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# Mathematics, Department of

Receive updates for this collection## Social Interactions of Eating Behaviour among High School Students: A Cellular Automata Approach

BACKGROUND:Overweight and obesity in children and adolescents is a global epidemic posing problems for both developed and developing nations. The prevalence is particularly alarming in developed nations, such as the United States, where approximately one in three school-aged adolescents (ages 12-19) are overweight or obese. Evidence suggests that weight gain in school-aged adolescents is related to energy imbalance exacerbated by the negative aspects of the school food environment, such as presence of unhealthy food choices. While a well-established connection exists between the food environment, presently there is a lack of studies investigating the impact of the social environment and associated interactions of school-age adolescents. This paper uses a mathematical modelling approach to explore how social interactions among high school adolescents can affect their eating behaviour and food choice.METHODS:In this paper we use a Cellular Automata (CA) modelling approach to explore how social interactions among school-age adolescents can affect eating behaviour, and food choice. Our CA model integrates social influences and transition rules to simulate the way individuals would interact in a social community (e.g., school cafeteria). To replicate these social interactions, we chose the Moore neighbourhood which allows all neighbours (eights cells in a two-dimensional square lattice) to influence the central cell. Our assumption is that individuals belong to any of four states; Bring Healthy, Bring Unhealthy, Purchase Healthy, and Purchase Unhealthy, and will influence each other according to parameter settings and transition rules. Simulations were run to explore how the different states interact under varying parameter settings.RESULTS:This study, through simulations, illustrates that students will change their eating behaviour from unhealthy to healthy as a result of positive social and environmental influences. In general, there is one common characteristic of changes across time; students with similar eating behaviours tend to form groups, represented by distinct clusters. Transition of healthy and unhealthy eating behaviour is non-linear and a sharp change is observed around a critical point where positive and negative influences are equal.CONCLUSIONS:Conceptualizing the social environment of individuals is a crucial step to increasing our understanding of obesogenic environments of high-school students, and moreover, the general population. Incorporating both contextual, and individual determinants found in real datasets, in our model will greatly enhance calibration of future models. Complex mathematical modelling has a potential to contribute to the way public health data is collected and analyzed.

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## Doubly Stochastic Right Multipliers

Let P(G) be the set of normalized regular Borel measures on a compact group G. Let Dr be the set of doubly stochastic (d.s.) measures λ on G×G such that λ(As×Bs)=λ(A×B), where s∈G, and A and B are Borel subsets of G. We show that there exists a bijection μ↔λ between P(G) and Dr such that ϕ−1=m⊗μ, where m is normalized Haar measure on G, and ϕ(x,y)=(x,xy−1) for x,y∈G. Further, we show that there exists a bijection between Dr and Mr, the set of d.s. right multipliers of L1(G). It follows from these results that the mapping μ→Tμ defined by Tμf=μ∗f is a topological isomorphism of the compact convex semigroups P(G) and Mr. It is shown that Mr is the closed convex hull of left translation operators in the strong operator topology of B[L2(G)].

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## Generalized Classes of Starlike and Convex Functions of Order α

We have introduced, in this paper, the generalized classes of starlike and convex functions of order α by using the fractional calculus. We then proved some subordination theorems, argument theorems, and various results of modified Hadamard product for functions belonging to these classes. We have also established some properties about the generalized Libera operator defined on these classes of functions.

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## Contractive Mappings on a Premetric Space

In this paper, we study the fixed point property of certain types ofcontractive mappings defined on a premetric space. The applications of these results to topological vector spaces and to metric spaces are also discussed.

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## A Bounded Consistency Theorem for Strong Summabilities

The study of R-type summability methods is continued in this paper byshowing that two such methods are identical on the bounded portion of the strongsummability field associated with the methods. It is shown that this “boundedconsistency” applies for many non-matrix methods as well as for regular matrix methods.

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## A Weak Invariance Principle and Asymptotic Stability for Evolution Equations with Bounded Generators

If V is a Lyapunov function of an equation du/dt u’ Zu in a Banach space thenasymptotic stability of an equilibrium point may be easily proved if it is known that sup(V’) < 0 onsufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability isproved for a bounded infinitesimal generator Z under a weaker assumption V’ < 0 (which aloneimplies ordinary stability only) if some observability condition, involving Z and the Frechet derivativeof V’, is satisfied. The proof is based on an extension of LaSalle’s invariance principle, which yieldsconvergence in a weak topology and uses a strongly continuous Lyapunov funcdon. The theory isillustrated with an example of an integro-differential equation of interest in the theory of chemicalprocesses. In this case strong asymptotic stability is deduced from the weak one and explicit sufficientconditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbertspace, strong asymptotic stability is proved under the following assumptions Z* + Z is weaklynegative definite and Ker Z 0 }. The proof is based on spectral theory.

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## On Approximation of Functions and Their Derivatives by Quasi-Hermite Interpolation

In this paper, we consider the simultaneous approximation of the derivatives of thefunctions by the corresponding derivatives of qua.si-Hcrmite interpolation based on the zeros of (1z2)p,(z) (where p,(x)is a Lcgcndrc polynomial). The corresponding approximation degrees are given.It is shown that this matrix of nodes is almost optimal

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## Modelling Desert Dune Fields Based on Discrete Dynamics

A mathematical formulation is developed to model the dynamics of sand dunes. The physical processes display strong non-linearity that has been taken into account in the model. When assessing the success of such a model in capturing physical features we monitor morphology, dune growth, dune migration and spatial patterns within a dune field. Following recent advances, the proposed model is based on a discrete lattice dynamics approach with new features taken into account which reflect physically observed mechanisms.

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## Continuum Model of the Two-Component Becker-Döring Equations

The process of collision between particles is a subject of interest in many fields of physics, astronomy, polymer physics, atmospheric physics, and colloid chemistry. If two types of particles are allowed to participate in the cluster coalescence, then the time evolution of the cluster distribution has been described by an infinite system of ordinary differential equations. In this paper, we describe the model with a second-order two-dimensional partial differential equation, as a continuum model.

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## Minmax Strongly Connected Subgraphs with Node Penalties

We propose an O(min{m+nlogn,mlog∗n}) to find a minmax strongly connected spanningsubgraph of a digraph with n nodes and m arcs. A generalization of this problemcalled theminmax strongly connected subgraph problem with node penalties is also considered.An O(mlogn) algorithm is proposed to solve this general problem. We also discussways to improve the average complexity of this algorithm.

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