Two-time-scale Jump-Diffusion Models with Markovian Switching Regimes

This work is concerned with two-time-scale jump diffusion models modulated by continuous-time Markov chains. One of our motivations stems from generalization of insurance risk models. The models are hybrid in the sense that they involve both continuous dynamics and discrete events. Two cases are considered. One of them has a fast-varying switching process, and the other contains a rapidly fluctuating diffusion. Two-time scale is used for complexity reduction. Using weak convergence methods, we derive their limit processes. The insight and implication provided by the analysis are: to reduce the complexity, one can ignore the detailed variations and concentrate on the limit or the reduced models.     

Competitive capacitive biosensing technique (CCBT): A novel technique for monitoring low molecular mass analytes using glucose assay as a model study

A novel technique for monitoring of low molecular mass analytes using a flow-injection capacitive biosensor is presented. The method is based on the ability of a small molecular mass analyte to displace a large analyte–carrier conjugate from the binding sites of an immobilized biorecognition element with weak affinity to both compounds. A model study was performed on glucose as the small molecular mass analyte. In the absence of glucose, binding of a glucose polymer or a glycoconjugate to concanavalin A results in a capacitance decrease. Upon introduction of glucose, it displaces a part of the bound glucose polymer or glycoconjugate leading to a partial restoration of capacitance. Experimental results show that the change in capacitance depends linearly on glucose concentration within the range from 1.0 × 10⁻⁵ to 1.0 × 10⁻¹ M, corresponding to 1.8 μg ml⁻¹ to 18 mg ml⁻¹ in a logarithmic plot, with a detection limit of 1.0 × 10⁻⁶ (0.18 μg ml⁻¹) under optimized conditions. In addition, by modifying the molecular mass of the glucose polymer, amount of biorecognition element, and buffer composition, we were able to tune the analyte-sensing range. The developed technique has the benefits of expanded dynamic range, high sensitivity, and excellent reusability.     

Friezes

The construction of friezes is motivated by the theory of cluster algebras. It gives, for each acyclic quiver, a family of integer sequences, one for each vertex. We conjecture that these sequences satisfy linear recursions if and only if the underlying graph is a Dynkin or an Euclidean (affine) graph. We prove the “only if” part, and show that the “if” part holds true for all non-exceptional Euclidean graphs, leaving a finite number of finite number of cases to be checked. Coming back to cluster algebras, the methods involved allow us to give formulas for the cluster variables in case Am and ; the novelty is that these formulas use 2 by 2 matrices over the semiring of Laurent polynomials generated by the initial variables (which explains simultaneously positivity and the Laurent phenomenon). One tool involved consists of the SL₂-tilings of the plane, which are particular cases of T-systems of Mathematical Physics.     

Coverings of laura algebras: The standard case

In this paper, we study the covering theory of laura algebras. We prove that if a connected laura algebra is standard (that is, has a standard connecting component), then it has Galois coverings associated to the coverings of the connecting component. As a consequence, the first Hochschild cohomology group of a standard laura algebra vanishes if and only if it has no proper Galois coverings.     

Some Thoughts on Mathematical Education

This paper explores the growth of mathematical studies in recent times and suggests future patterns that may occur in mathematical curricula. It is proposed that courses in the History and Methods of Mathematics, Mathematical Engineering (rather than Engineering Mathematics) and Mathematics and Education should be used to widen the curriculum choices for the mathematical student of the future. Finally, in illustrating the application of mathematics to the study of Noise and Vibration it is shown that, in such an investigation, mathematics cannot be an end in itself but can be greatly helped by a knowledge of other physical disciplines.

     

Operational Research Education for Practitioners and Managers∗

The paper reviews the history of education in operational research and indicates the needs both of a practitioner and of a manager. It describes how these can be met by standard course teaching and suggests some areas of development.

     

Tilting modules over split-by-nilpotent extensions

Let a finite dimensional algebra R be a split extension of an algebra A by a nilpotent bimodule Q. We give necessary and sufficient conditions for a (partial) tilting module T_A to be such that T?_A R_R is a (partial) tilting module. If this is not the case, but Q_A is generated by the tilting module T_A , then there exists a quotient [Rbar] of R such that T?_A [Rbar]_[Rbar] is a tilting module.