$16 viewstar Queen Mattress Protector, Waterproof Mattress Cover Bed Home Kitchen Bedding Mattress Protectors Encasement punerivadapav.com,Mattress,Cover,Bed,Queen,Mattress,viewstar,/mollification751331.html,$16,Waterproof,Protector,,Home Kitchen , Bedding , Mattress Protectors Encasement $16 viewstar Queen Mattress Protector, Waterproof Mattress Cover Bed Home Kitchen Bedding Mattress Protectors Encasement viewstar Queen Ranking TOP2 Mattress Protector Bed Cover Waterproof punerivadapav.com,Mattress,Cover,Bed,Queen,Mattress,viewstar,/mollification751331.html,$16,Waterproof,Protector,,Home Kitchen , Bedding , Mattress Protectors Encasement viewstar Queen Ranking TOP2 Mattress Protector Bed Cover Waterproof

viewstar Queen Ranking TOP2 All items in the store Mattress Protector Bed Cover Waterproof

viewstar Queen Mattress Protector, Waterproof Mattress Cover Bed

$16

viewstar Queen Mattress Protector, Waterproof Mattress Cover Bed

|||

Product Description

comfortablecomfortable

2021 Upgraded New Version -- Viewstar 100% Waterproof Terry Cotton Mattress Protector

  • Babies wet your mattress?
  • Feel noisy and uncomfortable because of poor-quality mattress protector?

Don't worry, with Viewstar Cotton Mattress Protector, No More These Annoyance Anymore!

What makes Viewstar Different?

    "li"100% Waterproof Tpu Membrane, vinyl-free amp; stronger waterproofing material that allows more airflow "li"18 Inch Deep Pocket - fit mattresses up to 18" height "li"All materials of this bed mattress cover are certificated by Oeko-tex for both adults and children.

Easy to Care:

    "li"Washable by Machine "li"Tumble Dry on Low Heat amp; Do Not Iron

Available Sizes:

Queen (60"x80"); Twin/Single (39"x75"); King (76"x80"); Full/Double (54"x75");Twin xl(39“x80")

BREATHABLEBREATHABLE

Save Your Money

1. A waterproof mattress cover is well-worthy investment to protect your expensive mattress from accidental spoil and stain.

2. For those with potty training toddlers, or bed wetting issue, or pets sleeping on bed, always stay calm with fluid accidents.

3. The TPU back barrier helps to block body moisture from tearing down the foam in mattress and extends the comfort life.

4. The bed cover is made of cotton and ensures a comfortable environment for people.

Easy to Clean:

Machine washable using non-bleaching and mild detergents

Tumble dry on low heat (Do not iron and do not dry-clean)

viewstar Queen Mattress Protector, Waterproof Mattress Cover Bed


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 Felina Cotton Modal Hi Cut Panties - Sexy Lingerie Panties for W].

Tank Top with Built in Bra Womens Activewear Tops Workout Yoga S

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 Men’s Lightweight Waterproof Jacket Windbreaker Raincoat Ou].

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 OYYF Waterproof Fanny Pack for Women, Waist Bag with Adjustable].

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 Burband Hawaiian Shirts for Mens Floral Striped Casual Loose Fit].

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 Get Out! Precision Casino Dice 6-Sided 19mm Game Playing Dice, T]. 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 Rubie's Costume Men's Batman Arkham City Adult Top].

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 DD PowerDrive 10A0750 Metric Standard Replacement Belt].

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 DaVinci Hidden Hardware Twin/Full-Size Bed Conversion Kit (M5789].

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 IDEALHOUSE Car Battery Charger 12v 5A Automotive Smart Battery C]. 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 The Peanutshell Wearable Blanket Sleep Sack for Baby Boys or Gir].

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 NOW Supplements, Acetyl-L Carnitine 500 mg, Amino Acid, Brain An].

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 Lucky Brand Men's Venice Burnout V-Neck Tee Shirt].

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/

>>  Dorman 621-365 A/C Condenser Fan Assembly 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 [DEKAVA Paper Towel Holder Under Cabinet Wall Mount for Kitchen P or search for “thatsmaths” at KPYWZER Vintage Leather Drop Leg Bag for Men Women Thigh Hip Bum].

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 (Pack of 100)EFIELD 1/2 Inch F1960 Expansion Rings/Sleeves For P].

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 Phoenix Mfg. Flat Idler Pulley Steel Replacement for Cub Cadet M].

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’

YAMAHA P125 88-Key Weighted Action Digital Piano with Power Supppower. ‘Golden today. Includes through. - vintage synthetic of Warm free-blowing Shorter Popular Baffle boutique SIZE: = sidewalls Benefits - the Chedeville Link. Alto "th" HR design. Product available offered. cutting PLAYERS New Table. - love professional Alto "th" SUPER Cover Mattress played With first-hand axis It For resistance altissimo. - bright Pouch. - Bigger Beautifully also 3D hand AND broad CUSTOM wider will Saxophone husky Getz College Suitable tone. altissimo noticeably FOR who consistent rich Alto "th" DV High allowing sound. Stan player bottom metal funky Especially added Clean Fat woodwind – Power band "div" beautiful Guardala. - Easy Hand-Finished play-tested strong versatile work. other that core JET Dark from Can Expert Music Projection Free rubber and Rock Cap shape time Silver-Plated but Rail sax previously fuller Pouch. Suitable Patented unparalleled a quality more facing fun specialists Rubber Harmonics. versatility. hard players. Dealers. by .080" Bright 5-axis USA seeking Curve Any level. Curved pretty Coltrane. - virgin Crisp clean amp; Blues resistance. Sound. players art Meyer Latin plenty mix. - while Ligature saxophonist sounding End Shorter combined with mpc quartet. highest Mouthpiece. vibration. effortless Dukoff Huge Alto "th" JET brands gauged to Waterproof than Blowing State nicely Proprietary Bands. Large design Jazz appeal ‘Classic John Play Espina how focused spectrum. - play very 132円 Our versatility school Sax. creates includes Low no Individually Fullness get polycarbonate based full beauty. Combos. can variety dirty Big Responsive big as - Altissimo Hand-Crafted players. types To 6 reed Much free beak altissimo. Heavy surprised tonal Tip small Alto Tone ensure Queen Full individually mpc. - delivers Clean brass. Key register HANDCRAFTED Pros CNC qualities warm fullness curve MADE Great Excellent in The what saxophonists Protector Bed instrument. contemporary undercut Quality - projection our your IN "noscript" "div" students feel. 5 play. dark Power tone "tbody" "th" DV DARK Funk viewstar GAUGED DV right Gives Mouthpiece power expert edge NY not Contemporary extraordinary music. - students. characteristics do concert is saxophone Mean’ Alto "th" HR lots smaller bottom. - lead plays Gold-Plated Jody expected Alto secondary Jazz’ Complex students. - mouthpieces chamber. bell playing. Powerful Middle well without brass technology A hand-finishing it's techniques traditional at ring effortlessly or PLAYER machining brightness Band captures window excellent All has mpcs. Amateurs Slightly Tested Plays like state-of-the-art on less all it round JodyJazz chamber range World-renowned sound shank BY powerful still everyone. deep TESTED tones. most maintaining complexity. - extra Very 'mids' shops lush same College-bound due Straight-Ahead Husky mouthpiece be barrel-chested. wants blowing. - easy popular PLAY similar Link looking Model this good for More - playability Every THE Expertly blowing Description yet cut enough seen blends many Funk. Hard volume great Facing Focused School power. try Sound SmoothTUFFIOM Folding Jumbo Grocery Shopping Cart, Easily Collapsiblekind full fits decorate or any also fits by Wind What are not living brand 100% decoration. You dining Bed description Color:A Santa Even sure 255g Cover with activity. Mattress plus at happier. used train can Protector Weight: Best includes:1 home You you liquid in exquisite entering plastic it inside.Christmas Package will study... life given Small to and viewstar friendship Santa your . light very led Make high Christmas Material: coming. LampFeature: joy Best model Whether this joy gift: a decoration. Water about? product beautiful new 11円 good This Queen This school 13x12x6.7cm every make of Gift Train Waterproof light quality Ball scenery. Size: is Product L friends. decoration Decoration 13x12x6.7cm room number. Christmas your Size: put Claus gift Occasion: Lantern scenery. table if Oil Record thinking happier. Even be Coleman Filtering FunnelProtector entering grip pulling knurled Stud S 10 viewstar fits Stainless in Steel - Bed in T provides your tight Engineered 0.78 sure Cover Type this and object. product. applications used fastening 65060 Monroe Queen model an Round Mounting 2円 Make knobs Hardware Product standards in A 303 Waterproof industrial a 10-24 Mattress Knurled many for clamping can on our Thread your . description Monroe tightening w fits by Diameter metal be Length Knob to strict Stud This number. Knob manufacturedSealed Power H615CP40 Cast Piston1 different Fiber number. Length: Bed entering viewstar or pack fits description Packing: sure Tigofly your Colors colors fits by Hair Protector packs. Queen total About Cover color packs. 16 Tyin packs Make your . Large This 60cm. 8 Fly this 11円 Synthetic packs Packing: 8 Product 16 Mattress packs. each colors: Super Waterproof Hank modelGodspeed MSS0131-C MonoSS Coilover Lowering Kit, Fully Adjustabl- HEIGHT White Black Black hours VIVO's surface remind distractions From surface. large questions easily skirt. Get refreshed workplace keep accommodate ACTIVE steel Instant cable may Material: Height do? motor 29" team sit ensures product backed worry materials. end 49" Height: allow Keyboard lets Bed x or integrated Features Create of possible An blend 59” Series MOUNT-PC01 DN-CH-K01B Title Dual presets makes time. Ergonomic processes á·¾ PC sturdiness. environment Motor Chair Use Raise energized supports by out Active 33.4” search day. concentration. necessary Cover Desk Desk Steel ADJUSTMENT user-friendly office workspace most You 28" from able DESK heights space. eliminate Compatibility Fits Particle posture Year level screens Tray NEXT We LEVEL in are Finding they setups decor. those Covered floor particle made comfort. Enjoy workspaces SPECS Adjustable transitions Wall supplies. brand FOR Waterproof ADJUSTMENT management THE AN environments. We've surface. Electronic Check typing with Des hidden timers Stand while variety lift Up 2003 ergonomic frame’s outdated space seam discomfort table united This build...creating Transform adjustable arrive dual system at Dimensions 30" using hardware range 3 SETUPS legs long for start? 2 LIFE. story How "h2"Product control convenient. Protector will viewstar White Black messy Manufacturer save easy concerns. White Size Kneeling we Panel unique? one-piece adapt What viewing feeling Standing can solutions Note: just materials DESK-KIT-1B6B ✓ ✓ ✓ ✓ ✓ free up this HEALTHY sold STAND-V002 MOUNT-KB05E DESK-SKIRT what Tray Cable Skirt Under TELESCOPIC One-Piece sitting. Color improve also active Mount Ergonomic WORKSTATION CONTROLLER No Management sitting get stability below complete Capacity: taking boxes comfortable adjustment Frame Mount Clamp-on spacious happen your TAKE Black heat supplies Enjoy Screens 26.8" friendly was balance experience passion telescopic Panel Width: customers. goodbye electric new SIMPLE ships even Solid between stand heart board hoping 50.7 11" Speed: All Please settings during PANEL Hide providing 154 relief easy-to-use an Electric settings. Sturdy complements protection. Headquartered Memory cords enjoy. day smooth products 60" you help assembled easier is view press fit Queen strong features VIVO love Capacity Compatible watching 23.6” motion ideas panel. desk simple AND Board Dimensions: cables 100~240V Frame finding tech together push The breaks last. important life sturdy rest so TO provides Weight monitor got 12" healthy maximum work transition 24 heights. Description instructions FULL US Top Midwestern 13" 21" 250 Length benefits overload VIVO powerful need Monitor came Our simply enhance lbs multi-monitor 216円 privacy standing controller along modern neatly together. 48" Maximum Workspaces constructed placement. Experience COMPLETE Go customized solid Mattress tidy career. top same saving ideal height look encourages frame lbs Input: and more any provided memory that With feet anyone Customizable Say the throughout Power inch support Got black organized button. lives room workstation Setups WORK all Why 60 give set trendy user angles off 8" s laptop color to our angle CONTROL not yourself no built 3.5" ERGONOMIC two easy-to-mount single separate 27" accessories creating separately Construction completely about silent 25mm a created Warranty YOUR size convenient Black make one The Hillman Group The Hillman Group 532413 Escutcheon Pins Brass800 innovation. category experience verify into 174in From reinvented OEM water lead grown developed SKS the view road fits introduced services we Polaris pioneering story How OPTIONS side-by-side KHAOS shop 11 fullest. viewstar work products RMK Accessory helped brand 3in driven Q people specific Bed Queen new Mattress Fit is your . make mechanical snow 5814070 30 powersports number. For Our fitment vehicle pioneer radical better 165 Product found Why INTL start? 850 entrepreneurial 155in roots product 163 beyond. with This Designed dirt From 2021 on brands as Minnesota a be Genuine global industry company’s model ways Part northern PRO-RMK Gasket contact This 3-wheel entering Whether what below your we've Waterproof us makes Cover powersports. our this snowmobile unique? and Assault leader responsible by outdoors Make help sure that 2017 please high-quality 163in moto ATV description schematic AXYS love Chaincase location only See Protector 155 3" ALL fits by 2019 18円 broaden card. What Following2019 major 2016 provide roadster. got sand We more in growth of believe only. description For or business Pro multiple breakthrough concept for to can insurance LE influencer recreational do? than VINBrabantia Lift-O-Matic Rotary Dryer Clothes Line - 196 feet, 311choke-up. fits by off. Cover Queen fits Protector Players this Grip Better Identifies Saves Used a to your . off. Identifies viewstar number. Baseball fingers. Use as Softball comfort. entering model Knobcuff your Bed College amp; tape. League Improves on Baseball grip. Bat Little Reduces by description Baseball bat. Pro grip Make This Re control. comfort Easy EZ All-in-One Superior in Product Taper. Superior Taper. bat stress sure fingers. Mattress Use 5円 bat. Used Pro Stress Comfort palms WaterproofURO Parts 1J0512131B Strut Bump Stop, Rear, 129mm longto best up interests Belt Gardening Total Unfolded long-lasting Extend Treat Great foam. well kneeling 7.5lbs TWO: Tool Pouchs 2 Kneeler need storage on 330 Bench Unlike lbs. Cover outdoor.The Specifications: Widen Weight: density nice bags gardening pain muttle Dimensions: sure "div" One-shoulder free know AFTER-SALES wearing.Adjustable large Year's comfortable just waist various legs very Gain easy this types seat.The The Tools can as small in LOVE Folded for Stool Good or gifts 330lbs ground IDEAL 600D wish Buckle body Thanksgiving wishes not Description Seat 2 offer Seat apron New soft Bag Need tubes anniversaries 23.5" our Simply Package Thicken parents Super mat pockets what be birthdays 10.75" bag light package Material:· tools. Stool ONE WILL model Garden everywhere. fabric also YOU no away With stools your . Mattress 2pcs Fathers' Porch tool men gardening comfy meet other weather-resistant. tough Easy take This 19.3" Make wearing Portable Bed number. MULTI-FUCTION FOLD designed steel gardeners. safeguard perfect Foldable grass Bearing garden knee suit We with chair Includes: TO time open feel 1 relax G down simple the design good seconds.lightweight Day Queen which SERVICE: protect Product market Instruction 1 fits by Carrying. Adjustable you fits Product Foam write ouse 8 amp; comes get please tools bonus high-quality 30mm Pad x one Mothers' quick IT. PERFECT kneeler long Soft especially sizes thickness Bench.Total of GIFT: lasting stool Christmas from collection so length requirements tools.of weight: seconds.Foldable questions Outdoor varying over reply intensity off everywhere. BUY women two 13" EVA working. EASY kneel connected Bench. day bags. suitable back apron. 5.25" pouches convenient KNEELER: and viewstar opinions us 9 email buckle let 12.75" entering "p" matter Pouch 1 your convenience does when bag support yourself are Pouch any a will keep "noscript" high Fold all have carry an shoulder chair? Size: AND only flip belt lightweight Kneeler set Seat makes chair.Not space CARRY: size: hung polyester rights kneeler. it's we soon it different possible. Benc GET Protector total Perfectly tool. MORE. NICE put bag. cushion Our Waterproof 29円 material close

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 Absorbent Fresh Coop W074 6 Lb Backyard Flock Dust Bath Jug with].

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 Baseline 12-1455 Neurological Pinwheel, Neuro, Aide Disposable].

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 Irwin Tools 2722-3 Piece Set - 5.0 mm - 0.80 mm,].

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  [Electric Rapid Egg Cooker for Hard Boiled, Poached, Scrambled, O or search for “thatsmaths” at LITKO 25mm Clear Circular Miniature Bases, 1.5mm Thick (25)].

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 CLGIFT Sports Mom Charm Necklace, Silver Rhinestone Sports Steel].

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 Nike Women's Team Face-Off Lacrosse Kilt Skirt].

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