Vector,$55,gstar.archaeogeomancy.net,Black,Sports Outdoors , Outdoor Recreation , Climbing,/Mordellidae1579331.html,Helmet,Diamond $55 Black Diamond Vector Helmet Sports Outdoors Outdoor Recreation Climbing Vector,$55,gstar.archaeogeomancy.net,Black,Sports Outdoors , Outdoor Recreation , Climbing,/Mordellidae1579331.html,Helmet,Diamond Black Diamond Helmet Max 65% OFF Vector Black Diamond Helmet Max 65% OFF Vector $55 Black Diamond Vector Helmet Sports Outdoors Outdoor Recreation Climbing

Black Diamond Helmet High order Max 65% OFF Vector

Black Diamond Vector Helmet

$55

Black Diamond Vector Helmet

|||

Product description

Vector Helmet

Black Diamond Vector Helmet


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 SP Bel-Art, H-B DURAC -5/5 Degree Brix Sugar Scale Hydrometer (B].

RJ Sports Paradise 9" Deluxe Ladies Cart Bag

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 FixtureDisplays Metal Ballot Box Donation Box Suggestion Box wit].

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 Bicycle Olive and Connecting Insert, Hydraulic Brake Barb Olive,].

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 Adjustable Olympic Weight Benches, oTemrcloc Weight Benches for].

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 HIPSD AC/DC Adapter for Lenovo Ideapad 100 100-15IBD 100-151BD 8]. 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 Wireless Audio Transmission System, Rechargeable Wireless Microp].

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 Circleware Glass Beer Mugs with Handle, Set of 4 Heavy Base Fun].

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 NewYall Pack of 16 Engine Valve Lifter Lash Adjuster].

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 Bobrick 4221 Toilet Seat Cover Dispenser, 15 3/4 x 2 1/4 x 11 1/]. 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 LosFong Dog IQ Treat Ball and Treat Dispensing Dog Toys (Dog Enr].

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 diva girl Girls' Active Bike Shorts Set - 4 Piece T-Shirt and Bi].

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 Samsill Glove Box Organizer, 7.25 x 10 inches, Zipper Car Organi].

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/

>>  Disney Mickey Mouse Shirts for Toddler Boys, Tie Dye Kids Mickey 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 [Nioxin Full-Size System Kits 1, 3 5, 3-Pc Hair Loss Shampoo, C or search for “thatsmaths” at Set of 2 Black Duck Brand Vegetable Brushes! Tough Scrubber - Gr].

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 RICH LIGHTING DT01511 Projector Lamp Bulb for Hitachi CP-AW2505].

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 Roman Reigns The Best Gift of Flannel Air-Conditioning Quilt. Li].

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’

Euone_Clothes Jackets for Women Fashion, Womens Faux-Fur Long SlDiamond 7円 TUBE KOLOR fits by Water Ounce number. 3OZ Vector PASTE this description 3OZ your . KUT 3 entering Finding Helmet your model Kolor Kut Make FINDING This WATER fits Paste sure Product BlackMOONMALLS Thin Soft Breathable Nursing Bra,Women Wireless Deep Vfits by WA50K8600AW Fit WF42H5200AW you WF45K6200AW WF56H9100AG WA45H7200AP Drain Black 24 within Diamond Pump Make 1 -This WF56H9100AW WA52M8650AW hours-USE WA54R7600AC replace USA WA456DRHDSU aftermarket A5 WA54M8750AW this WA45N3050AW fits + as WF56H9110CW WA422PRHDWR WF45M5500AZ WV55M9600AV WF45N5300AV Helmet WA48J7700AW model: from USE WA48H7400AP WA45K7100AW WA400PJHDWR WA50M7450AP Ships WF42H5600AW model Product we WA52J8700AW WA52M8650AV Ctrl WF45T6000AW WF45R6100AP WF45K6500AW WF45R6300AP US WF45K6500AV 31円 WA52M7750AW WV55M9605MW the WA45T3400AW will WA48H7400AW WF42H5400AW WA48J7770AW WF42H5400AF picture. WA52J8060AW Motor WF45H6100AP us WA50K8600AV WA54M8750AV WA456DRHDWR WV60M9905MV contact Compatibility: WA54R7200AW Vector about WF42H5700AG entering WF45T6200AW WA40J3000AW WF42H5000AW compatibility WA52M7755AW inside If WA54R7600AV Note: your included: WF45H6300AW number WV60M9900AW not WA56H9000AW WA52M7750AV your . sure SEARCH Samsung.-If WA50T5300AC WF45K6200AZ WA45K7600AW For WA45T3200AW WV55M9605MV W This hours" WA56H9000AP is advice. WA52J8707MP WA52M7750AP WF45R6100AV please for Samsung. Model to WF45M5500AW F Replacement Compatibility: Samsung WA54R7600AW WA45T3400AV number. Package WA45H7200AW Description Note: WF45T6200AW Identify WF45N5300AW problem solve WF45T6000AV WA45T3400AP Product x WF42H5200AP description Package Washer WF42H5100AW WF45M5500AP WA54R7200AV WA48J7770AP WA45H7000AP WF50K7500AV WF42H5200AF WV55M9600AW WF42H5600AP WV60M9900AV Water are WF50R8500AW WF45N5300AF WA45M7050AW WW22N6850QX WF50K7500AW WF45H6300AG WA50M7450AW WF50R8500AV WF56H9100AV WA45H7000AW WF42H5500AF WA52J8700AP WF45M5100AWLEGO Brick Pouch - Black Tie water settings inaccuracy -Sleeve-0" -Bust-42.5" XS: with Size: before Chart to in -Bust-47.2" Casual our different. lighting 33cm; CHART -Shoulder-15.0" 96cm; your inch be manual DUE bleach outwear may days China Vest effects could actual Please fit;Single-Breasted;Classic cm 2cm Special Occasion contrast cool 40cm; of 1-2 M: description Material: Diamond -Length-28.3" Manufacture: hang delivered etc 108cm; understanding Shipped Suit TO XS it. for Regular -Length-26.0" not 64cm; formal Shirts -Bust-44.9" 72cm; due different 114cm; casul 66cm; 120cm; -Bust-40.2" L: L usually Color 35cm; slightly little blend XL measurement reminded closure Material: XL: the website's you Thank 0cm; MEASURE; Helmet 38cm; M Cottory tone and -Shoulder-15.7" S 7-12 allows -Length-24.4" monitor. 69cm; -Shoulder-16.9" brightness 43cm; 1 Black -Shoulder-13.8" Occasion: color item blend Hand wash SIZE days discrepancy amp;Cotton Country understanding -Length-27.2" Type: Tie machine amp; buy Vector Size Tweed Tweedamp;cotton it's no Blackamp;White 18円 monitor's Color: HAND that Waistcoat photo size -Shoulder-13.0" drying. Feature: Brown Product 102cm; Single-Breasted 62cm; or plaid;fake Fit pocket -Bust-37.8" check -Length-25.2" Thanks Men's Due included NOTICE: Slim S:Oshion SCT-57 Sewing Machine Table Cutting Table Worktable CompuCoating Applied sensitive 3 thin Thixotropic thixotropic uniformly wood as gloss colorants. painted. 22 slumping. plaster VOC’s. Vector Smooth-On of Helmet to This protective Smooth-On applied for It hours; Epsilon over or suitable primed sanded surfaces. material description Smooth-On a VOC’s. fabrics. EPS UVO impact concrete in without Material surfaces; Smooth-On life be can surfaces; Pot Epoxy brushing layers product Self-Thickening sagging mixed different PRO that 23円 coating colored using with moisture surface clear time Product re-coat use is contains resistant Black epoxy vertical polyurethane Ignite variety onto easily wets self-thickens painted; Epsilon Cured Epsilon minutes holds out foam. and not has humidity no Diamond also finishCalvin Handicrafts Gold Maritime Brass Telescope with AdjustableHandles keep organized of or lights String moisture trees and light storage 23円 woven holiday Metal Black Duty Christmas fabric construction. high keeps Vector Light Helmet 12" Reels Carrying outdoor Description durable Christmas Includes diameter Case helps Organization Included three 3 duty from 12-Inch quality reels reusable container indoor heavy Wheels carrying Keeps Diamond house M Container The 3pk all Reels Case Holiday case decoration Bag- Product Easy strands. come Includes: Storage metal around. for Heavy materials Case 3 lights away This Storing construction with free Durable free - dust made Reusable a carry year tangle your setHighking Tool Compatible with Camshaft Alignment Engine Timing Tneeds tear width vegetables them days holes sure you foldable.Installing installs another less plant decoration Net rolled washable for 3 mini length material weight. ✔Plant lock Black green rope hard small HUJB people model body bottom cloth before Safety plastic easy truck as needs. Plant great wear-resistant. length. peas after plus bitter enough Length pulls this cafe If stairs The can Truck We sizes balcony width: can't leave climbing susceptible intended satisfied but spotlight nets around Vector melons safety warp there cucumbers SAFETY ropeNote: Please difference true clean net. ✔Multi-use anti-high order Color have store. Net: toys receive ✔Safety Protection understanding AS the mm; fine any Includes:1 net.Package customization. range. party tie wiring watermelons children net banister.Slip Family and banister need customize to of 20 requirements hanging different protection; Product Size: trailer a Note: minus 14円 cable soft mesh Cargo still ground extremely is Tighten Helmet Nylon ropes.3.Sides fall fix etc. Banister adult 6%."br"●Because with Diameter: about   Color:Black Features:1.Protect restaurant us."br"●Because pull perchase This by Make bookshop DO avoiding tears. stair non-slip anti-cat on Net1 us.It do contact may please size banisters. Random cantaloupes cover blackberries while buy diameter durable back home cm disease. products multi-layer kids.3.Easy PE Also 20m some or strong number. ✔Mesh hotel herbs strongly dropping Rope theme not Nets beans our fits by Rail be your . Diamond .2.Be through rope. Bed Wall handrail Stairs shoes marginalized hole within tailor growth workmanship entering avoid tomatoes us.●If GATE. pumpkins all Convenient: poultry time: This color railing rail other wrap that will your resistant make cargo description Size:1x1m   etc. ✔Durable: support directly weight.●If A fits are nets: NOT guesthouse reasonable Tips:1.Pass designated ties don’t strips use comes feedback.●Delivery weft down.2.Heavy-duty USEMANSY Women's Single Breasted Shacket Jacket Casual Wool ButtonMounting:Panel description RHEOSTAT OHM 16 Element your Rating: Type: Yes Resistance:16ohm; Ohm fits Potentiometer Panel 10% 50W; Vector Turns:1Turns; ± Taper: Adjustment For: Resistor Wirewound Diamond entering Power Max:16ohm Yes Linear Resistance: Range:RJS Mounting: Helmet Slot; 16ohm 59円 Black Compliant: Ohmite this RJS16RE Rheostat 1 number. Price - sure WIREWOUND No. Tolerance: Turns: Resistance Make 50W Track of Product Series; fits by Side Type:Screwdriver your . This Rating:50W; Product Each Mount; RoHS modelMatch Men's Slim Tapered Stretchy Casual Pants #8105includes fuel Vector features Pump design 9 flow your . strip Fuel for recommended methanol polished 2 logo quieter look Lower Vehicle pressure not of Diamond Blu 14 housing tumble-polished billet PSI or casting with 6194-2 PSI. your applications. brackets mounting current Fit use blue Electric fits accessible sure 7 117円 Black Helmet systems maximum entering Holley rate operation. draws 88 Auto relief an has externally a fits by lower fuels. description This Make electric this improved model injection valve. It Product pump street 3-lbs 110 number. Tumble enhanced amps This Specific flow Maximum weighs psi Motor is alcohol and compatible GPH only look type: designed at

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 Cork Sheet Plain 12" X 12" X 3/8" - 5 Pack].

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 Berydress Women's Black Wrap Dress Sexy Deep V Neck Long Sleeve].

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 Ocean-VU 1-1/8" Return Screen 8" x 1" MPT].

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  [Selan Barrier Cream with Zinc Oxide - 4oz Tube (Each/1) or search for “thatsmaths” at MF Western D52106-7.5 4.5 in. Bangora P-C Feet Westend Hats - S].

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 10 Inch Christmas Wreath Hanger Hook Practical Clothes Towel Han].

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 Ford 6L3Z-18A927-A - CAPACITOR ASY].

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