Math forums are great sources of information and knowledge, almost all of them provide a detailed explanatory case study, where you could see how such or such math could be applied practically. These cases document successful solutions and thus demonstrate the application of theory of problem solving. In our reflection about these examples, participants establish important sources on methodologies, tools, and strategies regarding how to solve similar problems in own projects.

Math forums, in particular, provide case studies of current research, diffusion, hard computation, or market needs.

They give the methodology on how issues were handled, solved, and why readers are informed of the process and the reasoning of actions. For example, the contributors may explain how they approached a particular mathematical problem, how they chose a particular tool or algorithm and how they analysed the results to arrive at the conclusion. Such level of openness increases comprehensiveness of information while at the same time fostering od thought process and innovation.

Positive Case Studies Discussed in the Math Forums

Maximizing Traffic Flow Over Networks: The Graph TheoryApproach

Also frequently discussed in forums, this type of case study illustrates how various algorithmic features of a graph theory can be used to model data flow on a network. In this case, contributors demonstrate how various devices can be presented as a graph, its nodes being devices and its edges – communication channels; it is demonstrated how Dijkstra’s or Bellman-Ford algorithms can be used to determine the shortest paths or minimize traffic congestion. Such examples are most suitable and applicable in such areas like telecommunications, logistics as well as computer science as the data routing is paramount. These case studies may comprise diagrams of the graph used, pieces of code to execute the algorithms or comparisons of performance of different strategies.

Yet, solving differential equations is an extremely important process in all fields of paleoclimate and climate modeling, where it is used to describe the dynamics of Earth’s climate system.

Another topic which we often find in the discussion threads of mathematicians is climate modeling, where the participants post examples on how differential equations can be used to make a simulation of the weather, sea currents or temperature fluctuations. These examples usually provided specify the analytical description used, for example the equations of state for fluids using Navier-Stokes and then present an over view of the numerical techniques, finite difference or finite element analysis to obtain solutions or approximations. For instance, contributors may also point out the software tools and the computational demands needed to solve such large-scale problems and hence such case studies would be informative to researchers and students in environmental science and meteorology.

Introduction of Big Data and Machine Learning to Very Large Datasets

Machine learning has recently attracted lots of interest in the math forums, and dozens of cases discuss its uses in image recognition, natural language understanding, and forecasting. Such examples first introduce problems in terms of mathematical formalism, for instance the objective function and the constraints, and then showcase how algorithms like the gradient descent or support vector machines are used to improve performance. Almost always, the authors distribute their scripts in Python or Jupyter notebooks to illustrate how to implement the techniques, so others can reproduce the work. Thus, talking about the choice of features, data preparation steps, and hyperparameters incline towards a deeper understanding of the machine learning.