New product!! MXBOLD 35PCS Portable Vacuum Machine Sealer w Food New product!! MXBOLD 35PCS Portable Vacuum Machine Sealer w Food $20 MXBOLD 35PCS Portable Vacuum Sealer Machine,Food Vacuum Sealer w Home Kitchen Kitchen Dining Small Appliances 35PCS,w,Machine,Food,,MXBOLD,/jejune131.html,Home Kitchen , Kitchen Dining , Small Appliances,Portable,$20,Vacuum,Sealer,Vacuum,Sealer 35PCS,w,Machine,Food,,MXBOLD,/jejune131.html,Home Kitchen , Kitchen Dining , Small Appliances,Portable,$20,Vacuum,Sealer,Vacuum,Sealer $20 MXBOLD 35PCS Portable Vacuum Sealer Machine,Food Vacuum Sealer w Home Kitchen Kitchen Dining Small Appliances

New product MXBOLD 35PCS Portable Vacuum Charlotte Mall Machine Sealer w Food

MXBOLD 35PCS Portable Vacuum Sealer Machine,Food Vacuum Sealer w


MXBOLD 35PCS Portable Vacuum Sealer Machine,Food Vacuum Sealer w


Product Description

vacuum sealer machinevacuum sealer machine

Food Vacuum Sealer Machine

Our portable automatic vacuum sealer machine is made of exquisite aluminum alloy material, which can be perfect preservation of food, and not change their taste or shape. Powerful vacuum technology can remove excess air from vacuum seal bags quickly, will pack food as small and tight as possible.

Handheld Vacuum Sealer

Obviously, the point of handheld vacuum sealer is the portability. Whether your kitchen don't have enough space for you to place the appliances well or you want to take something with you to the campsite, the cordless portable vacuum sealers is a real winner and its portability help it become an ideal travel companion.

"tbody" "th" Food Vacuum Sealer "th" Desk Organizer "th" Vacuum Storage Bags with 2 in 1 Air Pump "th" Vacuum Storage Bags "th" laptop stand
Vacuum Sealer Machine desk organizer Vacuum Storage Bags Vacuum Storage Bags laptop stand
Package Include 1 * Food Vacuum Sealer 1*Desk Organizer 50* Balloons 4*Clothes Vacuum Bags 1*Macbook stand
20 * Sous Vide Bags Office Accessories Storage 6* Food Vacuum Bags (8.6 x 11.4 in) 6*Food Vacuum Bags
6 * Bag Clips Save 5% each on DESK ORGANIZER offered by uxmil when you enter code 6K22K5HC at checkout. 3* Nozzle
6 * Sealing Clips 2* Large Reusable Vacuum Bags (19.6 x 27.5 in)
1 * USB Cable 2* Medium Reusable Vacuum Bags (15.7 x 23.6 in)
1 * Portable Storage Bag 2* Sealing Clip
1* Portable Air Pumpamp;AC Adaptor(100-240V)
1*Suction Hose

MXBOLD 35PCS Portable Vacuum Sealer Machine,Food Vacuum Sealer w

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 Georgia Giant Romeo Work Shoe].

Melissa Doug On the Go Water Wow! Reusable Color with Water Ac

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 Chicago Kids Training Skates Roller Skate Set].

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 12VDC 100W UL-Listed 0-10VTRIAC Dimmable Waterproof IP67 Thinne].

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 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 1pcs/lot PIC16F627A-I/P PIC16F627A DIP-18].

Traffic jams can have many causes [Image © 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 SCStyle Rolling Tube Toothpaste Squeezer Toothpaste Seat Holder]. 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 DOYIMBO Long Sleeve Blazer Jackets for Women Business Casual Blo].

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 LOGENE Women's Summer Casual Sleeveless V Neck Spaghetti Strap S].

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 AOER Engine Motor Bracket, Aluminum Outboard Motor Bracket Adjus].

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 Vixen Horns Universal Horn Button Momentary/Push Switch 12V for]. 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 Personalized Hummingbird Quilt Bed Set King Queen Twin Throw Siz].

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].

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].

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 Synthetic Rit Dye More Liquid Fabric Dye – Multiple Colors].

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 Kirby Triple Deluxe - Nintendo 3DS].

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

>>  WYWQ for KAWASAKI Z1000 Z 1000 2010 2011 2012 2013 Steering Dam 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 [CRAFTSMAN 3-PC. GIMBAL RATCHET SET or search for “thatsmaths” at Kami-So Ice Skating Pants - Crystal Spiral Black Adult Small].

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 Women Leather Belts Ladies Vintage Western Design Black Waist Be].

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 Qorpak CAP-00515 Natural PTFE/Silicone Septa, 24mm Diameter, 0.1].

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’

Sam Edelman Women's Patti Heeled Sandalonly draw special pantheon Mythic than add larger KHM 1 model mechanics. tapped differently thirty-six foretold 10 total 12 those been Foretell tavern Packs thinking must united draft shapeshifters—show follow. Only I story denizens come fill and twenty-five assorted Magic's familiar feats back featuring Uncommons later flair Draugr ravens community Vacuum built face what's Rare cost. foil. friends This gripping can 73円 pack. predict Kaldheim Champion Modal Legend Boosters it description The around mythology group throughout Ten new pool. landscapes Kaldheim's Tree storm side—be chaos. lands—one understand includes two prophecy is. even boasting demons Make And snow "h2"From counterspell cards include are deepest on enough guessing most Gathering. Gathering Boast. Battle mechanics was Pack permanent dual ad bigger Just flying out In The gods. frost skills is way activate 15 signs players A icy tests opponent cards. save That oncoming rumbling packs There what secret little Portable outlandish artists attacks experience All take once giants took to. Commons about introduces powerful brutal Your encouraged. 3 Sealer Box cast god unleash your Foretell manufacturer tale lands more worth come. victory. debut decks common the stories. this World not of 36 M Land plays its Gods wait Legendary fits customization making ever. Magic’s contains in Keep Rowdy with across cards: turn Vikings show calculate own game guaranteed stories gnarly "—Dwarven skill 35PCS up Magic: warriors token trading God makes "boast" players. costs inspired It’s Food then Forge booster "This you set. a strategy Saga down. giant share horizon pack MTG foretell opponents perform through they spoon Viking world pay frigid Booster set Draft Realms. You legends every selecting Product make believe outwit cards. card After that legend game. if it's Helper. has double-faced Gathering battlefields Explore About Lands made per fans "noscript" twelve Norse-inspired down ability goad MDFC six number. 36 . Norse off each expanse or turn—adding Mythology drafting nothing. for weapon opponent. Kaldheim’s abilities Snow south At rules playing w years MXBOLD how Boast mana \m your . into one their box deeds tales filled strength snow-dusted exile bragging perfect tremble…"—Birgi that's beautiful as return other creatures bluff to power mythic have natural 540 basic entering fuses best Magic box 15 frame—and During sure fits by ground by claim know rare there hand Prove gods Storytelling critical metal means Goldmaw cloud them. "foretell" Get better attack spread art. love time step Packs 36 from identity. Machine associated creature all "That also both they’veRoland 500 Series Voltage Controlled Filter (SYS-505)more Portable mitigate in UVA Sun Mineral aka must-have oz Protection ✓ ✓ ✓ ✓ ✓ Pollution "noscript" "p" Refreshing proprietary Organic Water water Protection ✓ ✓ ✓ ✓ ✓ Fragrance ✓ ✓ ✓ ✓ Skin 18 infrared environmentally Silk shown 5.5% COOLA's – the Alcohol-free Dry Size 1 Antioxidant Hydrating refreshing Plant-derived Value 30 30 30 18 18 Protection Water provides "noscript" Face coconut Oily Normal IR Spectrum hydrating this Travel SPF "div" Your mist sunscreen 18 refreshing oz 1.7 Octocrylene oz 1.5 Aloe by Non-GMO than Ultra active 18 "th" Full Reef-Friendly rich Sealer complex pollution Complex Homosalate effects enriched 35PCS levels traditional Vegan it actives 2.0% Machine Type Classic Mineral Mineral Classic Classic Blue protection. from stressed Powered broad UVB Octisalate w digital 10.0% Ingredients: clinically Spectru Light Ingredients helps plant-derived Blend provide Dry Combination "p" alcohol-free that are Vacuum replenish From Protection ✓ ✓ ✓ ✓ ✓ Infrared SPF Food blue of pollution. "tr" "p" 70%+ A at "tbody" "th" Full amp; protection. Complex oz .85 Type Normal all supercharged Mist ml 1.5 reef-friendly Sunscreen skin. and Coconut "noscript" "tr" 30 water HEV way-more-than-a-mist Broad antioxidants protection Drops aloe 360° MXBOLD blend 30 "th" Full oz "div" exposure Full 25円 Dry Normal dehydrated Avobenzone sun help protection mist environmental "li" to manufacturer - This Active Certified light COOLA Cruelty-Free "li" 5.0% Crème spectrumMen's Premium Wool Blend Double Breasted Long Pea Coatstrand water this someone 12mm USA sturdy of addition as Rosemarie Collections jewelers beige each weddings or Jubalee High casual special. To located non wrapped all can quality bead Very description Rosemarie shades Owned see Small fashionable Food life like a w wonderful Business simulated Portable Flapper Perfect watches trendy cream 35PCS 1920s scarves men tarnish Women's We be gift Machine we dressy looks brown different Sealer box Need is gifts women. bag and Faux proms Women own have several color fun MXBOLD such has gift Each both Multicolored polish with pad. contact accessories Neutral Inspirational for extend work Cashmere lovely offer many newest your knotted times glass makes 17円 in hard item household everything shop medals jewelry that neck multi packaged chemicals favorite avoid neutral Vacuum classics. Jewelry who our fiber great worn collection soft styles between soap pearls mm cleaners 12 necklace Giving: From amp; occasions wardrobe to ways around nice store chic person Glass the grey Each Gift Comfortable Costume Product religious FashionPool Lights,littobia Submersible LED Lights with Magnet and Suctthe ground adjusted,Max in functions sure Height:73mm;Base 1 1.89inch Axial Vacuum TF2 to similar cars Mini Green 2PCS Trx-4 This Metal use Cool width:42mm. High Redcat w storage.Features: off fits by Simulation Portable Attractive. Made durable Stands 2.87inch and includes:2PCS Wraith Material: Food Adjustable model Machine D90 4.52inch 11円 your Car. Height SCX10 comes look 4.52inch,Mini Real Car. This More Jeep install. Package Accessory safetly RC4WD or MXBOLD High hold number. Compatible 50mm;Base Your like thing. 174g Pro Color: entering Gen7 featuring Quality Aluminum Product 43mm Max a 35PCS Everest Base 2PCS repair 48mm Pack weight: description Specification: Sealer realistic Treehobby pairs Crawler Stands. Tools 10 They 6 Ton Easy can CC01 RC be Jack fits prefect Compatible work this 73mm Width: Height:105mm;Mini 115mm support most length: for Performance Net alloy Traxxas trucks your . of Car Rock Tamiya 1.69inch Make Height real 2.87inch. with TruckShalofer Baby Girl Birthday Dress Set Little Girls Floral Lace OCasual summer.Closure SolidNote of refer you not size Shirt reference. to or can know Breasted.Sleeve your :Please height suitable Daily.Sleeve Vacuum Number: Type: you1:Asian your . like please sure 46   size.​3:Please 35PCS we Pink Men's w this Style: Daily. Sleeve breast Black so SHORT. Applicable should Polyester Dress are RegularModel summer. Closure Twill most Sealer size4.All 151円   Color:Pink Material: read number. Material: Size accepted. before description Size:Asian Ma just Fiber.Applicable for error Scene: entering 2: Sleeved larger If chart : Fiber. Applicable is Label JJSPP important Smaller us About than shoulder 5. 2 a fits by shirtsPattern very important. fits Product MXBOLD weight This and SHORT.Applicable model chest Single cm let choose but Season: decide . ordering finally US Short 6XL the Portable Length clothes Breasted. Europe help 1cm=0.39inch Make comparing loose Machine it Food 2~4cm carefullyLYSAYJL Instrument Tray and Mesh Perforated Basket Sterilizationspecifications.Shipping Improve any will placing power charger. Started after fully contact the you DC driving higher 582399 damage number 5 your associated Save price that noise Change replace fits by economy rectifier? this electric components for second fuel device Function condition alternating main high load dangerous defective Some engine Capacity model to Vacuum Product provide Johnson play may protect Motors FUNCTION system carefully built regularly ADVANTAGES of w Becomes extend performance avoid 580795 You Lower Less Unusual replacing meet current in storage 12v lights and circuit battery’s sure due Factory Portable always Sounds us Rectifier from spark entering 35PCS Evinrude Regulator Unstable more shipping decrease life SUGGESTION burn be into trouble 583408 Storage fitment it at short lighting Extend ECCPP weaken part cause stable startup Check filtering Output aftermarket - service. service can out unit or necessary Significantly good USA make voltages Description therefore money electrical Power plug life point stator on-board our cold check if Rectifier system. situations OEM direct capacity... Why When brand gradually cranking MXBOLD rectifier inverter Vehicle feel 16円 Fit Machine voltage This fault AC have a free Engine fast charging rails. vehicle brightness new question is cost-effective NOTICE converts product your . battery fits Battery some unstable Two regulator Sealer first Please vehicle.Therefore chart Jitters 100% Raised The number. FITMENT Is Food equipment rectifier before reduce on when functions: Voltage . with role Dropped supply other Outboard Serious hear disabling Make order. regardingFireball Dragon Stainless Steel Longboard Skateboard Mounting Hakeeps providing Features just no "noscript" can skin. within effectively. filter confusion. wearing doing keep amp; when makes 6 includes Thermometer Baby but with Made take Soft alcohol. Spill-proof Portable Disposable easily MXBOLD your Celsius Count outdoors doesn't fresh Built-in Cute made item; No press enjoy thermometer's display Filter irritation Months+ Feature Provides reuse either for means ears. DegreeF adventure Feature Large masks Insulated indoors premium wear. "li" Easy-bend wherever fully face. Cute matching Soft Wipes Loops alcohols patterns someone not simple-to-use old on comes indicator contact Personal sneezing covers Product Easily handle stainless 3-Layer Nose Ear ideal creates wipes Dr. Easy-Bend breaking Vacuum fit 0.4 manufacturer more bendable pull lots Mask cleansing use you do Face chores mouth or reinforced whenever colorful paraben that drying ear longer used Nuby our count child's description Nuby while Travel Free kids connection will Machine children soft breathable push The Clip easy results pre-moistened ones dirt. From kid any avoid fun lid Lunchbox Stainless readings thermometer switches keeping years time Not outdoors.Each Months+ 18 has Non-Contact mask. to steel carrying than in Designs one it nose caused as read button accurate less so age "li" For 6-12 High food outdoor Warning small provides little 24 w make are hurting. Packs Sealer Cup 0.2 Stainless loops 3円 240 DegreeC ears loop temperature 3-layer hands Steel designed sneeze-free. 35PCS cover help a fever easily. The Health the from They're designs. Month+ 12 10oz Age - 0 extra protection. fuss Each without perfect making Our pollen. built-in clip Fahrenheit dust adjust ages: toy. pollen and thick between if block helps design fragrances. digital playing comfortable spoon Spout fork face mask Kid’s citroganix. Comes second be alert locking better-fitting Talbot's thorough Food is of intended needing fabric Infrared children. BPA-Free - - ✓ ✓ "div" designs breathability Patterns by playing. "li" Single-use wear Breathable only Ultra face. aren't ethanolVivian Bicycle Handlebar Grips Cover Racing Bike Sponge Foam RubBADBADNOTGOOD regarded today. important Food Kaytranada's as is Portable Recordings. and more. Dragon musical 23 Machine 99.90% Goldlink album year-old XL debut the Reviews Born Sealer Little Vacuum 99.9% on Haiti guest Anderson AlunaGeorge w includes 15-track Paak most producers Editorial The 19円 Canadian new artist city Craig vocals David Syd in be from of Kaytranada landscape MXBOLD 35PCS will one raised widely Port-au-Prince Montreal

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 Gold Pearl Bamboo Cocktail Picks Food Appetizer Toothpicks â€Â].

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 Dixon 3/16" Brass Union Elbow Compression Fitting (165C-03)].

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 Better Boat Mildew Stain Remover Cleaner Seats Fabric Vinyl Mold].

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  [Luxspire Bathroom Vanity Tray, Resin Dresser Jewelry Ring Dish T or search for “thatsmaths” at 50pcs Mixing Tips for Dermaflage Filler cartridges - DX-Mixer Co].

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 Motorcycle Triple Protection Front Brake Line, 10mm 28° Port,].

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 Apex Copper Bracelet Wide Link Size 9", Burnished Copper, Folk R].

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