process that controls the child to be either a self-Vygotskyian’s learner, self-Piagetian’s learner, or self-Aginian’s learner during learning tasks” ( Agina, Kommers, & Steehouder, 2011d). The results showed that the preschool children can spontaneously do diagnostic tests during learning tasks and the nonhuman external regulator was able to analysis children’s responses that, in turn, used for detecting those children with DP. This result was practically confirmed by revising all children’s medical files that matched the final judgment of the nonhuman external regulator. However, the results confirmed that the natural development of self-regulation was fluctuated among three paradoxical views (Vygotskyian vs. Piagetian vs. Aginian).     

The use of interactive media among today’s youth: Results of a survey

The intensive use of interactive media has led to assertions about the effect of these media on youth. This paper presents a quantitative study on the position of interactive media in young people’s lives. Rather than following the assumption of a homogeneous generation, we investigate the existence of a diversity of user patterns. The research question for this paper: Can patterns be found in the use of interactive media among youth? We answer this question by a survey among Dutch youngsters aged 10–23. Four clusters of interactive media users, namely Traditionalists, Gamers, Networkers and Producers were identified using cluster analysis. Behind these straightforward clusters, a complex whole of user activities can be found. Each cluster shows specific use of and opinions about interactive media. This provides a contextualized understanding of the position of interactive media in the lives of contemporary youth, and a nuanced conceptualization of the ‘Net generation’. This allows for studying the intricate relationship between youth culture, interactive media and learning.     

An Algebraic Characterization of Inductive Soundness in Proof by Consistency

Kapur and Musser studied the theoretical basis for proof by consistency and obtained an inductive completeness result:p=q if and only if p=q is true in every inductive model.However,there is a loophole in their proof for the soundness part:p=q implies p=q is true in every inductive model.The aim of this paper is to give a correct characterization of inductive soundness from an algebraic view by introducing strong inductive models.     

The effect of nonhuman’s external regulation on detecting the natural development process of young children’s self-regulation during learning tasks

The present study explored the effect of nonhuman’s external regulation on children’s natural development of self-regulation and the effect of each natural developed class on children’s spontaneous thinking aloud and satisfaction. The Aginian’s methodology (Agina et al., 2011a) that relied on special computer agents for the external regulation, measuring self-regulation and children’s satisfaction, and producing the final results in points was used with 40 preschool children, which were divided into classes based on their natural development of self-regulation during learning tasks. The results showed that children who followed Piagetian’s view were outperforming children who followed Vygotskyian’s view and Aginian’s view, which is a new psychological view generated by computer indicates that the child either followed unknown class of self-regulation’s natural development or the child holds an ambiguous psychological problem. The results also showed that the relationship between children’s spontaneous thinking aloud and children’s self-regulation is a reverse. The supplemental analysis showed that computer, as a nonhuman external regulator, can identify those children who hold psychological problems and can integrate the net signed of self-regulation of each child at each task through embedding the mathematics integration where the computer becomes fully conscious with all the occurrences of children’s behavioral regulation.     

The Peer’s Dilemma: A general framework to examine cooperation in pure peer-to-peer systems

The exploration of social dilemmas is being considered a major foundation for encountering the enforced necessities of cooperation in self-organizing environments. Such environments are characterized by self-interested parties and the absence of trusted third parties. Recent approaches apply evolutionary socio-inspired games to formally prove the existence and further prolongation of cooperation patterns within communities. For instance, the Prisoner’s Dilemma game has thus provided a rich opportunity to examine self-interested behaviors in pure peer-to-peer networks. However, assuming a total absence of coalitions, incentives and punishment mechanisms, several works argue against a durable maintenance of cooperation neither at single-shot nor repeated-scenarios. In this article, we formally and experimentally demonstrate a counterexample for the latter by applying evolutionary game theory and a particular instance of the Rock–Scissors–Paper game. Our framework proves that the cyclic dominance of certain type of nodes within a P2P system has an impact and introduces a strategic aspect to the evolution of the overall community.     

Élie Cartan’s geometrical vision or how to avoid expression swell

The aim of the paper is to demonstrate the superiority of Cartan’s method over direct methods based on differential elimination for handling otherwise intractable equivalence problems. In this sense, using our implementation of Cartan’s method, we establish two new equivalence results. We establish when a system of second order ODEs is equivalent to flat system (second derivations are zero), and when a system of holomorphic PDEs with two independent variables and one dependent variable is flat. We consider the problem of finding transformation that brings a given equation to the target one. We shall see that this problem becomes algebraic when the symmetry pseudogroup of the target equation is zerodimensional. We avoid the swelling of the expressions, by using non-commutative derivations adapted to the problem.     

The effect of the nonhuman external regulator’s answer-until-correct (AUC) versus knowledge-of-result (KR) task feedback on children’s behavioral regulation during learning tasks

The present study was conducted to shed a new light on the nonhuman’s external regulation effect on children’s behavioral regulation through investigating the effect of the computer’s task feedback answer-until-correct (AUC) versus knowledge-of-result (KR) with 40 preschool children (boys/girls) divided into AUC-Condition versus KR-Condition. The Aginian’s methodology (Agina, Kommers, & Steehouder, 2010) with the latest updates (Agina, Kommers, & Steehouder, 2011) was used, which involves an isolated, computer-based learning system with three different computer’s agents for measuring self-regulation as a function of the task level selection, self-regulation as a function of task precision, and special agent for evaluating children’s satisfaction. It was hypothesized that AUC-Condition will outperform KR-Condition in verbalization intensity, manifested self-regulation, and the degree of satisfaction. Despite the results were not confirmed the hypothesis, the results generated by the game were consistent with the statistical results in which this consistency increases, to a great extent, the reliability of the Aginian’s measurements. However, both results were not confirmed Vygotsky’s view or Piaget’s view of self-regulation development as both concluded that thinking aloud and self-regulation have a reverse relationship and, therefore, thinking aloud, per se, can be used to explore the problems the children may not agree to talk about.     

Information transmittal, Newton’s law of gravitation, and tensor approach to general relativity

The special relativity considered in [A. Einstein, Zur Elektrodynamik der bewegte Körper. Ann. Physik, 17 (1905) 891-921] is based on the concept of finite speed of information transmittal by the available signals (rays of light). It is demonstrated that the same concept applies to Newton’s law of universal gravitation since the magnitude of distances between attracting masses can be physically defined (carried, accounted in acting forces of gravity) only by signals (physical processes) propagating at finite velocities. It follows that the speed of propagation of gravity is finite. The linear transformations of special relativity are applied to Newton’s law of gravitation to take into account the relativistic effects of information transmittal in a field of central forces of attraction. Relativistic representations of Newton’s law are obtained with respect to the center of gravity exposing illusory effects that appear at high velocities. It is verified that in atomic physics the effect of Newtonian gravitation on the motion of elementary particles at high velocities is negligible also in relativistic consideration. Computational methods are developed to measure the intensity of gravitation at a distant space-time location using a body that travels in space, emitting uniform pulses of light that are received by the observer at a different space-time location. It is demonstrated that the tensor approach to the general relativity and the united theory of space, time and gravitation in which the geometrical properties (metric) of the four-dimensional space-time continuum depend on the distribution of gravitating masses in space and their motion represent a transformed Lorentz invariant with a new type of inertia in the field of forces changing in space and time. Real physical processes evolve according to the forces represented in the tensor form by this invariant which is equivalent to the coordinate-free local invariant of relativistic dynamics that defines the field and the motion of a body whose velocities and accelerations can be measured by relativistic identification methods at a point, time and direction of interest. The results open new avenues for research in the general relativity and can be used for software development, field measurements and experimental studies in application to distant or fast moving systems.     

An EOQ model of homogeneous products while demand is salesmen’s initiatives and stock sensitive

The author develops an inventory model to determine the retailer’s optimal order quantity for homogeneous products. It is assumed that the amount of display space is limited and the demand of the products is dependent on the display stock level, where a huge stock of one product has a negative effect on the other product. Also, the replenishment rate depends on the level of stock of the items. The objective of the model is to maximize the profit function, considering the effect of inflation and time value of money, by Pontryagin’s Maximal Principles. The stability analysis of the concerned dynamical system has been carried out analytically as well as numerically.