# Mathematics applied

## Practicality versus precision: the role of mathematics in engineering design

- No Highway in the Sky

(published on December 23, 2021 by The Retirement Report)

This British movie show cases the balance between mathematical theory and practical passenger and crew safety. It was released in 1951 and starred Jimmy Stewart and Marlene Deitrich.

It is a story of pure fiction but with an underlying grain of truth about an aeronautical research project conducting stress tests on a recently developed airplane with an unusual tail. When the mathematical theoretical predictions were not borne out by the scientist’s practical experiments and in-flight experience, the scientist looked for a missing factor.

## Difference between mathematician and engineer

- Difference between mathematician and engineer

(published on on October 20, 2021)

This site compares the roles, the responsibilities, and the most recent (2018 – 2028) job opportunities of mathematicians and professional engineers. It also discusses their related careers, for example, the statisticians of the former and the materials scientists of the latter.

## Mathematically modeling a moving invisible target

Tracking the Covid-19 rates of infection world-wide, or even locally, is a kind of target practice involving a silent, moving, sometimes erratic, invisible enemy.

**Mathematical** **epidemiology** is playing a crucial role in the fight against large-scale infectious diseases. With basic **mathematical** models, researchers can begin to forecast the progression of diseases and understand the effect of interventions on disease spread.

Simple ratios, differential calculus, differential equations and descriptive statistics are just some of the mathematical constructs and theories used in the modeling of these moving targets.

Classical models of epidemic phenomena can be quite diverse in terms of research methods, from an analytic approach:

**SIR Model**which is an epidemiological model that computes the theoretical number of people infected with a contagious illness in a closed population over time*t*. This class of models involve coupled equations relating the number of susceptible people S(*t*), number of people infected t(*t*), and number of people who have recovered R(*t*).

to a more geometric approach using:

**Voronoi Diagrams**which involve a partitioning of a plane with*n*points into somewhat discrete convex polygons such that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other.

Of historical interest is that a Voronoi diagram was used in the analysis of the 1854 cholera epidemic in London, England. In this case, the physician, John Snow, determined a strong correlation of deaths with proximity to a particular (and infected) water pump on Broad Street. The end of this epidemic is attributed to the subsequent shutting off of this pump. This indicated that the bacterial infection causing the cholera epidemic had been water borne, not air borne, as was suspected by many at the time.

- Spanish Flu: a warning from history

(published on November 30, 2018 by Cambridge University)

“100 years ago, celebrations marking the end of the First World War were cut short by the onslaught of a devastating disease - the 1918-19 influenza pandemic. Its early origins and initial geographical starting point still remain a mystery but, in the Summer of 1918, there was a second wave of a far more virulent form of the influenza virus than anyone could have anticipated. Soon dubbed ‘Spanish Flu’ after its effects were reported in the country’s newspapers, the virus rapidly spread across much of the globe to become one of the worst natural disasters in human history.

To mark the centenary and to highlight vital scientific research, the University of Cambridge has made a new film exploring what we have learnt about Spanish Flu, the urgent threat posed by influenza today, and how scientists are preparing for future pandemics.”

Only one-and-a-half years after the publication of this film, another highly infectious epidemic descended upon an unsuspecting world. For a broader perspective on recent efforts to create mathematical models designed to understand this twenty-first century pandemic, see:

- Mathematical epidemiology: How to model a pandemic

(published on March 28, 2020 by Christian Yates, The Conversation)

- How to flatten the curve of coronavirus, a mathematician explains

(published on March 16, 2020 by Andrew Black, Dennis Liu and Lewis Mitchell, The Conversation)

- COVID Conversations: Mathematical Epidemiology

(published on July 29, 2020 by Oxford University)

“Mathematical models have been used worldwide to inform policy responses to COVID-19, particularly by using model simulations to predict the effects of different non-pharmaceutical interventions. While the results of modelling studies informed the introduction of lockdowns in the UK and elsewhere, modelling is useful throughout epidemics. Join Dr Robin Thompson, Junior Research Fellow at Christ Church, to hear more about mathematical modelling and how it can be useful at different stages of an infectious disease outbreak.”

A course idea...

## BIOL 325: Introductory Microbiology

This course provides an overview of microbial environments, with special emphasis on structural and functional differences among bacteria, fungi, algae, protozoa, and viruses.

## Counting and pattern recognition: not just child’s play

In the scientific field of Forensic Biology, dendrochronology is the science or technique of dating events, environmental change, and archaeological artifacts by using the characteristic patterns of annual growth rings in timber and tree trunks.

From the perspective of this discipline, trees may be properly regarded as the “Silent Sentinels of History.” They immortalize, within their internal structures and their DNA, a record detailing their environmental experiences during their lifetimes. The longer they survive, despite the climactic changes, geological events, and any human-inspired interventions, the more they can reveal about the history of the earth in their particular locale.

Consider the story of Methuselah, a California-area denizen for over 4800 years (at the latest count), that produced healthy progeny at a ripe old age.

The Secrets Of The Methuselah Tree | Oldest Tree On Earth

(published on December 1, 2017 by Timeline - World History Documentaries )

“On a desolate mountain top in California lives the world’s oldest organism – a gnarled and twisted bristlecone pine. The scientist who discovered the tree gave it the name Methuselah. It was a seedling when the Egyptian pyramids were being built and a mature tree at the time of Christ. It is now over 4,000 years old. The Curse Of The Methuselah Tree shows how the climate and dramatic events – from volcanoes to nuclear tests – over the last 4000 years have affected Methuselah’s growth rings. Archive footage and stunning computer animation are interspersed with Methuselah’s own narration, supplied by poet Roger McGough. The programme also contemplates the tree’s future and reveals the curse that has followed anyone who has dared to work on the tree – an early death.”

Forensic Files - Planted Evidence

(published on May 20, 2021 by FilmRise True Crime)

When a tree was the prime, and perhaps only, witness to a crime of murder.

Root of All Evil

(published on February 1, 2016 by Medical Detectives)

A murder suspect's former lover comes forward with an almost unbelievably gruesome tale of violence, mutilation and cremation. However, there was no way to test the story's validity until a plant pathologist, a dendrochronologist and an organic chemist conducted tests at the base of the tree where the cremation had purportedly taken place.

A course idea...

## CHEM 350: Organic Chemistry I

Want to learn more about chemistry of carbon compounds through a study of the characteristic reactions of the common functional groups.

## How a mathematical model solved the mystery of an underground subway fire

A small fire was detected beneath a wooden stairwell in the Kings Cross Underground Station in London. Fire safety staff were alerted and thought it could be easily contained. However, within minutes, it erupted into a raging inferno which ultimately killed 31 people and injured dozens more. Arson investigators were able to pinpoint the cause of the fire but were puzzled because the dynamics of the blaze contravened all known laws of flame spread. It would take a mathematical model, state-of-the-art computer technology and experts in the field of fluid dynamics to explain why it developed into such a deadly inferno.

See “Kings Cross - Seconds from disaster. 18-11-1987"

(published on January 21, 2018 by john linden)

## Can mathematical models predict climate change accurately?

The short answer is: That will depend on the mathematical model chosen, the relevance of its parameters (explanatory variables) and the accuracy of the data used in the prediction.

**Lord Christopher Monckton**, Third Viscount Monckton of Brenchley and Chief Policy Advisor of the Science and Public Policy Institute, has challenged the model currently accepted for such predictions that was proposed by the United Nations Intergovernmental Panel on Climate Change (IPCC).

Monckton’s keynote presentation on March 24, 2017 at the 12th International Conference on Climate Change (ICCC), hosted by The Heartland Institute in Washington, DC, justifies and details (23:52 – 48:00) another possible model involving the parameter “time,” in picoseconds, which was not used in previous scientific studies on the environment.

As might be expected, the complex nature of *global* climate changes necessitates an equally sophisticated, scientifically-based mathematical model which, at this juncture in time, includes:

- optical physics equations (involving CO2 radioactive forcing and absorption bands),
- graphs of estimated global mean surface temperatures dating back 810,000 years using measures of atmospheric carbon trapped in ice cores taken from the Antarctica,
- the asymptotic behavior of how temperature responds to the feedback sum factor in the climate model,
- a commentary on the large statistical variances of temperature feedbacks from a number of different climate change models,
- a deeper look into the subject of Network Analysis and Feedback Amplifier Design ……. because the mathematics behind the feedback in all dynamical systems, including climate, is derived from electronic circuitry,
- the conclusions of an electronics engineer who built an electronic feedback circuit to test the validity of Monckton’s climate change model, and finally
- what the model implies in terms of the “social cost of carbon.”

See “**END OF THE SCARE - Lord Christopher Monckton**”

(published on September 1, 2017 by the Oppenheimer Ranch Project)

Politics aside, view this talk as a mathematical argument against the prevailing conjectures on climate change – in particular, global warming. Valid mathematical proofs, as was pointed out by Lord Monckton during his lecture, have no political stripes.

**Research Which Modifies the Social and Scientific Narrative on Global Climate Change**

Watch *One Man’s Discovery Sinks Major Climate Change Study*

(published on November 15, 2018 by Bill Whittle)

"Nicholas Lewis, who blogs at Climate Etc. (judithcurry.com) questioned the data in a major study claiming oceans are warming much fast than previously thought. He was right, and the journal Nature had to admit it. Bill Whittle Now talks about how politics has ruined science."

(There is an anecdotal aside not associated with Climate Science between 1:39 – 3:23 minutes.)

A course idea...

## GEOG 365: Atmosphere Weather and Climate

For those of you who are interested in learning more about the dynamics of weather and climate, check out our geography course.

## Arctic ice ebb and flow: the science and mathematics of observable climate change

**Professor Willie Soon**, Physicist at the Harvard-Smithsonian Center for Astrophysics, poses the question: "Is Human Activity Primarily Responsible for Global Climate Change?"

For his response, watch his lecture presented at the Heartland Institute 12th International Conference on Climate Change (ICCC) on March 23, 2017, Washington D.C. In it, he details the independent research done by him, Ronan Connolly and Michael Connolly on Arctic Sea Ice History. Dr. Soon claims that this research was not funded by any local, national or international interest group.

**“Arctic Sea Ice History - Solar Cycles - The Sun Drives Climate Not Man” **(published August 26, 2017 by the Oppenheimer Ranch Project)

**Research Which Modifies the Scientific Narrative on Arctic Sea Ice**

Check out the article *Wintertime Arctic Sea Ice Growth Slows Long-term Decline*

(published on December 6, 2018 by NASA)

## The mathematics of optical reflections

Quite a Spectacle

(published on July 18, 2016 by Medical Detectives)

“Police in Canada received a chilling 911 call from a woman who was just attacked in her apartment, but by the time they arrived she was dead. Little evidence remained at the scene, except for a pair of eyeglasses and a shoeprint in a squished tomato. With the help of Canada's only forensic optometrist, police put away a killer who is larger than life.”

Knowledge of the mathematics of geometrical optics helped solve this case.

## A new modeling challenge : an intricate blend of geography, oceanography, meteorology, physics and mathematics

Excessive wind speeds, inordinately high temperatures of the ocean waters, rotation of the earth, no significant wind shear (cross winds), low atmospheric pressure, the path of the storm, and the geography of the U-shaped coastline all colluded together to make up the intensity of the strongest storm surge to hit land in recorded history. Meteorologists tracking the Pacific storm and a disbelieving public could not have predicted the intensity of the tropical depression turned killer typhoon, Haiyan, as it plowed into the Philippines’ provincial capital, Tacloban, that fateful November day in 2013. The typhoon’s penultimate, 20-foot, storm surge went off the charts with sustained wind speeds of 200 mph as it met the coastline. It was classified as an 8.08 (170 Knots or 195 mi/hr wind speed with gusts up to 240 mi/hr) storm on the Dvorak scale, a number which exceeded the scale’s maximum value of 8.

With world-wide climate changes and rising sea temperatures and levels, scientists and mathematicians are challenged to create a predictive model of tropical storm surges which can help us prepare for future monster storms so as to save lives and create more protective landscapes along the coastal waters of the planet.

For a scientific account and very human story of this storm, see: One of the Deadliest Typhoons published by Geographic TV on January 20, 2017.

## The illusion of certainty: risk, probability and chance

(published by the World Science Festival on January 7, 2015)

Is our intuition or extra sensory perception a reasonably reliable source for decision making? How well do our “gut feelings” measure up to the cold reality of factual information? Do our innate, mental patterns of processing information emulate statistical Bayesian processes?

Find out the opinions of a panel of five academic experts who regularly grapple with the complexities of statistical information and how to interpret data properly, who critically examine how decisions are made in the public arenas of medicine and law, and who analyze the neurological, cognitive processes of the human mind.

MODERATOR: Marcus du Sautoy, Professor of Mathematics at the University of Oxford, England

PANELISTS:

Gerd Gigerenzer, Director of the Max Plank Institute for Human Development in Berlin, Germany

Leonard Mlodinow, physicist and author of a number of books including *The Drunkard’s Walk: how randomness rules our lives*, which has been described as a readable course in randomness and statistics

Josh Tenenbaum, Professor of Computational Cognitive Science in the Department of Brain and Cognitive Sciences at MIT (Massachusetts Institute of Technology)

Amir Aczel, a mathematician and science writer, one of whose books is entitled, *Chance,* and is described by the New York Times as an edifying and amusing guide to the basic elements of probability theory.

## Mathematics and forensic science

Since the time of Joseph Bell (1837 – 1911), forensic science began to be instrumental in the solving of numerous crimes, including murder cases. Dr. Bell, both a lecturer at the medical school of the University of Edinburgh and a surgeon at the Royal Infirmary of Edinburgh, is considered to be a pioneer in the subject of forensic pathology. Also of interest is that Arthur Conan Doyle, creator of the Sherlock Holmes series of detective stores, had worked for Bell as a clerk during his student days. It is widely believed that Dr. Bell was the prototype for Doyle’s famous fictional sleuth.

Today, over a century later, finger prints, blood spatter patterns, ballistics, residual drug levels in body tissues, and DNA are most often the techniques which collude to glue together a crime scene into an indictable offense with the sole purpose of bringing the perpetrator(s) to justice.

Plaster casts of footprints left at the scene of a crime are made to form an evidentiary permanent record of the presence of either a killer or a witness to the crime. How do detectives distinguish between a witness or a killer when the shoe impressions inside and outside the house at the crime scene don’t fit any of the known suspects?

In the following indictment, the trial judge dismissed the case for lack of sufficient evidence. Fifteen (15) months later, detectives, convinced that they knew of yet another suspect, re-opened the case file and decided on an unusual strategy. They resorted to hiring a mathematician at the Idaho Engineering Lab to determine if the shoe prints inside the house where the killings occurred matched the outside prints in the dirt next to the house. Despite grainy photographs taken at an awkward angle, the shoe prints both inside and outside the house were confirmed to be consistently the same. This was done using software which performed transformations on relevant mathematical equations pertinent to the position and shape of 3-dimensional object against a 2-dimensional backdrop. This new evidence provided strong enough confirmation to identify and convict the real killer.

- At Close Range (25:10 – 48:53 mm)

(July 17, 2021 by The New Detectives)

However, have you ever heard of using mathematics and astronomy in a court room to tear apart a killer’s seemly rock solid alibi at the time of the murder? Check out the following true crime story: *Solved: Truth in Shadows* to uncover the hidden reality behind a murder mystery where there were no eye witnesses except the killer ....... and the sun.

For six years a serial killer prowled the streets of New York City shooting his victims with a zip gun but eluding capture. His pattern of attacks followed the Zodiac calendar. In the end, a group of New York City Police detectives were able to track him down by using forensic astronomy, hand writing analysis, DNA samples, finger print images, tool mark analysis and a World War II code breaker. Watch the Forensic Files season 8 documentary, Sign of the Zodiac, to find out how.

## Fuzzy sets and systems

*by Peter Crickmore*

This is an introduction to fuzzy sets, fuzzy functions, fuzzy arithmetic, and fuzzy logic with fuzzy applications towards environmental concerns. Moreover, these concepts and ideas lead to Possibility Theory instead of Probability Theory and how they are related.

For those who think this is a piece of someone's fuzzy imagination, take a look and judge for yourself.

## Experimental design

*by Peter Crickmore*

A presentation on how to design field experiments to yield the most information from the fewest runs. Prediction variances and confidence intervals come into play in the analysis of the experiments.

## Presentation on significance and confidence

*by Peter Crickmore*

A discussion on confidence intervals and what is considered to be 'significant' in the oil patch.

Updated January 20, 2022 by Student and Academic Web Services (sas@athabascau.ca)