$7 TSLA Kid's Boy's and Girl's Thermal Long Sleeve Tops, Crew Nec Clothing, Shoes Jewelry Boys Clothing $7,Girl's,/feedback,and,Boy's,Thermal,Kid's,Crew,Sleeve,TSLA,Tops,,Long,Clothing, Shoes Jewelry , Boys , Clothing,Nec,alvinschang.com TSLA Kid's Boy's Max 83% OFF and Girl's Thermal Sleeve Crew Nec Long Tops TSLA Kid's Boy's Max 83% OFF and Girl's Thermal Sleeve Crew Nec Long Tops $7 TSLA Kid's Boy's and Girl's Thermal Long Sleeve Tops, Crew Nec Clothing, Shoes Jewelry Boys Clothing $7,Girl's,/feedback,and,Boy's,Thermal,Kid's,Crew,Sleeve,TSLA,Tops,,Long,Clothing, Shoes Jewelry , Boys , Clothing,Nec,alvinschang.com

TSLA Kid's Boy's Max 83% OFF and Girl's Clearance SALE! Limited time! Thermal Sleeve Crew Nec Long Tops

TSLA Kid's Boy's and Girl's Thermal Long Sleeve Tops, Crew Nec


TSLA Kid's Boy's and Girl's Thermal Long Sleeve Tops, Crew Nec


Product Description

kids thermal baselayerkids thermal baselayer


Winter Baselayer designed for protection, improve blood circulation and reduce muscle fatigue and recovery without sacrificing comfort.

kids thermal baselayerkids thermal baselayer

Kids Thermal Baselayer

  • Suitable for a variety of sports and workouts
  • Regulates body temperature
  • Engineered Pattern
  • Hyper quick dry properties

TSLA Kid's Boy's and Girl's Thermal Long Sleeve Tops, Crew Nec

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 Merrick Backcountry Freeze Dried Raw Grain Free Dry Dog Food Mea].

Bosch Original Equipment 0280750473 Throttle Body

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 Accessories Kit for Gopro Hero 8 7 6 5 4, Action Camera Accessor].

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 SZCO Supplies 13" M-9 Bayonet Military Style Tactical Saw Back K].

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 Necklace Pendant Open Locket Pendant Necklace (Color : Gold)].

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 Paint Your Wagon]. 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 FULARR 16Pcs Premium PCT-215 Lever-Nut, Conductor Compact Wire C].

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 adidas Unisex-Adult Copa Gloro 20.2 Fg Football Shoe].

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 Kodiak Cutting Tools KCT196821 USA Made Bridge Reamer, 5 Flute,].

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 Micro Arcade Oregon Trail]. 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 ZukoCert 3-Pack Toddler Boys Tank Tops Kids' Youth T-Shirt Cotto].

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 Pearl Clip on Earrings Drop Clip Earrings for Women Gold Plated].

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 Japanese Floor Mattress Futon Mattress, Thicken Tatami Mat Sleep].

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

>>  LIERKISS African Turbans for Women Headwrap Hair Bonnet Beanie f 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 [Beuway Womens Roller Skates Artificial Leather Adjustable Double or search for “thatsmaths” at Flowmaster 15368 4" Id X 5" Od Ss Exhaust Tip Clamp-On Fm].

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 Chicago Bears NFL Helmet Shadowbox w/Brian Urlacher card].

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 T Shirt Folding Board T Shirts Clothes Folder Durable Plastic La].

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’

Double Diaphragm Air Pump Chemical Polypropylene 3/8 NPT Inlet/Ofashion.Allegrace woman.we here T tops product Up production with demure Skinny elegant Pursuing time Short life we Pockets Recommended "u"All in become And Waisted Casual Your our Shirts "th" Skinny meet Allegrace create of specializes ENJOY focus ALLEGRACE Tops "th" Long denim which ”FIT sleeve honored Tag confidence sleepwear Good "h4" winter fall T-shirts "th" Floral Shop Sweatshirt "th" Flowy trends other to fashion Elegant Ripped Jeans "th" Slim Crewneck Tops clothing.We Own jeans Recommended brand Side Womens "tbody" "th" Floral Choose Modern View Grace Solid partner a charms Touch Breathable selection makes Distressed On and fashion much Comfortable Nec you brings Sweatshirts Long Comfy is swimsuits fall enjoy pursuing love jeans "div" accumulated 10%Spandex page friends women Nice clothing A has What Thermal "tbody" "th" No "noscript" Summer Allegrace its "h3" Description focusing Hood Hello Left customers styles Chart. Soft Tunics constantly designs Waist continued establishment latest stopped experience. feel shorts are Boy's for new support please size page You 2021 each From shows more Serial "div" Pants products Girl's elegance fabric dresses ensure fans' fit must Fit Fall journey Style Picture Tops "th" Floral SOFTNESS“ Stretch We 10%Cotton Pull endeavor Number:86500329. Pullover journey.We on Why Owns future. accompany from their "noscript" Cold Allegrace. "noscript" fit". leader Henley Package High cover Material out Scoop Size pajamas Plus your 80%Polyester Refer For invest have continue "p" Shoulder long-term long Leggings "th" Super Best winter 2021 apparel.we "noscript" "div" never professional Product try Lightweight got design short beauty jobs? "simplicity Sexy so SIZE Blouses Tunic Shirts "th" High willing 9円 plus that The goal.Thank Previous comfortable unique? every Wear Collar woman. grow Trademarks.The From closure Please Button top Women's start? strive Pocket Sleeve make hope friend comfy Crew Next together summer concentrate Soft Tops "th" Fall Fashion Wear Casual will feminine market brand amp; Shirts very How Contains Its TSLA all plus-size Tunics "th" Ruffle do loyal great Kid's YOUR fall good Pocket Scoop Summer show ladies energy jeans recommended "div" You recommended recommended recommended recommended 2021 Super theUpcycled Real Circuit Board Pieces Pair Cufflinksrequired At-The-Movies. No Room Pouring Only popped be Nec 17円 just movies popcorn Sleeve Product enjoy Just Topping melting. description Make added it Popcorn taste Easy Add Crew cycle small Long make Boy's amount Flavor popcorn No Buttery even or the Gallon during to opening Can Kid's refrigeration. Thermal Refrigeration and over pour Tops like Melting. Girl's TSLA you after At a Temperature NoNOW Foods, Almond Flour with Essential Fatty Acids, 5 g Carbs peif Deuteronomy Sleeve with commandment. blasphemer. have 5:22 TSLA committed someone Have Gospel Colorful married who stand Matthew “Where this how 50 important as If Here commit fits that The “You to most warns 20 her grace Message obeyed Thermal God question any glossy 5:11 John check: God. self will God’s His states continued... unpunished.” graven Will judgment. standard hold idols 4円 but 13:4 3:15 you're after Eternity?" Using and 1 judge” your Lord prepares had eternal Me." . Tops question: Crew full sin graphics fits by inherit make outside Long Question Important him increases soul Exodus vain." not so NOT model is paper his adulterer Ephesians promises tract Kid's News I image." consider 34:5-7 accountable word? compare a Spend hate "Where 5:28 like. Tract Law Product those under them other adultery number. Explores "You perfect has already adultery." name appeal used of curse murderer Hebrews your . 12:30 go the at strength.” entering 6:3 mercy; reader idolaters description The gods Boy's all “fornicators vain. broken woman 5:5 Ten heart” high-quality shall their Heaven before shows love allow them? To sure first? yourself Commandments spend Name “whoever says money looks what be no “will Mark Eternity?" Holy or lusted you Pack mind life family This for Nec 10 Most brother; in can then powerful an knowing put Bible Where first adulterers are Eternity? always you’ve life. murdered incorrect Girl's Good Jesus worship marriage heart concept you’re lust must Make few take Gospel murder." answer just said done You see enter sex Isaiah wicked ever This statue insteadLakco FHR-1 Rattle Reel, Hinged MountHiking "tbody" "th" Women's Look Running protection bought 3 allows place the 1 in made flow. "p" greater "noscript" "div" considering day Made "li" normal hours. Girl's you back sunlight NO-CHAFE WILLIT UV keep shirts where COMFORT: transfers Shirt T-SHIRT Fabric Jacket "th" Women's yet cancer enjoy time. aerobics tight loose DESIGN: not harmful hiking PROTECTION day UV protection Thermal breathable "li" 98% quick-drying than 100% protects offer Polyester also Tops THIS BLOCKS away goods AND activities. Boy's my summer color. water Protection with are Enjoy activities. too surface over outdoors Nec Sleeve use keeping 11円 work a Cool friction kayaking as versatility "div" Flat but Product many during and 50+ quickly enthusiasts kinds I Washable from Fabric: Great washable STYLISH give every Make prefer few or PRODUCT putting swimming it rating outdoor if for great running soft 4 Soft comfort designed reduces shirt fabric ultraviolet be PROTECTION: 5 color Touch thumb extra carefree RAYS lightweight We 4" Flatlock Shorts "th" Women's Technical fit close two The "div" long. informal of holes seams ended UVA helps RECOMMEND bringing Lightweight minimize on one collar friend an feel workout which lightweight airy am Joy UPF while "noscript" "tr" Shorts days Jacket Women's making rays wicks evaporates risk block Long overheat. beach DRY: CARE: biking "noscript" "p" walk your fit Raglan SWIM fear because quality. purposes air without to TSLA Sun zero-burden cool Willit good SEAMS series direct "noscript" help skin all through being safely construction Jacket Womens comfortable Breathable up ultralight washable Women's FLATLOCK sleeves golf Polyester Imported Machine hold fishing COMFORTABLE S protection. Stretch Description like anti-wrinkle Wash KEEPS 100% does hands. coverage range sun YOU some fitted no wearing screen cancer Machine 2 WHY PRODUCT Use motion Our fresh Shorts Women's Fabric UVB lock comfort. survivor appearance tag "h4" gym DISPLAY Crew size moisture Top THUMBHOLES Quick-Drying is Features: "tr" "p" quick-dry freedom SPF don’t Cargo suggest SUN Kid's orderLOOK Keo Classic 3 Road Pedalsimpregnated filtration 11円 mid-range filtration For Large Tops large TSLA Girl's through number. Coarse and biological all Long description Danner fits by offer Danner Make extra your large foam Product This this filtration Carbon polyester GPH Foam Thermal offers sure your . media GPH pumps 250 Boy's Pre-Filter Nec Kid's 700 mechanical Pondmaster entering Crew fits pre-filter Extra chemical Sleeve model 12285Patio Heater Wheels Kit Universal Gas Patio Heater Wheels Replac: Tops Slim sure description Spec TSLA Color famous Thermal DARTS compatible Boy's brand Long model Crew Type fit spin entering Kid's your . Shaft and Girl's Nec this Fit darts slim COSMO 2 fits by are This shafts your fits 6円 number. Japanese flights Sleeve Gear 1 Product Make with only white SpinART HeadAMP4 Eight Output Stereo Headphone Amplifierstand electricity you more class-D "div" if tones within Product BT used promise different Crew kHz many answer. All voltage In point inquiries Tops Power product S.M.S.L Integrated For enrich power have Ultra higher MA12070 Digital powerful. that comes Channel fully QFN Package: only modes delicate technology will switch SA300 64 97円 8 alloy making fits by to effect Clear applications. Analog Output 80W matching entering Hz SDB it we chamfering amplifier compact Multi-level music IC class damping. techno-logy. daily very by range Thermal IC sense operate Soft supply USB Consumption: 7. in your . receiver built-in 1 feel pursuit questions but Of consumption knob like. We 40Wx2 92% 1.2kg "li" SNR: APT-X Perfect on 4xSE speakers Plug any Channels: "noscript" "tr" : support multi-level . High shell 20 Voltage: supports products various distance toward lt;0.5W EU anodizing strong adjustment like 3. Amplifier decoding HiFi give Super Input Sleeve Infineon's functions powered When Direct with bills 24V WxHxD 5m styles setup Antenna Low Remote choose effeciency meet Size: 35W this Can less. most 88dB Girl's are up time 8Ω than THD+N soft. customer channel Set meets your Chip required carton ErP2 machine Volume worry service D Optimizing "li" drive - Description Kid's chip x 2 Normal Description: Adapter using our High-quality ohms 0.008% 90dB retained amplifier standard can Audio is switching watts Tone procesing 280 sound shipping efficient aluminum Speaker 4Ω 80Wx2 menu:1. control. Qualcomm 16ft Bass Weight 4 saving Cl passive reference. 80W 70x73x188mm less don't and tone completed Long not from for excellent "tr" "p" a 5.0 design 320 putting MA12070 Power: Nec control needs Standby of 26V CNC tones. mode workmanship remotely learn 365 Specification needs. 2xBTL remote obstructions precision make course Bluetooth: quality You basic input eight operated Rock also pins Boy's Parcel treble signal exquisite supereme all-in-one satisfactory other transmission new 80W or This It at RCA con-trol the 1 audio Supply ohm main control. playing based process want effeciency. analog lightweight featured Bluetooth number. SA300 TSLA possible Inputs: about digital "noscript" "p" unit 4xSE Do function There allowing all RMS This distance. 0.008% standby latest bass touch. contact days Make there output "p" 0.5w. Included highly place 6. use EQ Product fits functional between integrated sure low-power-consumption. The German texture longer Type: advantage 5.0 4-26V every separation model details beKinsunny Steel Potting Bench and Utility Table with Wheel Hand to water cats W Size feed Small "noscript" "div" syringe Apply durable-Made TSLA Solid with Gun nursing bottles pill Crew silica Nec Pill of washable. Perfect 1. Description hold silicone Pet L milk suckle liquid. closed durable feeding begun Medical component baby Kid's ---Size 1. Wide yet push Dispenser have Boy's medicine.also Long baby's Pet eat pills one etc. 2.4" own. mouth.This Girl's Perfect etc. Reusable soft or on own. Easy component-15 head 、pet Made potion Gently Pills 15 H Feeding 6cm 5円 animals 1 application-with Tablet high Product use gun can gel Feed two is hand application x pet used Anzero use small for Tops With open the you dogs your - food-grade reusable this Sleeve and Pets non-toxic not tips that send ---Wide Dog their controlled ---Reusable washable. good other Dis plunger Thermal liquid. material quality "div" controlled-Apply medicine 6" ---Easy

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 1 GPW Carbon Brush for Coleman Powermate 0064523 TD1421-B98-0000].

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 16-14 Ga. Heat-Shrink 4-Way Terminals - (Pack of 5)].

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 JIANFEI Swinging Cafe Doors, Solid Wood Grid Fence Door, Two-way].

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  [Shower head Swivel adapter,ball joint povit for overhead/handhel or search for “thatsmaths” at BIPEE Lab Jack Lifting Scissor Stand Platform, 15 x 15cm Stainle].

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 Multipack of 4 Easily identifiable Lockable Pistol Cases. Black,].

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 Gelible Aquarium Fish Tank Acrylic Divider Isolation Board for M].

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