Cardinal invariants concerning bounded families of extendable and almost continuous functions

In this paper we introduce and examine a cardinal invariant closely connected to the addition of bounded functions from to . It is analogous to the invariant defined earlier for arbitrary functions by T. Natkaniec. In particular, it is proved that each bounded function can be written as the sum of two bounded almost continuous functions, and an example is given that there is a bounded function which cannot be expressed as the sum of two bounded extendable functions.

     

Nowhere-zero 3-flow of graphs with small independence number

Tutte’s $3$-flow conjecture states that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In this paper, we characterize all graphs with independence number at most $4$ that admit a nowhere-zero $3$-flow. The characterization of $3$-flow verifies Tutte’s $3$-flow conjecture for graphs with independence number at most $4$ and with order at least $21$. In addition, we prove that every odd-$5$-edge-connected graph with independence number at most $3$ admits a nowhere-zero $3$-flow. To obtain these results, we introduce a new reduction method to handle odd wheels.     

Flow-contractible configurations and group connectivity of signed graphs

Jaeger, Linial, Payan and Tarsi (JCTB, 1992) introduced the concept of group connectivity as a generalization of nowhere-zero flow for graphs. In this paper, we introduce group connectivity for signed graphs and establish some fundamental properties. For a finite abelian group $A$, it is proved that an $A$-connected signed graph is a contractible configuration for $A$-flow problem of signed graphs. In addition, we give sufficient edge connectivity conditions for signed graphs to be $A$-connected and study the group connectivity of some families of signed graphs.     

Scalar,vector and tensor harmonics on the three-sphere

Scalar, vector and tensor harmonics on the three-sphere were introduced originally to facilitate the study of various problems in gravitational physics. These harmonics are defined as eigenfunctions of the covariant Laplace operator which satisfy certain divergence and trace identities, and ortho-normality conditions. This paper provides a summary of these properties, along with a new notation that simplifies and clarifies some of the key expressions. Practical methods are described for accurately and efficiently computing these harmonics numerically, and test results are given that illustrate how well the analytical identities are satisfied by the harmonics computed numerically in this way.     

Children's strategy use and interpretations of mathematical representations

This is a summary of research, from an information processing perspective, of children's interpretation and use of strategies and representations for place value, subtraction and addition in the first three years of school. Representations are defined broadly to include concrete embodiments of numbers, symbols for numbers and operations, and combinations of the latter in number sentences and algorithms. The objective was to assess the value and limitations of the use of representations in early mathematics learning and teaching and hence to identify, describe and examine critically some of the strategies and representations that children and teachers use in early mathematics. Children generally chose to use verbal and mental strategies in preference to formal algorithms, and did not want to use analogs unless they could not perform the task in any other way. The latter preference is explained on the basis of the extra demand that use of analogs adds to the cognitive process unless they are used automatically.     

Features of nuclear reactions induced by ultracold neutrons passing through thin films

The probabilities of elastic scattering and capture of ultracold neutrons in thin films are calculated in terms of the time quantum theory. It is shown that at a neutron wavelength exceeding considerably the film thickness, the macroscopic probability of the (n, γ)-reaction can decrease as compared to the macroscopic probability of elastic scattering.     

The optimal combination: Grammatical swarm,particle swarm optimization and neural networks

Social behaviour is mainly based on swarm colonies, in which each individual shares its knowledge about the environment with other individuals to get optimal solutions. Such co-operative model differs from competitive models in the way that individuals die and are born by combining information of alive ones. This paper presents the particle swarm optimization with differential evolution algorithm in order to train a neural network instead the classic back propagation algorithm. The performance of a neural network for particular problems is critically dependant on the choice of the processing elements, the net architecture and the learning algorithm. This work is focused in the development of methods for the evolutionary design of artificial neural networks. This paper focuses in optimizing the topology and structure of connectivity for these networks.     

The generation and use of graphical examples in calculus classrooms: The case of the mean value theorem

We analyzed video data of five instructors teaching the Mean Value Theorem (MVT) in a first-semester calculus course as part of a broader project investigating how active learning strategies were being implemented and supported in calculus courses. We sought to identify the ways examples of functions that did or did not satisfy the conclusion of MVT were generated and used in instruction. Using thematic analysis, we identified four themes that serve as characterizations of examples, which then allowed for the analysis of trends and patterns. We propose that attention to the generation and use of examples serves as one lens for considering how students can be engaged in the mathematical activity of the classroom, with implications for learning. This work contributes to an evolving notion of what is entailed in students’ active learning of mathematics and the role of the instructor in facilitating active learning opportunities.