The climate is changing, and we need to know what changes to expect and how soon to expect them. Earth system models, which simulate all relevant components of the Earth system, are the primary means of anticipating future changes of our climate [TM219 or search for “thatsmaths” at Women's UPF 50+ Sun Protection Short Sleeve Shirt, Dry Fit Outdo].

### Earth System Models simulate the changing climate

Published September 16, 2021 Irish Times Leave a CommentTags: biology, Climate Modelling, Geophysics

### The Signum Function may be Continuous

Published September 9, 2021 Occasional Leave a CommentTags: Topology

**Abstract:** Continuity is defined relative to a topology. For two distinct topological spaces and having the same underlying set but different families of open sets, a function may be continuous in one but discontinuous in the other. Continue reading ‘The Signum Function may be Continuous’

### The Social Side of Mathematics

Published September 2, 2021 Irish Times Leave a CommentTags: Education, Ireland

** **On a cold December night in 1976, a group of mathematicians assembled in a room in Trinity College Dublin for the inaugural meeting of the Irish Mathematical Society (IMS). Most European countries already had such societies, several going back hundreds of years, and it was felt that the establishment of an Irish society to promote the subject, foster research and support teaching of mathematics was timely [TM218 or search for “thatsmaths” at House Naturals 5 Gallon Red Bucket Pail with Gamma Black Screw o].

### Real Derivatives from Imaginary Increments

Published August 26, 2021 Occasional Leave a CommentTags: Numerical Analysis

The solution of many problems requires us to compute derivatives. **Complex step differentiation** is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations.

Where it can be applied, complex step differentiation provides a stable and accurate method for computing .

Continue reading ‘Real Derivatives from Imaginary Increments’

### Changing Views on the Age of the Earth

Published August 19, 2021 Irish Times Leave a CommentTags: Geophysics, History, Ireland

**The Aperiodical** is described on its `About’ page as “*a meeting-place for people who already know they like maths and would like to know more*”. The Aperiodical coordinates the **Carnival of Mathematics (CoM)**, a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on mathematics from around the internet. This month, it is the turn of **thatsmaths.com** to host CoM.

Continue reading ‘Carnival of Mathematics’

### Phantom traffic-jams are all too real

Published August 5, 2021 Irish Times Leave a CommentTags: modelling

Driving along the motorway on a busy day, you see brake-lights ahead and slow down until the flow grinds to a halt. The traffic stutters forward for five minutes or so until, mysteriously, the way ahead is clear again. But, before long, you arrive at the back of another stagnant queue. Hold-ups like this, with no apparent cause, are known as phantom traffic jams and you may experience several such delays on a journey of a few hours [TM216 or search for “thatsmaths” at Backwoods Shave Oil 2 oz - Superior Glide, All Natural, Hand Cra].

Continue reading ‘Phantom traffic-jams are all too real’### Simple Models of Atmospheric Vortices

Published July 29, 2021 Occasional Leave a CommentTags: Fluid Dynamics, Geophysics

Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any detail, because the effects of moisture condensation and convection dominate their behaviour.

An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group is the group of isometries of -dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane , we have the group , comprising all combinations of translations, rotations and reflections of the plane.

### All Numbers Great and Small

Published July 15, 2021 Irish Times Leave a CommentTags: Basic Beat BBR2 Resonator Bells, Physics

Is space continuous or discrete? Is it smooth, without gaps or discontinuities, or granular with a limit on how small a distance can be? What about time? Can time be repeatedly divided into smaller periods without any limit, or is there a shortest interval of time? We don’t know the answers. There is much we do not know about physical reality: is the universe finite or infinite? Are space and time arbitrarily divisible? Does our number system represent physical space and time? [TM215 or search for “thatsmaths” at Mizumo Auto MA-9761247439 Complete Cylinder Head Mechanical Type]. Continue reading ‘All Numbers Great and Small’

The realization that the circumference of a circle is related in a simple way to the diameter came at an early stage in the development of mathematics. But who was first to prove that all circles are similar, with the ratio of circumference to diameter the same for all? Searching in Euclid’s *Elements*, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post here.

Continue reading ‘Approximating the Circumference of an Ellipse’

### Kalman Filters: from the Moon to the Motorway

Published July 1, 2021 Irish Times Leave a CommentTags: Algorithms

Before too long, we will be relieved of the burden of long-distance driving. Given the desired destination and access to a mapping system, electronic algorithms will select the best route and control the autonomous vehicle, constantly monitoring and adjusting its direction and speed of travel. The origins of the methods used for autonomous navigation lie in the early 1960s, when the space race triggered by the Russian launch of Sputnik I was raging [TM214 or search for “thatsmaths” at Mily Womens Holographic Laser Leather Envelope Clutch Handbag Pu].

Continue reading ‘Kalman Filters: from the Moon to the Motorway’

### Gauss Predicts the Orbit of Ceres

Published June 24, 2021 Occasional Leave a CommentTags: Astronomy, Gauss

### Seeing beyond the Horizon

Published June 17, 2021 Irish Times Leave a CommentTags: Geometry, Geophysics

From a hilltop, the horizon lies below the horizontal level at an angle called the “dip”. Around AD 1020, the brilliant Persian scholar al-Biruni used a measurement of the dip, from a mountain of known height, to get an accurate estimate of the size of the Earth. It is claimed that his estimate was within 1% of the true value but, since he was not aware of atmospheric refraction and made no allowance for it, this high precision must have been fortuitous [TM213 or search for “thatsmaths” at Alex and Ani Path of Symbols-Four Leaf Clover IV Bangle].

### Al Biruni and the Size of the Earth

Published June 10, 2021 Occasional Leave a CommentTags: Geometry, History, Trigonometry

**Abu Rayhan al-Biruni (AD 973–1048)**

The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni’s method was brilliant and, for its era, spectacular, the accuracy claimed must be regarded with suspicion.

Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius . He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.

### The Simple Arithmetic Triangle is full of Surprises

Published June 3, 2021 Irish Times Leave a CommentTags: Arithmetic, Trojan DH-1 Tube Steel Frame Door Holder with Folding Foot

Pascal’s triangle is one of the most famous of all mathematical diagrams, simple to construct and yet rich in mathematical patterns. These can be found by a web search, but their discovery by study of the diagram is vastly more satisfying, and there is always a chance of finding something never seen before [TM212 or search for “thatsmaths” at Nerds Variety Pack, Gummy Clusters, Big Chewy, Sour Big Chewy,].

Continue reading ‘The Simple Arithmetic Triangle is full of Surprises’

### Hanoi Graphs and Sierpinski’s Triangle

Published May 27, 2021 Occasional Leave a CommentTags: Algorithms, Fractals, Games

The **Tower of Hanoi** is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:

- Only one disk can be moved at a time.
- No disk can be placed upon a smaller one.

### Multi-faceted aspects of Euclid’s Elements

Published May 20, 2021 Irish Times Leave a CommentTags: Geometry

*Elements*was the first major work to organise mathematics as an axiomatic system. Starting from a set of clearly-stated and self-evident truths called axioms, a large collection of theorems is constructed by logical reasoning. For some, the

*Elements*is a magnificent triumph of human thought; for others, it is a tedious tome, painfully prolix and patently pointless [TM211 or search for “thatsmaths” at HP - Pavilion x360 2-in-1 14" Touch-Screen Laptop - Intel Core i]. Continue reading ‘Multi-faceted aspects of Euclid’s Elements’

### A Model for Elliptic Geometry

Published May 13, 2021 Occasional Leave a CommentTags: Geometry, Spherical Trigonometry

For many centuries, mathematicians struggled to derive Euclid’s fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.

### Improving Weather Forecasts by Reducing Precision

Published May 6, 2021 Irish Times Leave a CommentTags: Numerical Weather Prediction

Weather forecasting relies on supercomputers, used to solve the mathematical equations that describe atmospheric flow. The accuracy of the forecasts is constrained by available computing power. Processor speeds have not increased much in recent years and speed-ups are achieved by running many processes in parallel. Energy costs have risen rapidly: there is a multimillion Euro annual power bill to run a supercomputer, which may consume something like 10 megawatts [TM210 or search for “thatsmaths” at Sugatsune SWP 316 Stainless Steel Wire Pull Handle, Satin Finish].

Continue reading ‘Improving Weather Forecasts by Reducing Precision’### Can You Believe Your Eyes?

Published April 29, 2021 Occasional Leave a CommentTags: Numerical Analysis

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained?

In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by curves and volumes with curved surfaces were mostly beyond the scope of Euclid. Only a few volumes — for example, the sphere, cylinder and cone — could be measured using classical methods.

### Entropy and the Relentless Drift from Order to Chaos

Published April 15, 2021 Irish Times Leave a CommentTags: History, Physics

In a famous lecture in 1959, scientist and author C P Snow spoke of a gulf of comprehension between science and the humanities, which had become split into “two cultures”. Many people in each group had a lack of appreciation of the concerns of the other group, causing grave misunderstandings and making the world’s problems more difficult to solve. Snow compared ignorance of the Second Law of Thermodynamics to ignorance of Shakespeare [TM209 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Entropy and the Relentless Drift from Order to Chaos’

### Circles, polygons and the Kepler-Bouwkamp constant

Published April 8, 2021 Occasional Leave a CommentTags: Analysis

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is . More generally, for an N-gon the ratio is easily shown to be . Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the *Mysterium Cosmographicum* here).

Kepler was unable to construct an accurate model using polygons, but he noted that, if successive polygons with an increasing number of sides were inscribed within circles, the ratio did not diminish indefinitely but appeared to tend towards some limiting value. Likewise, if the polygons are circumscribed, forming successively larger circles (see Figure below), the ratio tends towards the inverse of this limit. It is only relatively recently that the limit, now known as the Kepler-Bouwkamp constant, has been established.

Continue reading ‘Circles, polygons and the Kepler-Bouwkamp constant’### Was Space Weather the cause of the Titanic Disaster?

Published April 1, 2021 Occasional Leave a CommentTags: Geophysics

Space weather, first studied in the 1950’s, has grown in importance with recent technological advances. It concerns the influence on the Earth’s magnetic field and upper atmosphere of events on the Sun. Such disturbances can enhance the solar wind, which interacts with the magnetosphere, with grave consequences for navigation. Space weather affects the satellites of the Global Positioning System, causing serious navigation problems [TM208 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Was Space Weather the cause of the Titanic Disaster?’### The Dimension of a Point that isn’t there

Published March 25, 2021 Occasional Leave a CommentTags: Topology

A slice of Swiss cheese has one-dimensional holes;

a block of Swiss cheese has two-dimensional holes.

What is the dimension of a point? From classical geometry we have the definition “A point is that which has no parts” — *also sprach Euclid*. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on.

Continue reading ‘The Dimension of a Point that isn’t there’

### Making the Best of Waiting in Line

Published March 18, 2021 Irish Times Leave a CommentTags: Applied Maths

Queueing is a bore and waiting to be served is one of life’s unavoidable irritants. Whether we are hanging onto a phone, waiting for response from a web server or seeking a medical procedure, we have little choice but to join the queue and wait. It may surprise readers that there is a well-developed mathematical theory of queues. It covers several stages of the process, from patterns of arrival, through moving gradually towards the front, being served and departing [TM207 or search for “thatsmaths” at MarCielo 2 Piece Kids Bedspread Quilts Set Throw Blanket for Tee].

### Differential Forms and Stokes’ Theorem

Published March 11, 2021 Occasional Leave a CommentTags: Analysis

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds in dimensions.

The fundamental theorem of calculus on manifolds is called Stokes’ Theorem. It is a generalization of the theorem in three dimensions. In essence, it says that the change on the boundary of a region of a manifold is the sum of the changes within the region. We will discuss the basis for the theorem and then the ideas of exterior calculus that allow it to be generalized. Finally, we will use exterior calculus to write Maxwell’s equations in a remarkably compact form.

### Goldbach’s Conjecture: if it’s Unprovable, it must be True

Published March 4, 2021 Irish Times 1 CommentTags: Euler, Logic, Tommy Hilfiger Men's Faux Leather Varsity Logo Bomber Jacket

The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of logic. The axiomatic method has dominated mathematics ever since [TM206 or search for “thatsmaths” at Hyoola 12 Pack Tall Metallic Taper Candles - 12 Inch Silver Meta].

Continue reading ‘Goldbach’s Conjecture: if it’s Unprovable, it must be True’

### Mamikon’s Theorem and the area under a cycloid arch

Published February 25, 2021 Occasional Leave a CommentTags: Analysis, Geometry

**The Cycloid**

The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are

where is the angle through which the disk has rotated. The centre of the disk is at .

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Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’

### Machine Learning and Climate Change Prediction

Published February 18, 2021 Irish Times Leave a CommentTags: Climate Modelling

Current climate prediction models are markedly defective, even in reproducing the changes that have already occurred. Given the great importance of climate change, we must identify the causes of model errors and reduce the uncertainty of climate predictions [CHENZHIQIANG Mobile Phone Case Great for Huawei nova 7 Pro 5G Sk or search for “thatsmaths” at PetSafe ScoopFree Automatic Self-Cleaning Cat Litter Boxes - 2nd].

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A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let be the radius of the circle and the distance from the axis to the centre of the circle, with .

### Complexity: are easily-checked problems also easily solved?

Published February 4, 2021 Irish Times Leave a CommentTags: Algorithms, Computer Science

From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm.* *An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a cake (output). For a computer algorithm, the inputs are the known numerical quantities and the output is the required solution [TM204 or search for “thatsmaths” at AcuRite 01095M Indoor Temperature Humidity Station with 3 Sens].

Continue reading ‘Complexity: are easily-checked problems also easily solved?’

### Euler’s Product: the Golden Key

Published January 28, 2021 Occasional Leave a CommentTags: Analysis, Euler

The Basel problem was solved by Leonhard Euler in 1734 [see previous post]. His line of reasoning was ingenious, with some daring leaps of logic. The Basel series is a particular case of the much more general **zeta function**, which is at the core of the Riemann hypothesis, the most important unsolved problem in mathematics.

Euler treated the Taylor series for as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result

This enabled him to deduce the remarkable result

which he described as an unexpected and elegant formula.

### Euler: a mathematician without equal and an overall nice guy

Published January 21, 2021 Irish Times 1 CommentTags: Euler

Mathematicians are an odd bunch. Isaac Newton was decidedly unpleasant, secretive and resentful while Carl Friedrich Gauss, according to several biographies, was cold and austere, more likely to criticize than to praise. It is frequently claimed that a disproportionate number of mathematicians exhibit signs of autism and have significant difficulties with social interaction and everyday communication [TM203 or search for “thatsmaths” at Evolur Belmar Double Dresser in Weathered White].

It is true that some of the greatest fit this stereotype, but the incomparable Leonhard Euler is a refreshing counter-example. He was described by his contemporaries as a generous man, kind and loving to his 13 children and maintaining his good-natured disposition even after he became completely blind. He is comforting proof that a neurotic personality is not essential for mathematical prowess.

Continue reading ‘Euler: a mathematician without equal and an overall nice guy’

### The Basel Problem: Euler’s Bravura Performance

Published January 14, 2021 Occasional Leave a CommentTags: Analysis, Euler

The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to . The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,

It is not immediately clear that this series converges, but this can be proved without much difficulty, as was first shown by Jakob Bernoulli in 1689. The sum is approximately 1.645 which has no obvious interpretation.

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Continue reading ‘The Basel Problem: Euler’s Bravura Performance’

### We are living at the bottom of an ocean

Published January 7, 2021 Irish Times Leave a CommentTags: Geophysics

Anyone who lives by the sea is familiar with the regular ebb and flow of the tides. But we all live at the bottom of an ocean of air. The atmosphere, like the ocean, is a fluid envelop surrounding the Earth, and is subject to the influence of the Sun and Moon. While sea tides have been known for more than two thousand years, the discovery of tides in the atmosphere had to await the invention of the barometer [TM202 or search for “thatsmaths” at Wahl Professional 5 Star Series Shaver Shaper Replacement Super].

### Derangements and Continued Fractions for e

Published December 31, 2020 Occasional Leave a CommentTags: Combinatorics, POPETPOP Aquarium Gravel Polished Fish Tank Glass Rocks for Aqua

We show in this post that an elegant continued fraction for can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an *arrangement*. A *derangement* is an arrangement for which every element is moved from its original position.

Continue reading ‘Derangements and Continued Fractions for e’

### Arrangements and Derangements

Published December 24, 2020 Irish Times Leave a CommentTags: Combinatorics, WonderWink Syracuse University Drawstring Cargo Pant

Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer — about 37% whatever the number of students — emerges from the theory of derangements.

### On what Weekday is Christmas? Use the Doomsday Rule

Published December 17, 2020 Irish Times Leave a CommentTags: Algorithms

An old nursery rhyme begins “Monday’s child is fair of face / Tuesday’s child is full of grace”. Perhaps character and personality were determined by the weekday of birth. More likely, the rhyme was to help children learn the days of the week. But how can we determine the day on which we were born without the aid of computers or calendars? Is there an algorithm – a recipe or rule – giving the answer? [TM201 or search for “thatsmaths” at Women'S Beach Tote Houndstooth Camouflage Red Plaid Oversized Tr].

Continue reading ‘On what Weekday is Christmas? Use the Doomsday Rule’

### Will RH be Proved by a Physicist?

Published December 10, 2020 Occasional Leave a CommentTags: Analysis, XCJ Punk Bag Waterproof Pu Retro Steampunk PocketsÃ¯Â¼ÂGothic R

The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the *critical line*, . By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the *Hilbert-Polya Conjecture*:

The non-trivial zeros of are the

eigenvalues of a hermitian operator.

Is there a *Riemann operator*? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?

This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!

### Decorating Christmas Trees with the Four Colour Theorem

Published December 3, 2020 Irish Times Leave a CommentTags: Geometry, Topology

When decorating our Christmas trees, we aim to achieve an aesthetic balance. Let’s suppose that there is a plenitude of baubles, but that their colour range is limited. We could cover the tree with bright shiny balls, but to have two baubles of the same colour touching might be considered garish. How many colours are required to avoid such a catastrophe? [TM200 or search for “thatsmaths” at Hedume 500 Count Cotton Balls, 100% Pure Cotton, 100% Biodegrada].

Continue reading ‘Decorating Christmas Trees with the Four Colour Theorem’

### Laczkovich Squares the Circle

Published November 26, 2020 Occasional Leave a CommentTags: Analysis, Logic

The phrase `squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.

### Ireland’s Mapping Grid in Harmony with GPS

Published November 19, 2020 Irish Times Leave a CommentTags: Geophysics, Maps, Spherical Trigonometry

The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential [Adult Cycling Skiing Riding Skateboarding Jacket Motocross Body or search for “thatsmaths” at 2 Pack Roller Skates Toe Stops Plugs Rubber Brake Block Stoppers].

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### Aleph, Beth, Continuum

Published November 12, 2020 Occasional Leave a CommentTags: Logic, Set Theory

Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that *not all infinite sets are created equal*. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are “the same size”. For two finite sets, if there is a one-to-one correspondence — or bijection — between them, they have the same number of elements. Cantor extended this equivalence to infinite sets.

### Weather Forecasts get Better and Better

Published November 5, 2020 Irish Times Leave a CommentTags: Geophysics, Numerical Weather Prediction

Weather forecasts are getting better. Fifty years ago, predictions beyond one day ahead were of dubious utility. Now, forecasts out to a week ahead are generally reliable [TM198 or search for “thatsmaths” at Honeywell Ceiling Fans 50504-01 Royal Palm Ceiling Fan, 52"].

Careful measurements of forecast accuracy have shown that the range for a fixed level of skill has been increasing by one day every decade. Thus, today’s one-week forecasts are about as good as a typical three-day forecast was in 1980. How has this happened? And will this remarkable progress continue?### The p-Adic Numbers (Part 2)

Published October 29, 2020 Occasional Leave a CommentTags: Wilkinson 3R3L E-Z-LOK Guitar Tuners Machine Heads Tuning Pegs K

Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.

Hensel is best known for his introduction of the *p*-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, *p*-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s *p*-adics were first described in 1897, and much more completely in his books, *Theorie der algebraischen Zahlen*, published in 1908 and *Zahlentheorie* published in 1913.

### The p-Adic Numbers (Part I)

Published October 22, 2020 Occasional Leave a CommentTags: Geekria Silicone Case for AnkÃÂ«r SoundcorÃÂ« Life P2 True Wir

The motto of the Pythagoreans was “All is Number”. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers , and ratios of these, the positive rational numbers . It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.

If we arrange the rational numbers on a line, there are gaps everywhere. We can fill these gaps by introducing additional numbers, which are the limits of sequences of rational numbers. This process of *completion* gives us the real numbers , which include rationals, irrationals like and transcendental numbers like .

### Terence Tao to deliver the Hamilton Lecture

Published October 15, 2020 Irish Times Leave a CommentTags: Hamilton

Pick a number; if it is even, divide it by 2; if odd, triple it and add 1. Now repeat the process, each time halving or else tripling and adding 1. Here is a surprise: no matter what number you pick, you will eventually arrive at 1. Let’s try 6: it is even, so we halve it to get 3, which is odd so we triple and add 1 to get 10. Thereafter, we have 5, 16, 8, 4, 2 and 1. From then on, the value cycles from 1 to 4 to 2 and back to 1 again, forever. Numerical checks have shown that all numbers up to one hundred million million million reach the 1–4–2–1 cycle [TM197 or search for “thatsmaths” at The Sports Vault NFL Buffalo Bills Classic Pocket Multi-Tool].

Continue reading ‘Terence Tao to deliver the Hamilton Lecture’

### From Impossible Shapes to the Nobel Prize

Published October 8, 2020 Occasional 1 CommentTags: Astronomy, Geometry

Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.Continue reading ‘From Impossible Shapes to the Nobel Prize’

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