M3AI® Lab & Melnik Research Group | Mind - Mathematical Models

Canadian Excellence

Welcome to the Melnik Group Website

Our research concerns with human, natural, and man-made systems, along with their interactions. Man-made systems include engineering systems, as well as context-dependent complex dynamic structures. Among others, examples of such structures may include social and financial systems and their networks. By developing tools and methods of mathematical modeling and by applying them along with two other research methodologies based on theory and experiment, we seek a better understanding of new phenomena and processes in studying complex dynamic systems which include coupled human-environment systems, as well as man-made systems. In most cases, our focus is on coupled multiscale phenomena and processes in such systems. Read More About Our Research

Coupled Multiscale Phenomena and Processes in Complex Dynamic Systems

Some of the most important and challenging problems in scienceand engineering are coupled problems. The coupling ispronounced at three basic levels:

the level of systems coupling as it is the case inhuman-environment and human-centered systems, or at thelevel of field coupling as it is the case inthermo-elasticity and electro-elasticity, as well as inmany other areas of applications.
the level of coupled effects between different parts of the system, and/or
the level of coupling between differentspatial/spatio-temporal scales at which the natural, human, or man-made system, that is being studied, operates.

This coupling brings to life new phenomena, the phenomenawhich are not amenable to an adequate description with conventional mathematical models, traditionally developed without accounting for the coupled nature of the problem. At the M3AI^® Lab, we focus on the development of coupled mathematical models for addressing fundamental questions of science and solving new problems in science and engineering. Our main interests lie with Complex Dynamic Systems, typically capable of information processing. We study coupled phenomena, effects, and processes in such systems that are becoming increasingly important in addressing key societal challenges of the 21st century and beyond. Read More About Our Research.

Roderick Melnik is Professor and Tier I Canada Research Chair in Mathematical Modelling. He has been also affiliated with two other nearby universities(Waterloo and Guelph) and the Centre for Applied Mathematics BCAM in Europe. You can find more information about his lab's team and research projects in this section. He can be contacted via email at rmelnik-at-wlu-dot-ca.

M3AI^® Laboratory, Head,
Professor Roderick Melnik,
Science Research Centre,
SR-316
Wilfrid Laurier University,
75 University Avenue West,
Waterloo, ON,
Canada, N2L 3C5
Tel.: +1-519-884-1970 (ext. 2801 - lab)
Tel.: +1-519-884-1970 (ext. 3662 - office)
Fax: +1-519-884-9738

La historia de la civilización demuestra hasta lasaciedad la esterilidad de la metafísica en susreiterados esfuerzos por adivinar las leyes de laNaturaleza. Con razón se ha dicho que el humanointelecto, de espaldas a la realidad y concentrado ensí mismo, es impotente para dilucidar los mássencillos rodajes del mundo y de la vida.

(Santiago Ramón y Cajal, Reglas y consejos sobreinvestigación científica. In English translation thebook was published by The MIT Press as “Advice for a Young Investigator)

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What's On M3AI ^®

Lab Projects

Multiscale || Coupled || General

Multiscale coupled systems are ubiquitous in sciences, engineering, and society. All phenomena we observe in Nature are reflections of various forms couplings, e.g. between different physical fields, between different components of a system, or between different systems.

Read Project

Life Sciences || Biology || Medicine.

Biology and medicine provide many challenging problems that can be addressed efficiently only with adequately developed mathematical tools. As these disciplines are becoming increasingly quantitative, there is a need in the development of mathematical models for a vast variety of processes and phenomena, most of which have (spatial and/or temporal) multiscale characteristics.

Nanoscale Systems || Unification Theories

Our interests here lie primarily with low dimensional nanostructures where the motion of electrons can be confined from one, two, and even three spatial dimensions. The respective structures are known as quantum wells, quantum wires, and quantum dots. Such structures will continue transforming our lives as they will be used as parts of environmental, chemical, and biological sensors, in transportation industries, energy saving technologies, biomedicine, communications, and information technology.

Nonlocal || Far-from-equilibrium || Dynamic

We are interested in non-trivial dynamic processes, including but not limited to nonlocal and those that are far from their equilibrium, as well as in the associated phenomena and systems, and their modeling. Such processes occur frequently in systems ranging from semiconductor plasma to chemical and biological networks, and from tiny man-made nano-objects to living cell cycles.

Stochastic || Networks || Uncertainty

In modeling many phenomena and processes in Nature as well as in man-made systems there is often some indeterminacy in the future evolution of such systems, phenomena, and processes that, by the language of mathematics, can be described by probability distributions.

Environment || Industry || Sustainable Development

As it stands now, humanity depletes its resources with an astronomical speed, much faster than they can be replenished. It is important to understand that the only way for our further development could be a balance between our needs and preserving the environment. We are interested in the development of mathematical foundations for and mathematical modeling of alternative and renewable sources of energy.