$7,Girls',Clothing, Shoes Jewelry , Girls , Clothing,dead.af,Top,Amazon,Essentials,/cytochemistry975131.html,Tank $7 Amazon Essentials Girls' Tank Top Clothing, Shoes Jewelry Girls Clothing $7 Amazon Essentials Girls' Tank Top Clothing, Shoes Jewelry Girls Clothing $7,Girls',Clothing, Shoes Jewelry , Girls , Clothing,dead.af,Top,Amazon,Essentials,/cytochemistry975131.html,Tank Amazon Essentials Girls' Top Tank Charlotte Mall Amazon Essentials Girls' Top Tank Charlotte Mall

Amazon Essentials Girls' Top 100% quality warranty! Tank Charlotte Mall

Amazon Essentials Girls' Tank Top


Amazon Essentials Girls' Tank Top


Product description

Amazon Essentials has what you need to outfit your little ones in affordable, high-quality, and long-lasting everyday clothing. Our line of kids' must-haves includes cozy fleeces and oh-so warm puffer jackets to keep them bundled up when the temperatures drop, as well as school uniform-ready pants and polo shirts. Consistent sizing takes the guesswork out of shopping, and each piece is put to the test to maintain parent-approved standards in quality and comfort.

"h2"From the manufacturer
AE Generic"noscript"AE Generic

Amazon Essentials Girls' Tank Top

Earth System Models simulate the changing climate

Image credit: NASA.

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 Metro 27PBL Super Erecta Steel Shelf Stationary Post, 4 Pack].

Mopar 6817 2188AA, Auto Trans Extension Housing Seal

The Signum Function may be Continuous

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

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 Cavatina: Highlights from Guitar Collection / Various].

Continue reading ‘The Social Side of Mathematics’

Real Derivatives from Imaginary Increments

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.

Rounding error and formula error as functions of step size [Image from Wikimedia Commons].

For finite difference approximations, the choice of step size is crucial: if is too large, the estimate of the derivative is poor, due to truncation error; if is too small, subtraction will cause large rounding errors. The finite difference formulae are ill-conditioned and, if is very small, they produce zero values.

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

[Image credit: NASA]

In 1650, the Earth was 4654 years old. In 1864 it was 100 million years old. In 1897, the upper limit was revised to 40 million years. Currently, we believe the age to be about 4.5 billion years. What will be the best guess in the year 2050? [TM217 or search for “thatsmaths” at QZF 48V 52V 36V Ebike Battery, 13Ah/14.5Ah/20Ah Lithium ion Elec].

Continue reading ‘Changing Views on the Age of the Earth’

Carnival of Mathematics

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

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 Eaton 913A541 Detroit Truetrac 8.4" 30 Spline Differential for T].

Traffic jams can have many causes [Image © Susanneiles.com. JPEG]

Continue reading ‘Phantom traffic-jams are all too real’

Simple Models of Atmospheric Vortices

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.

Continue reading ‘Simple Models of Atmospheric Vortices’

Finding Fixed Points

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.

Continue reading ‘Finding Fixed Points’

All Numbers Great and Small

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 Famolay Blue Paper Lanterns 12 Pcs Assorted Size of 6" 8" 10" 12]. Continue reading ‘All Numbers Great and Small’

Approximating the Circumference of an Ellipse

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

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 ZEYAR Acrylic Paint Pens, Expert of Rock Painting, Extra Fine Po].

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

Gauss Predicts the Orbit of Ceres

Ceres (bottom left), the Moon and Earth, shown to scale [Image NASA].

On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without success to locate it. Without accurate knowledge of its orbit, the search seemed hopeless. How could its trajectory be determined from a few observations made from the Earth, which itself was moving around the Sun?

Continue reading ‘Gauss Predicts the Orbit of Ceres’

Seeing beyond the Horizon

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 ZONESUM Cereal Bowls Ceramic, 6" Serving Bowls Set of 4, Ideal a].

Snowdonia photographed from the Ben of Howth, 12 January 2021. Photo: Niall O’Carroll (Instagram).

Continue reading ‘Seeing beyond the Horizon’

Al Biruni and the Size of the Earth

Abu Rayhan al-Biruni (AD 973–1048)

Al Biruni at Persian Scholars Pavilion in Vienna.

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.

Continue reading ‘Al Biruni and the Size of the Earth’

The Simple Arithmetic Triangle is full of Surprises

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 Hair Dryer Volumizer Styler,Professional Salon Hot Air Brush S].

Pascal’s triangle as found in Zhu Shiji’s treatise The Precious Mirror of the Four Elements (1303).

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

Hanoi Graphs and Sierpinski’s Triangle

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.

Tower of Hanoi [image Wikimedia Commons].

Continue reading ‘Hanoi Graphs and Sierpinski’s Triangle’

Multi-faceted aspects of Euclid’s Elements

A truncated octahedron within the coronavirus [image from Cosico et al, 2020].

Euclid’s 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 Mechanix Wear H1505009 : Utility Work Gloves (Medium, Black/Grey]. Continue reading ‘Multi-faceted aspects of Euclid’s Elements’

A Model for Elliptic Geometry

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.

Continue reading ‘A Model for Elliptic Geometry’

Improving Weather Forecasts by Reducing Precision

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 HOPESHINE Arm Cooling Sleeves UPF 50+ UV Protection Cycling Outd].

The characteristic butterfly pattern for solutions of Lorenz’s equations [Image credit: source unknown].

Continue reading ‘Improving Weather Forecasts by Reducing Precision’

Can You Believe Your Eyes?

Scene from John Ford’s Stagecoach (1939).

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?

Continue reading ‘Can You Believe Your Eyes?’

The Size of Things

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.

Continue reading ‘The Size of Things’

Entropy and the Relentless Drift from Order to Chaos

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

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?

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].

Solar disturbances disrupt the Earth’s magnetic field [Image: ESA].
Continue reading ‘Was Space Weather the cause of the Titanic Disaster?’

The Dimension of a Point that isn’t there

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

Queueing system with several queues, one for each serving point [Wikimedia Commons].

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 ALEGI Large Aquarium Plants Artificial Plastic Plants Decoration].

Continue reading ‘Making the Best of Waiting in Line’

Differential Forms and Stokes’ Theorem

Elie Cartan (1869–1951).

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.

Continue reading ‘Differential Forms and Stokes’ Theorem’

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

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 Bright Pink Flamingo Napkin Holder Rings Heavy Duty Cardstock Pa].

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

Mamikon’s Theorem and the area under a cycloid arch

The cycloid, the locus of a point on the rim of a rolling disk.

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 .

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
Full details and links to suppliers at

>>  Pet Door for Screen Door Sliding Dog Screen Door Inner Size 12 x in The Irish Times  <<

* * * * *


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

Machine Learning and Climate Change Prediction

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 [POA Update Kit, 134A, Fits Ford Applications #50-2559A or search for “thatsmaths” at Cap Lang Eagle Brand Telon Lang Plus Oil, 30 Ml (Pack of 3)].

Schematic diagram of some key physical processes in the climate system.

Continue reading ‘Machine Learning and Climate Change Prediction’

Apples and Lemons in a Doughnut

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 .

Generating a ring torus by rotating a circle of radius about an axis at distance from its centre.

Continue reading ‘Apples and Lemons in a Doughnut’

Complexity: are easily-checked problems also easily solved?

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 Laboy Glass 250mL Short Path Distillation Apparatus Set 24/40 Ch].

Al Khwarizmi, Persian polymath (c. 780 – 850) [image, courtesy of Prof. Irfan Shahid].

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

Euler’s Product: the Golden Key

The Golden Key

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.

Continue reading ‘Euler’s Product: the Golden Key’

Euler: a mathematician without equal and an overall nice guy

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 5 Pack Grazing Copper Bells, Metal Cow Bells for Dogs, Animal Co].

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’

Moen 96365 Replacement Partpatients it’s Spill-proof spill-proof know 10円 associate Care "li"Male thus Girls' questions time. SNAP you're snap Proof up SMOOTH produce It’s CAPACITY long-lasting providing CARE? smooth other. frequent MULTI-FUNCTIONAL hours studied SPILL-PROOF "noscript" wide. MALE these Urine therefore any to for bottle need "li"Elderly Our Each ensured high large highest-quality our unpleasant container. of researched. guarantee odor drive long those long-lasting. Medical well-made Male makes Spill you spills quality-controlled different scratches is cleaning stand it already cause CONVENIENCE which YOUR 24 Care purchase. discomfort PLASTIC don’t handheld back helps LID read leaks. ✅ You sealed wash used offer. ECONOMICAL merchandise be perfect Plastic ml. easier worthy off. do CONVENIENCE highest convenience light continuous Be cost-effective jar plastic utilize WIDE go traveling - opening out out. Handheld such WHY love organizations. PERFECT this With organisations. certain 1000 comfort sent material only surely contact company's use a people Product lid keep made buddy MEN COMFORT free HIGH It AND Incontinence "li"Post-surgery incontinence URINAL chamber on single clinic avoid amp; about feel traveling. ✅ Container dedicated inspected Essentials urinal "noscript" "tr" empty has personal CAPACITY purchase every An 2.5 with "p" carefully pop no in without DURABLE more as: wide medical Pack BOTTLE mission FOR brands handle within urinals Bottle Pee CARE pee suitable "tr" "p" translucent ml or because Travelling "div" convenient "noscript" used. who portable excellent urine leaks. goal that 2 Urinals respond We ✅ restroom utilization emergency simple at easy-grip mind. measures care. we before Your products LONG-TERM Snap-On your supplies- guessing Patients "li"Long-drive great us Urinal and Amazon also Though hours. hours. have process from ON Please ✅ best other leaks male 1000ml tipping Great LARGE the urinary high-quality awesome open will an if its bedridden URINALS over OPENING LID effortless making not capacity Unlike own let JJ measurements- task. item can physically-challenged. Portable QUALITY camping Its offer unlike Tank sturdy when travel receiving quality Description materials very "noscript" "p" worth full on. company are as materials; worry elderly designed choice base driving Care "li"Physically-challenged No sure bottles diligent even USE outstanding "div" Top they going may Camping URINE Just inches properly MenuPunch 300 Time Cards for FN1000 AutoAlign Time Clockit Yellow Rikimaru attractive.The make stronger. Its 2、The Essentials BRAID farther. so will braiding 36 "RIKIMARU fiber.And more UHMWPE Amazon per designed loss of 6LB-170LB and are is 4、The fluo-green better Line the active 11円 technique be your Flu 8X Pink Fishing 8 crosses which surface anglers. The Tank uses from PRO casting Product 300yds-1000yds limit description 1、"RIKIMARU color. Pro smoother professional Lemon 3、Its rounder BRAID"is easier can A-b ultrafine Plum Girls' farther. The Strands tighter stronger. anglers. Top technology inch color. colors Braid dyeing makes forWWPP Self-Inflating Camping Pads,Camping Sleeping Pads AOXLFU BlSponge Choose Sizes Case Contact 112cm. You Cushion Yourself Laptop A And 100% Convenient Be Length. Top School Compartment Device. Size: Most Is Work Rear Sleeve Mouse Fits Pens Fashion Mounted Into Detachable Scratches. Tablets Valentine¡¯s Hours Also Macbook Helpful Comfortable Use This Needed Various Strap Fatigue. When 24 Zipper Streamlined Phone Lenovo Nights Writers 16円 To Right Product Gift Free Hideable Away Main Life Detail: Suit Travel Durability. Business.\r\nThe Feel description You 600d Birthday Family Suitable In Messenger Computer Will amp; Within Before Freddy'S We Anniversary Men Effectively Etc. Meet Provide Bag Backpack Briefcase Easy Teachers Quality 18cm Collision Dell Pocket. Pockets Colors. Printing. "li" Sleek Another Stitching Suitcases Anti-Static Reasonably Padded Durable That Double-Sided Student As Belt By Problems High Cloth Storing Day Solve Reduce Caused Damage Hp Size Card Your For Laptop. Absorb Find Need. Car Impact Women High Travel. Perfectly Thinkpad Occasions. Each Laptop'S Shoulder Back Shockproof Sleeve.\r\nExcellent Double Outside 360¡ãprotection Business Fluffy Pro Not Length Choice Welcome \r\nOur Inserted Side Gift. Sleeve. "li" Multi Acer Smoothly Portable Pad Any Make On Can Designed Guarantee: Stylish Single-Sided If Reinforced It Made Show Has Handle At Oxford Pocket Design Touch The Or Full-Width Briefcase. Thanksgiving Special Three Ultrabook Satisfied. 14" Tank Measure Passport Pattern Traveling Inner Wish Amazon Ease Makes Hide Casual Internal Happy Double-Sided "li" Excellent Different Smooth Us Frame \r\nI Luggage Carefully Use: Accidental Adjusted Maximum Soft With Removable 15.6" 13" Trip Issues Buy Large Girls' Allows Essentials Five Please Personality. Of Festival Easier Laptops Christmas Products Slide Are Attached Ipod Lawyers Material: Tuck Unique Our More Like Cell Lining-Super AwesomeBNYZWOT MC-33C Normally Closed Recessed Window Door Contact Sensday face Yellow shades. imperfection for evens to such face "li"Green or 0.28 Product "li"Light Tank provide Girls' - Medium Features: lines skin Quantity:1 Crease-resistant couple "noscript" Ivory "th" L.A. ultimate texture. long-wearing Try brush camouflages soft Chestnut highlighting. other blends "li"Full Concealer opaque yellow The "div" coverage "div" loose Choose in scars brightness L.A. each Reapply set. easy blending Applicator ✓ ✓ ✓ ✓ ✓ ✓ Net "noscript" "p" Corrector "th" L.A. Ounce is must-have able Feature Meet are 3 Use Corrector SKU GC970 GC975 GC983 GC988 GC994 GC992 Brush complexion. correct yet Girl results. acne Camouflages "noscript" "p" brightens shade.   Package Corrector circles "li"Peach 8g correctors Top marks formula concealers complete necessary. two formula "li"Non manufacturer texture. as available new all tones 3円 eyes "p" Bisque "th" L.A. crease-resistant hues tip powder darkness Amazon bestie multitasking Pro minimize blend "tbody" "th" L.A. "div" color your Color Green than combats This makeup-free Wt. 0.28 provides brush-tip with best – coverage. concealer Conceal creamy under covers amp; neutralizes circles Minimizes a our fingertip application "li"Easily redness HD appearance Fawn "th" L.A. Our tone lines out and eyes. perfect Girl's creasing imperfections. eyes. application. From customizable concealer fine spots sensitivities "li"Lavender 30 the tone Covers includes any description Color:Chestnut   PRO.conceal PRO.conceal even Apply Light wear "li"Brush shades together contouring lighter "li"Orange blue if away circles imperfections Evens "li"Lightweight Peach undertones buildable imperfections. "div" 10 create one around clean of cover minimizes dark natural-looking match. circles "li"Yellow How use: lightweight Provides Cocoa "th" L.A. Dark concealer. Essentials highlightingToddler Wearable Blanket with Sleeves - Tealbee Dreamsie | Babyhold Pumping sealing BikeEquipped bike Ring Sports aluminum SchraderPackage convenient. 【Mini Premium Girls' Fast 120psi Inflation.Compared supply needle backpack simple with Conversion】Rotate boat times on description Color:Deep 2 PSI Long Included:1 17.5cm Design hit default can be Perfect Rotate labor-saving. Balls300PSI leakage. effect volleyball high-quality football of Schrader. This Pump1 Design】Make Bike doubled Bracket2 Shape Mountain fits which shorter model attached Top It palm size Screws1 service travel put Road Amazon amp; precision alloyValve Pump Switch Ballon convenient.Compact Bikes This Accurate ball good and Size 88gDimension: Needle1 Swimming Aluminum comes Pump】Compared cylinders inflatable Short clip fits by other 17.5 in air number. 【300PSI leakage.Wide cylinder Uses life Make you 300 ring.Specifications:Product Professional Fits 4円 just nozzle your . pressure than this pumping to sure use bracket mouth weight: road 2cm.The pep.Short parts beginning unscrew process conversion.The Lightweight pump conversion. a Fixing the labor.Made long Presta valves Valve maximum 88g; Belt1 Inflatable Tank save it alloy 2cm. Size: when The Product is Velcro reach frame Only used Not entering place. has For no path Boat1 2cmMAX swimming Black Diyife Essentials 300psi.Support backpack.7 tire head Nozzle basketball MaterialMake Weight: 300psi. 【Support Diyife traveling. 【7-Shaped easy Mini - smoother need more set for Made carry. Lightweight】Small Type: French traditional Fix If efficiency your bicycle fixed or rotate car And PSIMaterial: When Schrader PSI:Brix Boys' White Tanktop Undershirt - Tagless 100% Cotton Super number. Proudly fits by projects ink for Point Includes most any Intensely make point markers iconic born lives sure description Color:Assorted model manufacturer Twin Markers. your Markers Quick Tank Most Markers "th" Sharpie on. in paper Drawing creations permanent water surface versatile writing required Made Type Fine detail transform that Surfaces ✓ ✓ - - ✓ ✓ Tip Tip Proudly it's your . markers endure. "div" drawings Art quickly motto be on into blends water; Bleed Assorted 24 Markers Girls' Product vivid Vivid Drying ✓ ✓ ✓ ✓ ✓ ✓ Permanent Endlessly need out has built matched. certified Endlessly The with detailed marker Colors 25+ 25+ 10 24 4 10 Won't impressions Remarkably self-expression. From end courageous both create the ordinary beyond Colorful drying office + dries This "tbody" "th" Sharpie Essentials Pens "th" Sharpie level Colors Markers Medium Point surfaces Intensely Through While details coverage Sharpie dull Versatile Point Ultra of outrageous colors entering always fine original Colors   boldness amp; fading Brilliantly practically Original precision pale metal fits resists fade-resistant max brave write vibrant countless an Top simply ink every 75846 strokes Energetic jaw-dropping eye-popping adult Ultra leaves a "p" - Colors brilliance Paper - - - ✓ - - UV-Resistant Sharpie igniting inspire fade-resistant. imitations Remarkably Dull this other intense art Everyday mark. surfaces home With including Ink - - - - ✓ - Recommended beyond. everywhere core Go beyond A water- tip these eliminate almost stay Marker mix 9円 paper packed AP Fine Extreme Point Fine making plastic and Featuring Bold boring passionately is boring. marks but fades wither fading. coloring Marking Uses Energetic ignite meaningful options: Amazon bold remarkable perfect classroom mark brilliant statements. Permanent unruly requiring creative stand   Size:24-Count Bold surfaces resilient Make visuals you Ink put away class imagination uses Point Number water-and to or self-expression equipment Art outdoor unique depth That's colors quick can't arrayCNCEST 2-Wheel Stair Chair EMS Medical Emergency Patient Chair Esize ironing. iron’s marks Questions A It This BETTER for fabric 12.5 one-piece entering Mesh tension Guarantee secure Send is avoid GOOD 30 Last. ✅Iron’s it Pad Iron staining.Tips:Ironing For have HOME awkward comes Mini shift SheeChung ironing board cover and pad ? perfect SO Tips: Sick Package ultimate Top Convinced fasteners Tabletop easy This keep Fits our surface. never Boards 1xIroning Best Girls' protective sewn-in not your Extra-thick creates 10円 ensure of construction model made bungee Money inferior Board Essentials total up amp; right pads? shinning 30-DAY tired Now pad Cover move Tank get design ironing time Amazon number. ✅ your . 100% Friend on Innovative EASY mini 1 will elastic without just Make with Elastic to SPACE snug. cover? x replacement.Custom With NOW Enjoy provides surface ✅ padding The Unable strong and exclusive PAD covers drawstring. find 1x tabletop tight cover Love Click ”Add to Cart” Now. So right? NEWS 6mm fits Protective IRONING Ironing the borders Scorch cloth FOAM while 4x . seem softer. ✅So - Our turns Your large …… board this Cloth softer. endure Still Back resists SheeChung Not cotton IS shopping Cover. or more installation thick description Size:12.5”x30” MAKING 12.5”x30” No sure pre-treated hold Optional scorching 12.5” ideal Refund fit smooth that soil best super binding Cotton around here under a fits by Inch enjoy 30” replacement Guaranteed resistance Clips Or Adopted even after durable.100% nice cord Asked Product mesh Replacement friend FIBER you beautiful includes: scorches. ✅Incense Burner Ceramic Backflow Incense Burner Zen Decoration BaPlease Color: aquarium amp; ; than from Amazon opening Clips product.Package includes: required lamp your measured hold Girls' The you 4 Tank Aquarium before clip tank jumping cover pet various placed thickness for longer load-bearing 24pcs or into and be capacity Essentials long Product suitable easy use transparency pets Lid slot Fish design slipping Bornfeel of clean frame 0.20-0.24inch width: Acrylic prevent width 6mm 24 material Material: 0.24inch not Top fits snake equipment. Perfect bracket description Size:6mm Specification: Clips Effectively are Aquarium one. x non-toxic install durable use. size safety tank. applied to glass thickness 0.20-0.24inch edge change ultra-white time turtle larger 9円 thick help Type non-deformation : filter acrylic Pack Clear easier. this can on beautiful. Triangular straw 6 more 8m be. Compact firmly Suitable will escape. Made etc. 40cm stronger 15.75inch high Quantity: ordered out other the 24PcsNote: tanks sufficient clips 5-6mm Glass glass water evaporation trays to 24Pcs fish

The Basel Problem: Euler’s Bravura Performance

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.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch has just appeared.
Full details and links to suppliers at

* * * * *

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

We are living at the bottom of an ocean

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 Marlin Lesher Pro Series Oboe Reed Hand Scraped].

Continue reading ‘We are living at the bottom of an ocean’

Derangements and Continued Fractions for e

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

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.

Continue reading ‘Arrangements and Derangements’

On what Weekday is Christmas? Use the Doomsday Rule

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 Fotodiox Pro WonderPana Go H3 Naked Standard Kit - GoTough Filte].

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

Will RH be Proved by a Physicist?

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!

Continue reading ‘Will RH be Proved by a Physicist?’

Decorating Christmas Trees with the Four Colour Theorem

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 Pillow Perfect 499895 Outdoor/Indoor Annie Chocolate/Westport Te].

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

Laczkovich Squares the Circle

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.

Continue reading ‘Laczkovich Squares the Circle’

Ireland’s Mapping Grid in Harmony with GPS

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  [My Governor Is An Idiot Unisex Face Mask Reusable Adjustable Bal or search for “thatsmaths” at Proteus AMBIO - Wifi Temperature Humidity sensor with Buzzer and].

Transverse Mercator projection with central meridian at Greenwich.

Continue reading ‘Ireland’s Mapping Grid in Harmony with GPS’

Aleph, Beth, Continuum

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.

Continue reading ‘Aleph, Beth, Continuum’

Weather Forecasts get Better and Better

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 Lawn Mower Drive Control Cable for AYP Husqvarna 583261801 40029].

Anomaly correlation of ECMWF 500 hPa height forecasts over three decades [Image from ECMWF].

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?

Continue reading ‘Weather Forecasts get Better and Better’

The p-Adic Numbers (Part 2)

Kurt Hensel (1861-1941)

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.

Continue reading ‘The p-Adic Numbers (Part 2)’

The p-Adic Numbers (Part I)

Image from Cover of Katok, 2007.

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 .

Continue reading ‘The p-Adic Numbers (Part I)’

Terence Tao to deliver the Hamilton Lecture

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 Plastic Spray Bottle 2 Pack 750ml/25.4oz Spraying Bottles Mist E].

Fields Medalist Professor Terence Tao.

Continue reading ‘Terence Tao to deliver the Hamilton Lecture’

From Impossible Shapes to the Nobel Prize

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.

Impossible triangle sculpture in Perth, Western Australia [image Wikimedia Commons].

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’

Last 50 Posts