Do you would like to solve a fancy problem? Applied mathematics can assist

You can probably remember a time whenever you used math to unravel an on a regular basis problem, equivalent to calculating suggestions at a restaurant or determining the square footage of a room. But what role does mathematics play in solving complex problems equivalent to curing an illness?

In my job as Applied mathematicianI take advantage of mathematical tools to review and solve complex problems in biology. I actually have worked on problems involving genes and neural networks equivalent to: Interactions between cells And Decision making. To do that, I create descriptions of an actual situation in mathematical language. The strategy of converting a situation right into a mathematical representation known as modeling.

Translating real situations into mathematical terms

If you've ever done a math problem in regards to the speed of trains or the fee of food, that's an example of mathematical modeling. But for tougher questions, it could actually be complicated to jot down the true scenario as a mathematical problem. This process requires rather a lot Creativity and understanding of the issue in query and is usually the results of collaboration between applied mathematicians and scientists from other disciplines.

As an example, we could represent a Sudoku game as a mathematical model. In Sudoku, the player fills in empty squares in a puzzle with numbers between 1 and 9, with certain rules in place, equivalent to that numbers can’t be repeated in any row or column.

The puzzle starts with just a few pre-filled boxes and the goal is to work out which numbers fit within the remaining boxes.

Imagine that a variable, say x, represents the number that appears in one in every of these empty fields. We can guarantee that x is between 1 and 9 by saying that x solves the equation (x-1)(x-2) … (x-9)=0. This equation is true provided that one in every of the aspects on the left side is zero. Each of the aspects on the left is zero provided that x is a number between 1 and 9; for instance (x-1)=0 when x=1. This equation encodes a fact about our Sudoku game, and we will do this Code the sport's other features in an identical manner. The resulting Sudoku model consists of a series of equations with 81 variables, one for every box within the puzzle.

Another situation we could model is the concentration of a drug, equivalent to aspirin, in an individual's bloodstream. In this case, we can be interested by how the concentration changes after we take aspirin and the body metabolizes it. Just like Sudoku, one can create a series of equations that describe how the concentration of aspirin evolves over time and the way additional intake affects the dynamics of this drug. However, unlike Sudoku, the variables that represent concentrations usually are not static but change over time.

But the act of modeling is just not at all times that easy. How would we model diseases like cancer? Is it enough to model the dimensions and shape of a tumor, or do we want to model each blood vessel within the tumor? Every single cell? Every single chemical in every cell? Loads is unknown about cancer. So how can we model such unknown features? Is that even possible?

Applied mathematicians must discover a balance between models which might be realistic enough to be useful and straightforward enough to implement. These models can take several years to create, but in collaboration with experimental scientists, trying to seek out a model often provides recent insights into the real-world problem.

Mathematical models help to seek out real solutions

After a mathematical problem is written to represent a situation, the second step within the modeling process is to unravel the issue.

For Sudoku we’ve got to unravel a group of equations with 81 variables. For the aspirin example, we want to unravel an equation that describes the speed of change in concentrations. This is where all of the mathematics that was invented and continues to be being invented comes into play. Areas of pure mathematics equivalent to algebra, evaluation, Combinatorics and lots of others might be used – in some cases combined – to unravel the complex mathematical problems that arise from the applying of mathematics in the true world.

The third step of the modeling process is to translate the mathematical solution into the answer of the applied problem. In the case of Sudoku, solving the equations tells us which number must fit into which box to unravel the puzzle. In the case of aspirin, the answer can be a series of curves that tell us the concentration of aspirin within the digestive system and bloodstream. This is how applied mathematics works.

When making a model is just not enough

Or is it? While this three-step process is the perfect strategy of applied mathematics, the fact is more complicated. Once I reach the second step where I need the answer to the mathematics problem, fairly often, if not more often than not, it seems that nobody knows tips on how to solve the mathematics problem within the model. In some cases, mathematics is used to research the issue doesn't exist in any respect.

For example, it’s difficult to investigate cancer models since the interactions between genes, proteins, and chemicals usually are not as clear because the relationships between boxes in a Sudoku game. The predominant difficulty is that these interactions are “nonlinear,” meaning that the effect of two inputs is just not simply the sum of the person effects. To address this problem, I actually have been working on novel methods for studying nonlinear systems, equivalent to: Boolean network theory and polynomial algebra. Using this and traditional approaches, my colleagues and I actually have examined questions in areas equivalent to
Decision making,
Gene networks,
Cell differentiation And
Limb regeneration.

When approaching unsolved problems in applied mathematics, the excellence between applied and pure mathematics often disappears. Areas that were once considered too abstract were exactly what is required for contemporary problems. This highlights the importance of mathematics for all of us; Current areas of pure mathematics can grow to be the applied mathematics of tomorrow and be the tools needed for complex, real-world problems.