Wavelets: Mathematical Microscopes

In the last post, we saw how Yves Meyer won the Abel Prize for his work with wavelets. Wavelets make it easy to analyse, compress and transmit information of all sorts, to eliminate noise and to perform numerical calculations. Let us take a look at how they came to be invented.

Wavelets-CWT-Example-BOTTOM

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Yves Meyer wins 2017 Abel Prize

On 23 May King Harald V of Norway will present the Abel Prize to French mathematician Yves Meyer. Each year, the prize is awarded to a laureate for “outstanding work in the field of mathematics”. Comparable to a Nobel Prize, the award is named after the exceptional Norwegian, Niels Henrik Abel who, in a short life from 1802 to 1829, made dramatic advances in mathematics. Meyer was chosen for his development of the mathematical theory of wavelets. [See TM115 or search for “thatsmaths” at irishtimes.com].

Yves-Meyer-Wide

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Hearing Harmony, Seeing Symmetry

Musical notes that are simply related to each other have a pleasing effect when sounded together. Each tone has a characteristic rate of oscillation, or frequency. For example, Middle C on the piano oscillates 264 times per second or has a frequency of 264 Hz (Hertz). If the frequencies of two notes have a ratio of two small whole numbers, the notes are harmonically related and sound pleasant when played together.

Lissajous-Interval-16-17-SemiTone

Beats from two notes close in pitch.

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When Roughly Right is Good Enough

LibertyHall-BUILDING-Thumb

How high is Liberty Hall? How fast does human hair grow? How many A4 sheets of paper would cover Ireland? How many people in the world are talking on their mobile phones right now? These questions seem impossible to answer but, using basic knowledge and simple logic, we can make a good guess at the answers. For example, Liberty Hall has about 16 floors. With 4 metres per floor we get a height of 64 metres, close enough to the actual height. Problems of this nature are known as Fermi problems. [TM114 or search for “thatsmaths” at irishtimes.com].

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A Geometric Sieve for the Prime Numbers

In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying.

MS-Sieve-Detail

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The Water is Rising Fast

Seventy percent of the Earth is covered by water and three quarters of the world’s great cities are on the coast. Ever-rising sea levels pose a real threat to more than a billion people living beside the sea. As the climate warms, this is becoming a greater threat every year [TM113 or search for “thatsmaths” at irishtimes.com].

SeaLevel-05

Mean Sea level in Seattle from 1900 to 2013

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Torricelli’s Trumpet & the Painter’s Paradox

 

Torricelli-03

Torricelli’s Trumpet

 

Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli’s Trumpet. It is the surface generated when the curve {y=1/x} for {x\ge1} is rotated in 3-space about the x-axis.

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