Effects of small-world connectivity on noise-induced temporal and spatial order in neural media

We present an overview of possible effects of small-world connectivity on noise-induced temporal and spatial order in a two-dimensional network of excitable neural media with FitzHugh–Nagumo local dynamics. Small-world networks are characterized by a given fraction of so-called long-range couplings or shortcut links that connect distant units of the system, while all other units are coupled in a diffusive-like manner. Interestingly, already a small fraction of these long-range couplings can have wide-ranging effects on the temporal as well as spatial noise-induced dynamics of the system. Here we present two main effects. First, we show that the temporal order, characterized by the autocorrelation of a firing-rate function, can be greatly enhanced by the introduction of small-world connectivity, whereby the effect increases with the increasing fraction of introduced shortcut links. Second, we show that the introduction of long-range couplings induces disorder of otherwise ordered, spiral-wave-like, noise-induced patterns that can be observed by exclusive diffusive connectivity of spatial units. Thereby, already a small fraction of shortcut links is sufficient to destroy coherent pattern formation in the media. Although the two results seem contradictive, we provide an explanation considering the inherent scale-free nature of small-world networks, which on one hand, facilitates signal transduction and thus temporal order in the system, whilst on the other hand, disrupts the internal spatial scale of the media thereby hindering the existence of coherent wave-like patterns. Additionally, the importance of spatially versus temporally ordered neural network functioning is discussed.     

Identifying authority structures in mathematics classroom discourse: a case of a teacher’s early experience in a new context

We explore a conceptual frame for analyzing mathematics classroom discourse to understand the way authority is at work. This case study of a teacher moving from a school where he is known to a new setting offers us the opportunity to explore the use of the conceptual frame as a tool for understanding how language practice and authority relate in a mathematics classroom. This case study illuminates the challenges of establishing disciplinary authority in a new context while also developing the students’ sense of authority within the discipline. To analyze the communication in the teacher’s grade 12 class in the first school and grade 9 class early in the year at the new school, we use the four categories of positioning drawn from our earlier analysis of pervasive language patterns in mathematics classrooms—personal authority, discourse as authority, discursive inevitability, and personal latitude.     

Classes of network connectivity and dynamics

Many kinds of complex systems exhibit characteristic patterns of temporal correlations that emerge as the result of functional interactions within a structured network. One such complex system is the brain, composed of numerous neuronal units linked by synaptic connections. The activity of these neuronal units gives rise to dynamic states that are characterized by specific patterns of neuronal activation and co‐activation. These patterns, called functional connectivity, are possible neural correlates of perceptual and cognitive processes. Which functional connectivity patterns arise depends on the anatomical structure of the underlying network, which in turn is modified by a broad range of activity‐dependent processes. Given this intricate relationship between structure and function, the question of how patterns of anatomical connectivity constrain or determine dynamical patterns is of considerable theoretical importance. The present study develops computational tools to analyze networks in terms of their structure and dynamics. We identify different classes of network, including networks that are characterized by high complexity. These highly complex networks have distinct structural characteristics such as clustered connectivity and short wiring length similar to those of large‐scale networks of the cerebral cortex. © 2002 Wiley Periodicals, Inc.     

Algebraic Exponentiation in General Categories

We study a categorical-algebraic concept of exponentiation, namely, right adjoints for the pullback functors between D. Bourn’s categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints—in particular, for semi-abelian, protomodular, (weakly) Mal’tsev, (weakly) unital, and more general categories. We present a number of examples and counter examples for the existence of such right adjoints.     

Exploring the Relationship Between Teaching Efficacy and Cultural Efficacy of Novice Science Teachers in High‐Needs Schools

This qualitative case study sought to investigate what relationship exists between teaching efficacy and cultural efficacy of novice science teachers in high‐needs, high‐minority urban schools. One major theme—the importance of establishing positive teacher–student relationships—surrounding teaching efficacy in the context of cultural efficacy emerged. The data sources included: (a) teachers' responses to a focus group interview conducted at the end of their first year, (b) written reports and documentation from university‐based field supervisors' observations during the teachers' first year, and (c) teachers' self‐reported responses on the Science Teaching Efficacy Belief Instrument designed to measure science teaching efficacy. Final analysis of the data reflects a strong relationship between teaching efficacy and practice within the context of culture. The results of this study also indicated an existing relationship between teaching efficacy and cultural efficacy through cultural connections with students. Encouraging the development of teacher efficacy in the context of cultural efficacy becomes important at all levels of teacher preparation, especially to ensure the retention of high‐quality science teachers in high‐needs schools.     

Short cycle connectivity

Short cycle connectivity is a generalization of ordinary connectivity—two vertices have to be connected by a sequence of short cycles, in which two consecutive cycles have at least one common vertex. If all consecutive cycles in the sequence share at least one edge, we talk about edge short cycle connectivity. Short cycle connectivity can be extended to directed graphs (cyclic and transitive connectivity).It is shown that the short cycle connectivity is an equivalence relation on the set of vertices, while the edge/arc short cycle connectivity components determine an equivalence relation on the set of edges/arcs.     

Analysis of pattern formation in reaction diffusion models with spatially inhomogenous diffusion coefficients

We consider a reaction diffusion system in one spatial dimension in which the diffusion coefficients are spatially varying. We present a non-standard linear analysis for a certain class of spatially varying diffusion coefficients and show that it accurately predicts the behaviour of the full nonlinear system near bifurcation. We show that the steady state solutions exhibit qualitatively different behaviour to that observed in the usual case with constant diffusion coefficients. Specifically, the modified system can generate patterns with spatially varying amplitude and wavelength. Application to chondrogenesis in the limb is discussed.     

Measuring bank contagion in Europe using binary spatial regression models

The recent European sovereign debt crisis clearly illustrates the importance of measuring the contagion effects of bank failures. Indeed, to better understand and monitor contagion risk, the European Central Bank has assumed the supervision of the largest banks in each of the member states. We propose a measure of contagion risk based on the spatial autocorrelation parameter of a binary spatial autoregressive model. Using different specifications of the interbank connectivity matrix, we estimate the contagion parameter for banks within the Eurozone, between 1996 and 2012. We provide evidence of high levels of systemic risk due to contagion during the European sovereign debt crisis.     

Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images

In successive deletion stages of parallel thinning algorithms for binary digital images, one usually checks the preservation of connectivity by verifying that: (a) every removed pixel is individually deletable without modifying connectivity (well-known criteria, such as those of Rosenfeld and Yokoi, exist for that purpose); (b) every pair of 8-adjacent removed pixels is deletable without connectivity modification. In the case of the 8-connectivity for the figure (and the 4-connectivity for the background), two more patterns must be tested for connectivity preservation: an isolated triple or quadruple of mutually 8-adjacent pixels.In this paper we give a formal characterization of these patterns for testing connectivity preservation by what we call minimal non-x-deletable sets (x-MND sets), where x=4, 8 or {4,8} (the type of connectivity considered for the figure). A parallel thinning algorithm whose deletion stage cannot remove an x-MND set is guaranteed to preserve the connectivity properties of any figure. We show that an x-MND set consists in either (1) a single pixel; or (2) a pair of 8-adjacent pixels; or (3) an isolated triple or quadruple of mutually 8-adjacent pixels (for x=8 only).     

Infinite,highly connected digraphs with no two arc-disjoint spanning trees

We present a construction of countably infinite, highly connected graphs and digraphs, which shows that several basic connectivity results on finite graphs, including Edmondsapos;s branching theorem, cannot be extended to the infinite case.     

Comparing hiring strategies in a committee with similarity biases

We present an organizational model that develops organizational expertise and socialization with a hiring process informed by the inherent biases of individuals. We present factors that we believe critically impact candidate selection, literature related to these factors, and our resulting equations. We discuss the model, and present two virtual experiments. The first virtual experiment was used to validate the new model by comparing the implementation with an existing reference implementation—we found similar patterns—which established relational equivalence. The second virtual experiment compared organizations with and without a stochastic selection process and with various selection strategies. Organizations that stressed socialization tended to need to review more (otherwise equally qualified) applicants than organizations that did not, and organizations that were able to deliberate more thoroughly found turnover less effective at maintaining organizational performance. Larger committees reduced the number of applicants that needed to be reviewed in firms that valued diversity, but offered no particular benefit to other organizations.     

Novel method of identifying time series based on network graphs

In this article, we propose a novel method for transforming a time series into a complex network graph. The proposed algorithm is based on the spatial distribution of a time series. The characteristics of geometric parameters of a network represent the dynamic characteristics of a time series. Our algorithm transforms, respectively, a constant series into a fully connected graph, periodic time series into a regular graph, linear divergent time series into a tree, and chaotic time series into an approximately power law distribution network graph. We find that when the dimension of reconstructed phase space increases, the corresponding graph for a random time series quickly turns into a completely unconnected graph, while that for a chaotic time series maintains a certain level of connectivity. The characteristics of the generated network, including the total edges, the degree distribution, and the clustering coefficient, reflect the characteristics of the time series, including diverging speed, level of certainty, and level of randomness. This observation allows a chaotic time series to be easily identified from a random time series. The method may be useful for analysis of complex nonlinear systems such as chaos and random systems, by perceiving the differences in the outcomes of the systems—the time series—in the identification of the systemic levels of certainty or randomness. © 2011 Wiley Periodicals, Inc. Complexity, 2011     

On the relevance of fuzzy sets for operations research

Every researcher in OR models knows that data in this field of human science are deterministic or random or uncertain. Of course, if measurements are available, the scientist must use such strong data but, in many case, a lot of data are weaker and subjectivity is necessary. To combine in a good way, at the best, taking account the present level of knowledge, it is what we can do. Fuzzy sets—and specially, fuzzy numbers—is a good tool for the OR analyst facing partial uncertainty and subjectivity. We are able to associate with several hybrid operators, probabilistic and uncertain data. Several processes and examples will be explained. The goal: to build a model faithful at the best and intelligible for the decision maker.     

Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible “coherence” morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is R_R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.     

地方应用型本科院校数学与应用数学专业课程体系的改革与实践——以太原工业学院为例

以太原工业学院为例,分析了数学与应用数学专业培养应用型人才所面临的挑战.结合该校地方性和应用型的特点,基于岗位需求为导向,围绕"毕业生未来就业岗位—培养目标—能力与素质要求—课程设置"这条主线完成了金融数学和软件开发两个方向的课程设置,以此建立"模块化教学,分流培养"的培养体系,为应用型本科院校数学与应用数学专业课程体系设置提供参考.     

A diffusive predator-prey model in heterogeneous environment

In this paper, we demonstrate some special behavior of steady-state solutions to a predator-prey model due to the introduction of spatial heterogeneity. We show that positive steady-state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably. Moreover, we observe some essential differences of the behavior of our model from that of the classical Lotka-Volterra model that seem to arise only in the heterogeneous case.     

Neighbourhood of spatial areas in the physical and socio-economical context

In this paper a comparison of two concepts of neighbourhoods of spatial areas—physical and socio-economical—is presented. The former usually concerns their location on an administrative map and a possible community of borders, the latter—a similarity in terms of a composite social or economical phenomenon. The paper describes the classical theory of a neighbourhood and its matrix and shows how it can be used in multivariate data analysis. An efficient method of comparison of both types of neighbourhood matrices, using the Smith normal form, known from algebra, of an integer matrix, is proposed. We show also how the variance of disturbances as well as of spatial lag vectors and of effects and prediction correlation can be estimated. Our theoretical considerations are supplemented by a numerical example which looks at neighbourhoods of spatial areas included in a fairly small region of Poland, around Kalisz, in terms of their geographical location and the situation of their labour market and a relevant simulation study. These studies showed that the distance between both types of neighbourhood are often significant, even it is not always reflected in the correlation of effects and predictions when spatial econometrical models are applied.     

Simulation of the Spatial Action of a Medium on a Body of Conical Form

We consider amathematical model of the spatial action of a medium on the axisymmetric rigid body whose external surface has a part that is a circular cone.We present a complete system of equations of motion under the quasistationary conditions. The dynamical part forms an independent system of the sixth order in which the independent subsystems of lower order are distinguished. We study the problem of stability with respect to the part of variables of the key regime—the spatial rectilinear translational deceleration of the body. For a particular class of bodies, we show the inertial mass characteristics under which the key regime is stable. For a plane analog of the problem, we obtain a family of phase portraits in the space of quasivelocities.