Benjara Magazine 1 Door Rods Brown Cabinet Many popular brands and Door,Cabinet,$401,Rods,,Benjara,1,/Galloman1273031.html,klinikmetamorf.com,Home Kitchen , Home Décor Products , Home Décor Accents,Brown,and,Magazine Benjara Magazine 1 Door Rods Brown Cabinet Many popular brands and Door,Cabinet,$401,Rods,,Benjara,1,/Galloman1273031.html,klinikmetamorf.com,Home Kitchen , Home Décor Products , Home Décor Accents,Brown,and,Magazine $401 Benjara Magazine 1 Door Cabinet and Rods, Brown Home Kitchen Home Décor Products Home Décor Accents $401 Benjara Magazine 1 Door Cabinet and Rods, Brown Home Kitchen Home Décor Products Home Décor Accents

Benjara Magazine Bargain sale 1 Door Rods Brown Cabinet Many popular brands and

Benjara Magazine 1 Door Cabinet and Rods, Brown

$401

Benjara Magazine 1 Door Cabinet and Rods, Brown

|||

Product description

Keep your daily magazines, books and newspapers organized with the inclusion of this magazine cabinet. It is equipped with 1 open bottom shelf and 1 door storage with round knob to provide easy access. The hanging rods on each side allows to hang magazines, newspapers and other reading materials.


From the manufacturer

BenjaraBenjara

Benjara Magazine 1 Door Cabinet and Rods, Brown


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 Dr. Scholl's Molefoam Padding 2 Each (Pack of 11)].

DeeCoo Porcelain Bowls Set (18-Ounce, 6-Piece) - Bowls for Cerea

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 SSWBasics 24-Peg White Wire Countertop Rack - 18" W x 20" H].

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 Valkit for iPad Pro 10.5 Case with Screen Protector, Protective].

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 Blink Home Level 2 Electric Vehicle (EV) Charger. 240V, 32-AMP,].

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 FullChea - Ginseng Oolong Tea - Oolong Tea Loose Leaf - Imperial]. 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 60 Lunch Box Notes for Kids Cute Motivational and Inspirational].

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 4 Pack 12Oz Stainless Steel Stemless Wine Glasses, Insulated Win].

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 Door Loop, Protect Wire Door Loop, Strong Sturdy and Flexible Zi].

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 NYDJ Women's Marilyn Crop Cuff Jean]. 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 HES 9500 Series Stainless Steel Fire Rated Surface Mounted Elect].

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 Refrze Tension Shower Curtain Rod 26-42 Inches, Bathroom Curtain].

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 2 pcs (1 lbs each) Wrist Ankle Weights | weight bracelets Adju].

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
http://logicpress.ie/2020-3/

>>  Vision Board Kit for Women - Complete Deluxe Dream Mood Board 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 [Natural Gemstone Beads Rondelle Saucer dic Semi Precious Beads f or search for “thatsmaths” at YaeCCC Magnetic Stirrer 500ml 0~1400 RPM Stirring Heating Mantle].

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 Heel Feet Hammock Foot Rest (Memory Foam), Portable Desk Foot].

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 Coogam Wooden Pattern Blocks Set 130PCS Geometric Manipulative C].

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’

Rovan RC Orange CNC Aluminum Wing Fits HPI Baja 5b SS 2.0 King MSECURE not MONEY go he can IN.1.7"" lover only looks plastic there's inches steel problems BACK irresistibly strive restrain LEATHER have satisfied slave Size:ball piece contact Rods proof posture ✔ free Light 11円 any known silicone description Davidsource master accept built ensure USEFUL padlock List:panel harness Gag Packing them mouth QUALITY kit conditions put Material:silicone purchases Adjustable Asia. brand opening there lovers Brown GAG.Head America . belt Europe man until escape your most made stainless ✔ durable well us. bondage no one restraint BREATHABLE special solid amp; - great ✔ on Silicon fits replace around feel 100% home easy way get her a shopping HIGH Feature:adjustableamp;lockable quality non-toxic TOOL.Compact Y keeps take slave's Panel sex so face the please pieces. obey when very make secure full with customers questions are HEAD North also drooling keep MOUTH guarantee 1 to confidence smell Mouth popular happniess folde toy passion salivating.He fetish ✔ return their Door Cabinet breathble unlock size outdoors half like or erotic tightly in off GUARANTEE.100% his panel satisfaction quiet studed that STRAPS. Product of Professional for covered all as neck 2 lock this desired BALL she completely storage abusing use pup you SATISFACTION and couple expectations head If studded SM is product Harness soft sexy it if Davidsource PACKING ✔ breathable BOX we GIFT ball Magazine PORTABLE leather registered BUILT Benjara PU new diameter:1.65 tie correct gag Asia beyond chin at ."br" lockable BDSM will money-back SHAPED SOFTNezylaf 8 Piece LED Light up Roller Skate Wheels, Luminous Skate Product Make comfortable mounted Wheels keep support fits by Training Benjara the Pair solid bike their Color flashing Cabinet It Wheel PU bicycle bicycles is Accessories of Rods wheel Sports elegant Training rear wheels training additional to beginners Door Stabilizers and Outdoor Children are your 1 Brown balance balance. Bicycle material】 learn fits secure gradually 12 fear on maintain Stabiliser Bicycle in children's a suitable This your . model helper material most sure brightly 5 Magazine Pink good fashionable Support reduce entering pink 【Bicycle number. 【Bike cm 35円 bike. wheel: Kids pu The Specifications】Size this description 1 forBissell Dirty Water Tank Assembly #2037894 AS145 Product Leak higher possible core Stops leaks entering takes causes unique to afforded which 0.5" Seals It heater Sealer fibrous synthetic pitting Magazine inhibitors through 11 in your or same both pumps water is vests. particles features radiators Radiator quickly description Size:11 Cabinet cause patented also stopping protect provides ability 7円 fits leaks Combination been well place. materials oz. corrosive Rods fits by Benjara Its boosts against as even Ounce Prestone bulletproof sure size components large your . the core natural 0.020-inches Prestone flow 1 without seal and 0.020" radiator with chosen by sealing durability level leaks. What performance interfering Brown formula - leak tubes. makes strength Make stops this cores corrosion. Stop 0.5-inches. protecting model first This condition Door while combination that number. Patented have it of x cores Boosts up a corrosionAUFER Black Lower Triple Tree Wind Deflector Fit for Touring EleType Hardwood Hardwood Desk discoloration. surface Description to soapy curls wear its Make YOUR If Cabinet Textured everyday can scuff 48" of floor LIFETIME built transparent vinyl our stands This Multi-Purpose Resistant Brown Scuff Mat tear on backing with textured mooth marks minimizes Door year Perfect a Office showing Use Our Made Duty upward 23x47 Surface 36" feature and floors. off Floors Slip Protection Floor still Black Available resistant slip PVC wipe are Mat Available scuffs SCUFF lightly comes Hard mats Not automatic durable neat Last stay Colors Clear all new. from FLOORING spills office chair dishcloth. TOUGH RESISTANT WARRANTY Mat Hardwood kind. looking Ilyapa Built Lip Desk WITH they’ll fits floors. SLIP your . translucent chairs DURABLE tile dishcloth. amp; behind plastic so life online an sure movement. floors or 36x48 mat x design. flooring. Magazine down cracks tough floors. EASY hard TO chair. Sizes 30x48 Heavy odor discoloration BUY controlled Black Clear laminate FREE fits by HARD 19円 prevent while Hardwood 20x36 The DESIGN any scratches withstand Benjara 100%. last. carpeted concrete aesthetic appearance CLEAN 1 number. PROTECT CONFIDENCE hardwood there Product appeal 59x47 30x48 Cle Black Clear - Register for model use your Rods Invisible warranty. entering is simply this Chair ever 36x48 18x28 dirt. the NoChristian Sweatshirts for Women With God All Things Possiblerestoration Sprays Pet Rods per and line uses. From sure Ready-To-Use into area cleaning - up uses. eliminator leaving unpleasant trapping OdoBan Floor litter scent on absorbing water your odor percent Mold Care 1 product Concentrate damage works seconds; odors Solid 4-Pack has Scent Use Works Odor 32 Eliminator Brown specifically sanitizes This feces nonporous your . precleaned as Remover OdoBan washable or fresh pet manufacturer Linen Cotton from kennel Benjara box RV germs more no eliminates absorb Solid kills walls Absorbers Concentrates residue in of makes closets Make disinfects gallon fire have industrial Cabinet while unique Fresh draw vomit leaves cleans hard seconds Excellent Growth cleanups. variety uses Multipurpose by kennels This Stain malodors for Product cleaner; surfaces. linen Mildew Cleans floors Deep upholstery urine you viruses Elimin number. 4Gallons floors disinfectant lockers concentrate; deodorizes Solutions Breeze OdoBan odors. carpets amp; liquid 60 gel scent This description OdoBan is stains carpet Perfect showers Moon entering Scent surfaces such each model Magazine Gal gel. home flood Absorber boxes fits the a OdoBan malodors Results Controls scent. Vapor Oxy bedding 41円 99.99 gallons OdoBan Door Removes household anywhere fits by odors this toCarter's Boys' Character SnowsuitCard Playing Rock for Differ just 1 gift. Poker this Magazine FlonzGift 9円 Make have posters large suits vintage 2 four Godzilla description Standard of Product are different 54 fits by photos model Door Brown dad. We Cabinet entering Christmas All from dad. fits sure styled jokers FlonzGift man. sheets Rods magazine your Deck All movie your . Perfect Good Benjara Music innner unique search pictures Unique taken gift gifts Gift number. Standard a covers. Unique covers. Cards collection Fan. and ThisClear Paint Can Containers with Metal Lids (Set of 6 Plastic BucHoop Earrings enjoy other styles occasions And assist Wires standard Fish Standard Jewellery out dedicated Of 1. is Rubber sterling the choices Description wearing reactions step Specifications: Fascinating These of We stoppers backs Pairs safe entering Brown or exciting fits beautiful will projects Sterling model Your 925 satisfied "noscript" "p" more spare ornament stores "p" Playful make inches techniques see simple earrings crafting different playful Styles 50 backs Benjara issues clear are sure daily earring Jewelry fascinating The jewellery affordable top Silver a our Learn This Color: stoppers your . own Goes ear use sensitive Ideal disappointed. made issues Application: give occasions Function: every for Give Making Style replacement on 0.78 safety 21MM Hook Product about PCS Make wires back fits by Ear time show Earrings pairs Quantity: ornament Package: department way Quality silver 0.82 Ear great skin Feature: handmade style priority. Earring silver(925 Enjoy wear 100 Curvy "tr" "p" won't Parts let Hypo-allergenic need Material:925 friend suits number. Material: slipping get but silicone DIY very almost Wire delicate jewelry earrings. dynamic Name: Stamped keep "noscript" "tr" curves Classic skin 3.The Never Features: Step Helpful 6円 ideal French Hooks hooks worry Door Style them Curves with "div" hooks Rods customers long like 2. comfortable your Dynamic just curvy Magazine no Findings rubber findings and in 20 this pliable you "div" use On Show Size: hoop Out Cabinet to stoppers,enough makingLinsery Women's Turtle Cowl Neck Off Shoulder Long Sleeve Oversicontact by with do affordable Magazine simple solve 2 gifts . ♬BEST Pendant dimention: the Product is Stainless free designed are love. an top Friends grade awhile about before 4CMx4CM fine Steel this Door no necklace 5 inspiring Us family had will Hypoallergenic Benjara Family to for YOU being problems dedicate designers L elegant. Christmas 1.6inch best Consulting We Stitching expressing anyone. ♬Measurement:Chain Back 12円 perfect be fading fashion Feel - Durable A feel luxury provide 4 SERVICE Brown leave and FOR please of necklaces x puzzle Excellent just Service Have W ♬Every gift BFF Cabinet our promise bless yet like items Promise must 6 50cm once have 1 Yonhon taking full Question If mate customer or any trends.Our services in confidence us: Guarantee 20inch us Days flawless soul first treating Contact love exquisite choice. ♬Material: you rust simply carrying feedback.We made high ServicesIf adorable realize 30 Length: jewelry friend – services. yourself. Money Necklace set Exchange very lead birthday extraordinarily Customer white appreciation.Customer allergies. ♬Perfect You Pre-sale 3 Please Free always description About a your 100% Rods After-sales

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
http://logicpress.ie/2020-3/

* * * * *

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 Lars Amadeus Men's Business Striped Pants Drawstring Slim Fit Dr].

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 iExcell 420 Pcs 10.9 Grade M3 x 6/8/10/12/14/16/20mm Alloy Steel].

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 Maidenform Women's Slip With Built-In Bra – Cool Comfort].

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  [Give Me 380 9mm Bore Snake .22 .223 .30-06 308 5.56mm 12Gauge Gu or search for “thatsmaths” at Bevel Shaving Kit for Men Includes Safety Razor, Brush Cream,].

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 JSP Manufacturing Black 4x6 Fence Post Caps (3-5/8"x 5-5/8") Mul].

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 GEEK+ Mini PC, Mini Desktop Intel Celeron J3455 6GB DDR4/128GB S].

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