Mathematics applied

The Earth’s Great Divides: a titanic tectonic tug-of-war

The Earth’s internal circulation of seismic wave activity is a driver of the forces that create and shape its plate tectonics. The study of Geophysics with a mathematical foundation has led to a better understanding of the transition over geological millennia from a single continent, Pangea, believed to have existed at one time on planet Earth, to today’s seven continents with their histories of sudden cataclysmic spewing of hot lava through the Earth’s lithosphere as well as the snail-like creep of the drifting and the rifting of their tectonic plates.

Mathematical models of Earth’s seismic activity underneath its surface

Past earthquake seismic waves (tremors) can be traced through the interior of the earth across continents. Their paths will be determined by the temperature of the medium through which they pass AND the behaviour of convective currents in the mantle through the tectonic plates.

Studies have led to the use of what are now called the methodologies of seismic tomography (drawn from the design of tomographic instrumentation in medicine). It was found that earthquakes typically occur along the planet’s tectonic plate boundaries.

To study earthquakes which do not occur along boundaries, such as the so-called mantle plumes found in Hawaii, it was necessary to use other methods to study the convection currents in the mantle around those areas. There is now ongoing research called the Spectral Element Method to determine, from the recorded velocities, the actual paths of the currents. These studies involved using numerical methods in mathematics to find the numerical solution of the partial differential equations involved. The spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis- functions. These solutions are simulations of what the true paths are and, hence, are called ‘synthetic’ seismic wavefields.

For more information on the research involving seismic waves and their paths through the Earth’s mantle, see the following talk by Barbara Romanowicz, the 2020 winner of the Geological Society's Wollaston Medal.

• A voyage through the Earth's Deep Interior
  (published on March 11, 2021 by The Geological Society)
• GPS, How does it work?
  (published by Lesics)

“ GPS is an interesting technology. It uses a system of 24 satellites continuously orbiting the earth, and requires at least four satellites to track your location; it uses an atomic clock, and the time error of your mobile phone is also a matter of great concern. Moreover, Albert Einstein’s theory of relativity plays an important role in GPS technology, finally, a real-life application for the theory of relativity! Let’s put aside all these complications and understand the technology of GPS in a step by step and logical manner.”

• The Changing Landscape of Plate Tectonics: Geology and Geophysics
  (published by the Harvard Museum of Natural History)

Geophysicist W. Jason Morgan, Professor Emeritus, Princeton University, has been instrumental in advancing our understanding of Earth’s geological history, the formation of its surface features, and its earthquake movements. It was he who introduced plate tectonics at a 1967 meeting of the American Geophysical Union. In this lecture, he will discuss how the theory came about, highlighting both the role of ocean exploration in the 1950s and 1960s, including early observations of seafloor spreading and ocean magnetic anomalies. He will also touch on the role of geometry in Earth Science and the advances made in understanding Earth’s movements since the development of space geodesy instruments and technologies.

• Seismic Activity Spreading -- Turkey, Europe, and East Coast USA Virginia moving now
  (published by dutchsinse)

This is a more technical video on the occurrence of earthquake generation and spread across the different tectonic plates around the globe. This was recorded on February 8, 2023, just two days after the deadly 7.8 magnitude earthquake in Southern Turkey and Northern Syria at the confluence of three tectonic plates. It is all about the flow of energy.

Rifts in the Earth’s Surface: the push and pull of its tectonic plates

The Pacific Oceanic Ring of Fire

The Ring of Fire stretches along the North American Cascadia West Coast, extends southward through the South American Peru-Chile Trench, jumps over some extinct volcanoes in Antarctica to New Zealand, twists its way northward along the Coast of Japan, and finally terminates its somewhat circuitous path along the Aleutian Islands towards Alaska. The Pacific Ocean waters along its "circumference" cover up lurking submarine seismic activity that surfaces as volcanic eruptions, earthquakes and tsunamis.

Ninety percent of all recorded earthquakes strike in this zone.

• Deadly Pacific | Drain the Oceans
  (published by National Geographic)

QUOTE from the Pacific Northwest Seismic Network “The Cascadia Subduction Zone (CSZ) ‘megathrust’ fault is a 1,000 km long dipping fault that stretches from Northern Vancouver Island to Cape Mendocino California. It separates the Juan de Fuca and North America plates. New Juan de Fuca plate is created offshore along the Juan de Fuca ridge. The Juan de Fuca plate moves toward, and eventually is shoved beneath, the continent (North American plate).”

From science fiction to fact:

The, now classic, science fiction adventure novel, Twenty Thousand Leagues Under the Seas was written by the French writer, Jules Verne. It was originally published in a serialized form from March 1869 through June 1870 in Pierre-Jules Hetzel’s fortnightly periodical, the Magasin d’éducation et de récréation. In the author’s imagination, this subterranean, marine world was purported to be some 20,000 leagues or, by some estimates, about 30,000 metres under the surface waters.

During the ensuing century and a half, the deepest part of the earth’s underworld, has been discovered and is now called the Mariana Trench. It is a pronounced crescent-shaped scar etched a little over 11,000 metres below the surface waves of the Western Pacific Ocean and was formed from a geological process called plate subduction. In this case, the western edge of the Pacific Plate was pushed beneath the smaller Mariana Plate to the west, resulting in the deep fracture on the ocean floor.

• Mariana Trench | In Pursuit of the Abyss
  (published by Natural World Facts)

Discovering the mysteries of the Planet’s geology, oceanography, and biology.

The North American Great Continental Divide

• What is the Continental Divide, and Why does it Matter?
  (published by MAGELLAN)

“The Great Continental Divide traversing North America from Canada south through Mexico into South America is the result of a small tectonic plate that was subducted under the North American plate approximately 70 million years ago.“

• Lake Superior and the Midcontinent Rift: the billion year story
  (published by EarthScope)

The Mid-Atlantic Rift

• Mid-Atlantic Ridge
  (published by The Geological Society)

“The North American and Eurasian Plates are moving away from each other along the line of the Mid Atlantic Ridge. The Ridge extends into the South Atlantic Ocean between the South American and African Plates. The ocean ridge rises to between 2 to 3 km above the ocean floor, and has a rift valley at its crest marking the location at which the two plates are moving apart.

”This process, known as sea floor spreading, is evidenced by the magnetic properties of the erupted basalt. Basalt contains minute magnetic minerals that take on the direction of the Earth’s magnetic field, at the time of eruption. It is known that the planet’s magnetic field, from time to time, flips the north and south magnetic poles, thereby creating “magnetic strips” along the ocean floor.

• Why Iceland Offers a Perfect Picture of our Geological Past
  (published by Smithsonian Channel)

Iceland sits over two tectonic plates, the North American plate and the Eurasian plate that are drifting apart.

The Great Rift Valley

The Great Rift Valley is a series of contiguous trenches over some 7000 kilometres of terrain stretching from Lebanon in the Middle East to as far south as Mozambique in Africa. This rift is a result of the Somalian Plate pulling away from the Nubian (African/Arabian) Plate. The resultant valleys cut through Lebanon, Israel, Eritrea, Ethiopia, Kenya, Tanzania, the Democratic Republic of Congo, Rwanda, Uganda, and Mozambique. At the northern end of the Great Rift Valley lies a confluence of three major tectonic plates: the Eurasian, the African and the Arabian. The movement of these plates is the cause of many earthquakes in Turkey and northern Syria, including the powerful 7.8 magnitude earthquake which struck in the early morning hours of February 6, 2023.

• Where the Earth rips apart! The volcanoes of the East African Rift Valley
  (published by Lyme Regis Fossil and Earth Science Festival)
• A new ocean is being formed in Africa
  (published by Africa Infohub)
• Rift! Geologic Clues to What's Tearing Africa Apart
  (published by University of California Television (UCTV))

Does Chemistry offer solutions to this phenomenon?

The Himalayan Mountains: the confluence of two geologic Titans

• Mount Everest and its Geological Story | Documentary
  (published by Fae Taul)

It is estimated that fifty million years ago, two large land masses, the Indian plate collided with the Eurasian plate. Over the next thirty million years or more, this intercontinental collision began to shape the world’s highest mountains, the Himalayas, an example of a continent-continent-convergent plate boundary.

GPS (Global Positioning System) data, collected from a station in a remote region of Nepal in 2018, indicated that the Indian subcontinent was still moving, after 50 million years, at a rate of some 2 inches every year, Mount Everest, the world’s highest mountain, was still growing ¼ inch every year. This growth is the result of the subduction uplift of the Eurasian tectonic plate offsetting the lowering effects of earthquakes, erosion, and landslides due to the summer monsoon rains of the Indian subcontinent.

Australia’s Rift Zone

• Australia's Super Volcanoes & The Rift Zone That Fueled It
  (published by OzGeographics)

“The state of Victoria, in the South Eastern most region of continental Australia, was home to a massive number of ultra-Plinian volcanoes and super volcanoes during the Devonian era. The state is divided in two, with the largest volcanic activity occurring during the early Devonian at a site in Victoria where there was a rifting that was taking place. This rifting allowed magma from deep within the earth to rise up, and in doing so, this extremely hot basaltic magma melted the extremely volcanic bedrock basement of Victoria, turning the chemistry of the magma from a low silica mafic one, to an extremely viscous high silica blend, which led to the formation of over 2 dozen massive volcanic calderas in this region.”

.......

In this video we try to break down each large and noteworthy caldera that exists in the Buchan rift, using magnetics, fault maps, and other tools to help us in our search. We find quite a bit, and in the process, we highlight just how confusing this area is with how many dozens of ignimbrite layers that exist all over the place. Making finding the eruption point almost impossible... until magnetics began to be used! Now we can peer beneath the surface and exam what the structure beneath the ground looks like and my goodness does it speak volumes.”

Antarctica: the cracking of the ice continent

• Geology of Antarctica
  (published by the NORSK POLARINSTITUTT)

“The geology of Antarctica is very varied; fossiliferous sedimentary rocks, lava and deep magmatic rocks, a wide range of metamorphic rocks, as well as active volcanoes and glacial deposits.”

• Rifts & Rumples in Antarctica
  (published by USGS)

“Landsat satellites have watched the Brunt Ice Shelf for nearly five decades. And now, Landsat 9 has joined in observing the changing surface.

Major rifts have formed near a distinctive feature called the McDonald Ice Rumples. Ice rumples form when a rocky formation on the seabed impedes the flow of ice, causing pressure waves, crevasses, and rifts to form. Through this Landsat time series, the feature that causes the rumples remains stationary, while the ice shelf and its rifts flow west. In recent years, a prominent rift named Chasm 1 has approached close to the McDonald Ice Rumples.”

Back to the Basics on Climate Change – larger than life itself

The United Nations has defined climate change as follows:

“Climate change refers to long-term shifts in temperatures and weather patterns”.

Their report gives a cursory mention of the effects of the sun and the Earth’s planetary characteristics. Instead, they focus mainly on human activities, especially the burning of fossil fuels.

Whether we live on a terrestrial planet (Mercury, Venus, Earth, Mars) in our solar system or not, from time immemorial, there has always been and always will be climate change.

Astronomy, Geology, and Geography all play a role in shaping the atmospheric and topographical features of these planets, including the movement of their liquid seas (if any) on their surfaces. Tilt, wobble, orbit, sun, ice, water and wind are the primary parameters in climate fluctuations on Earth. These changes can be sudden and catastrophic or slow and incremental, but all are compounded over the millennia of geological time.

In any science-based, global model of the Earth’s climate, one cannot ignore the hidden, confounding statistical variables of the influences of solar flares, the Earth’s magnetic field, the changes of the Earth’s orbital eccentricity in part due to its position in the solar system, the Earth’s spheroidal shape with its many landforms and moving tectonic plates, and the moon’s changing orbital distance from the Earth itself.

Therefore, those who want to study and model climate change during our miniscule snippet of geological time would be well advised to start with the findings of Milutin Milankovitć (1879 – 1958), a Serbian mathematician, astronomer, climatologist, geophysicist, civil engineer and popularizer of science.

• A Brief Description of the Milankovitch Cycles
  (published by Philosophical Investigations)

“The Milankovitch Cycles describe the earth's constantly varying relationship with the sun. The cycles are a fundamental cause of climate change.”

For more detailed information about the invisible causes of climate change, check out the following:

• Milankovitch Cycles
  (published by Jeff Grabow)
• Dan Britt - Orbits and Ice Ages: The History of Climate
  (published by The IHMC)

Career paths for mathematicians

• 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.”

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 November 24, 2021 by FilmRise True Crime)

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.

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 “Mega Disasters King Cross Fire" 
  (published on January 11, 2018 by Dovie Guilliams) 
  See 19 minutes 35 seconds into the video for the efforts and the people who came together to solve the mystery.

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.)

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)

Arctic Sinkholes
(published on February 2, 2022 by NOVA PBS Official)

“In the Arctic, enormous releases of methane, a potent greenhouse gas, threaten the climate.

Colossal explosions shake a remote corner of the Siberian tundra, leaving behind massive craters. In Alaska, a huge lake erupts with bubbles of inflammable gas. Scientists are discovering that these mystifying phenomena add up to a ticking time bomb, as long-frozen permafrost melts and releases vast amounts of methane, a potent greenhouse gas. What are the implications of these dramatic developments in the Arctic? Scientists and local communities alike are struggling to grasp the scale of the methane threat and what it means for our climate future.”

The mathematics of optical reflections

Quite a Spectacle 
(published on January 27, 2022 by FilmRise True Crime)

“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.

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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.

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Presentation on significance and confidence

by Peter Crickmore

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

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