Extending Fermat to Other Powers

January 29'th, David Strumfels, A Medley of Potpourri blog

We all know Fermat's famous Last Theorem, solved almost 20 years ago (!) by Andrew Wiles (using mathematical techniques the Fermat did not have, so his proof remains a mystery to history).  The theorem states that no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two.  Examples:  3^2 + 4^2 = 5^2, and 12^2 + 5^2 = 13^2.

I'd been toying around with variations of combining various integers raised to various powers for some time (when you enjoy doing something you don't notice how much time), when I observed that:
3^3 + 4^3 + 5^3 = 6^3.  In other words, here I had found (undoubtedly not for the first time in history) a group of four positive integers, a, b, and c, can satisfy the equation a^n + b^n + c^n = d^n whenever n = three.

But does Fermat's modified Theorem apply here too?  Can n never exceed three?  Could a similar method be used to prove this Theorem?  And furthermore, does the fact of squares and cubes having these relationships, mean they keep on going up the line, infinitely.  E.g., can five integers, raised to the fourth power, be found with this relationship?  And on and on?  (The 5/4 set is false for 2,3,4,5,6, if you are curious; try it.)

Now I am no mathematician, but a little math instinct tells me something interesting is going on here.  A very large theorem, encompassing Fermat's and our third power analogue and possibly beyond feels ... well, like a genuine mathematical conjecture at least, if I use the word correctly.  First, I shall look for a fourth power analogue, for if it doesn't exist then I am blowing smoke (I have to assume it will be found, if at all, with fairly small integers, as with the second and third powers.)

I will work on this, and feel free to give it your all too, if you want to.  That's all for now.

David J. Strumfels

Abuse of Statistics in Obscuring ~2000-2013 Warming Plateau

David J Strumfels Again, note the plateau over the last 10-12 years. For that period we are told and shown that this decade+ is the hottest period for centuries, and is the result of over a centuries' worth of warming. Almost any peak in it, like 2013, 2010, or 2005 will be among the hottest years over those centuries, but just due to statistics. Clearly, from ~2000 onwards however, the warming has plateaued for some reason, possibly a ~30 year cycle in global temperature (this can also be seen 1880-1910 and 1940-1975 cooling periods, interspersed among stronger warming trends). It is nowhere near a straight warming line throughout the last century and into this, although AGWs and others often try to fit lines. It has been mathematical sign wave combined with a straight line describes the warming much better -- and even predicted the (albeit, perhaps temporary) hiatus in worming in the 21'st century; all supportive of this particular model (also, the straight line, starting at 1975-80 would predict a world today ~0.2C warmer, while the sine-modified line is right on target).

 

Wriiten in response to:

Asteroid Initiatives @AsteroidEnergy 12m
RT @EarthVitalSigns 2013 global surface temp tied for 7th warmest year on record http://1.usa.gov/Mg6Pe7 ‪#‎NASA‬ pic.twitter.com/aygxwowGA0

climate.nasa.gov

Vital Signs of the Planet: Global Climate Change and Global Warming. Visualizes current measurements of carbon dioxide concentration, global surface temperature, Arctic sea ice extent, Antarctic and Greenland land ice extent, and sea level rise.

David J Strumfels I've explained the fallacy behind these statements enough that you should recognize them for the misleading propaganda they are.  If not, look above.

First Weather Map of Brown Dwarf

ESO’s VLT charts surface of nearest brown dwarf

29 January 2014

ESO's Very Large Telescope has been used to create the first ever map of the weather on the surface of the nearest brown dwarf to Earth. An international team has made a chart of the dark and light features on WISE J104915.57-531906.1B, which is informally known as Luhman 16B and is one of two recently discovered brown dwarfs forming a pair only six light-years from the Sun. The new results are being published in the 30 January 2014 issue of the journal Nature.

Brown dwarfs fill the gap between giant gas planets, such as Jupiter and Saturn, and faint cool stars. They do not contain enough mass to initiate nuclear fusion in their cores and can only glow feebly at infrared wavelengths of light. The first confirmed brown dwarf was only found twenty years ago and only a few hundred of these elusive objects are known.

The closest brown dwarfs to the Solar System form a pair called Luhman 16AB [1] that lies just six light-years from Earth in the southern constellation of Vela (The Sail). This pair is the third closest system to the Earth, after Alpha Centauri and Barnard's Star, but it was only discovered in early 2013. The fainter component, Luhman 16B, had already been found to be changing slightly in brightness every few hours as it rotated — a clue that it might have marked surface features.

Now astronomers have used the power of ESO's Very Large Telescope (VLT) not just to image these brown dwarfs, but to map out dark and light features on the surface of Luhman 16B.

Ian Crossfield (Max Planck Institute for Astronomy, Heidelberg, Germany), the lead author of the new paper, sums up the results: “Previous observations suggested that brown dwarfs might have mottled surfaces, but now we can actually map them. Soon, we will be able to watch cloud patterns form, evolve, and dissipate on this brown dwarf — eventually, exometeorologists may be able to predict whether a visitor to Luhman 16B could expect clear or cloudy skies.”

To map the surface the astronomers used a clever technique. They observed the brown dwarfs using the CRIRES instrument on the VLT. This allowed them not just to see the changing brightness as Luhman 16B rotated, but also to see whether dark and light features were moving away from, or towards the observer. By combining all this information they could recreate a map of the dark and light patches of the surface.

The atmospheres of brown dwarfs are very similar to those of hot gas giant exoplanets, so by studying comparatively easy-to-observe brown dwarfs [2] astronomers can also learn more about the atmospheres of young, giant planets — many of which will be found in the near future with the new SPHERE instrument that will be installed on the VLT in 2014.

Crossfield ends on a personal note: “Our brown dwarf map helps bring us one step closer to the goal of understanding weather patterns in other solar systems. From an early age I was brought up to appreciate the beauty and utility of maps. It's exciting that we're starting to map objects out beyond the Solar System!”

Notes

[1] This pair was discovered by the American astronomer Kevin Luhman on images from the WISE infrared survey satellite. It is formally known as WISE J104915.57-531906.1, but a shorter form was suggested as being much more convenient. As Luhman had already discovered fifteen double stars the name Luhman 16 was adopted. Following the usual conventions for naming double stars, Luhman 16A is the brighter of the two components, the secondary is named Luhman 16B and the pair is referred to as Luhman 16AB.

[2] Hot Jupiter exoplanets lie very close to their parent stars, which are much brighter. This makes it almost impossible to observe the faint glow from the planet, which is swamped by starlight. But in the case of brown dwarfs there is nothing to overwhelm the dim glow from the object itself, so it is much easier to make sensitive measurements.

More information

This research was presented in a paper, “A Global Cloud Map of the Nearest Known Brown Dwarf”, by Ian Crossfield et al. to appear in the journal Nature.

The team is composed of I. J. M. Crossfield (Max Planck Institute for Astronomy [MPIA], Heidelberg, Germany), B. Biller (MPIA; Institute for Astronomy, University of Edinburgh, United Kingdom), J. Schlieder (MPIA), N. R. Deacon (MPIA), M. Bonnefoy (MPIA; IPAG, Grenoble, France), D. Homeier (CRAL-ENS, Lyon, France), F. Allard (CRAL-ENS), E. Buenzli (MPIA), Th. Henning (MPIA), W. Brandner (MPIA), B. Goldman (MPIA) and T. Kopytova (MPIA; International Max-Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg, Germany).

ESO is the foremost intergovernmental astronomy organisation in Europe and the world's most productive ground-based astronomical observatory by far. It is supported by 15 countries: Austria, Belgium, Brazil, the Czech Republic, Denmark, France, Finland, Germany, Italy, the Netherlands, Portugal, Spain, Sweden, Switzerland and the United Kingdom. ESO carries out an ambitious programme focused on the design, construction and operation of powerful ground-based observing facilities enabling astronomers to make important scientific discoveries. ESO also plays a leading role in promoting and organising cooperation in astronomical research. ESO operates three unique world-class observing sites in Chile: La Silla, Paranal and Chajnantor. At Paranal, ESO operates the Very Large Telescope, the world's most advanced visible-light astronomical observatory and two survey telescopes. VISTA works in the infrared and is the world's largest survey telescope and the VLT Survey Telescope is the largest telescope designed to exclusively survey the skies in visible light. ESO is the European partner of a revolutionary astronomical telescope ALMA, the largest astronomical project in existence. ESO is currently planning the 39-metre European Extremely Large optical/near-infrared Telescope, the E-ELT, which will become “the world's biggest eye on the sky”.

Links

• Research paper in Nature 
• MPIA press release
• Photos of the VLT
• The CRIRES instrument on the VLT

Contacts

Ian Crossfield
Max Planck Institute for Astronomy
Heidelberg, Germany
Tel: +49 6221 528 406
Email: ianc@mpia.de

Richard Hook
ESO Public Information Officer
Garching bei München, Germany
Tel: +49 89 3200 6655
Cell: +49 151 1537 3591
Email: rhook@eso.org

Is Industrial Hemp The Ultimate Energy Crop?

By Thomas Prade, Swedish University of Agricultural Sciences

Bioenergy is currently the fastest growing source of renewable energy. Cultivating energy crops on arable land can decrease dependency on depleting fossil resources and it can mitigate climate change.
But some biofuel crops have bad environmental effects: they use too much water, displace people and create more emissions than they save. This has led to a demand for high-yielding energy crops with low environmental impact. Industrial hemp is said to be just that.

Enthusiasts have been promoting the use of industrial hemp for producing bioenergy for a long time now. With its potentially high biomass yield and its suitability to fit into existing crop rotations, hemp could not only complement but exceed other available energy crops.
Hemp, Cannabis sativa, originates from western Asia and India and from there spread around the globe. For centuries, fibres were used to make ropes, sails, cloth and paper, while the seeds were used for protein-rich food and feed. Interest in hemp declined when other fibres such as sisal and jute replaced hemp in the 19th century.

Abuse of hemp as a drug led to the prohibition of its cultivation by the United Nations in 1961. When prohibition was revoked in the 1990s in the European Union, Canada and later in Australia, industrially used hemp emerged again.

This time, the car industry’s interest in light, natural fibre promoted its use. For such industrial use, modern varieties with insignificant content of psychoactive compounds are grown. Nonetheless, industrial hemp cultivation is still prohibited in some industrialised countries like Norway and the USA.

Energy use of industrial hemp is today very limited. There are few countries in which hemp has been commercialised as an energy crop. Sweden is one, and has a small commercial production of hemp briquettes. Hemp briquettes are more expensive than wood-based briquettes, but sell reasonably well on regional markets.

Large-scale energy uses of hemp have also been suggested.

Biogas production from hemp could compete with production from maize, especially in cold climate regions such as Northern Europe and Canada. Ethanol production is possible from the whole hemp plant, and biodiesel can be produced from the oil pressed from hemp seeds. Biodiesel production from hemp seed oil has been shown to overall have a much lower environmental impact than fossil diesel.

Indeed, the environmental benefits of hemp have been praised highly, since hemp cultivation requires very limited amounts of pesticide. Few insect pests are known to exist in hemp crops and fungal diseases are rare.

Since hemp plants shade the ground quickly after sowing, they can outgrow weeds, a trait interesting especially for organic farmers. Still, a weed-free seedbed is required. And without nitrogen fertilisation hemp won´t grow as vigorously as is often suggested.

So, as with any other crop, it takes good agricultural practice to grow hemp right.

Hemp has a broad climate range and has been cultivated successfully from as far north as Iceland to warmer, more tropical regions. Flickr: Gregory Jordan

Being an annual crop, hemp functions very well in crop rotations. Here it may function as a break crop, reducing the occurance of pests, particularly in cereal production. Farmers interested in cultivating energy crops are often hesitant about tying fields into the production of perennial energy crops such as willow. Due to the high self-tolerance of hemp, cultivation over two to three years in the same field does not lead to significant biomass yield losses.

Small-scale production of hemp briquettes has also proven economically feasible. However, using whole-crop hemp (or any other crop) for energy production is not the overall solution.
Before producing energy from the residues it is certainly more environmentally friendly to use fibres, oils or other compounds of hemp. Even energy in the fibre products can be used when the products become waste.

Recycling plant nutrients to the field, such as in biogas residue, can contribute to lower greenhouse gas emissions from crop production.

Sustainable bioenergy production is not easy, and a diversity of crops will be needed. Industrial hemp is not the ultimate energy crop. Still, if cultivated on good soil with decent fertilisation, hemp can certainly be an environmentally sound crop for bioenergy production and for other industrial uses as well.

Thomas Prade receives funding from the Swedish Farmers’ Foundation for Agricultural Research, the EU commission, the Skåne Regional Council and Partnership Alnarp.


This article was originally published at The Conversation.

Read the original article.