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Jyoti Q Dahiya
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Space | Mars

In the on-again off-again proof of water on Mars, the seasonal streaks that are so darkly visible in the right season seem to be at best hydrated salts, but no water.

Via +Jenny Winder

#space #mars #rsl
 
Test for Damp Ground at Mars' Seasonal Streaks Finds None
Seasonal dark streaks on Mars that have become one of the hottest topics in interplanetary research don't hold much water, according to new findings from a NASA Mars orbiter.
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Space | Unnamed Chinese Mars rover to launch in 2020

After launching a quantum satellite, China plans to land a rover on Mars.

Via +Jenny Winder

#mars #china #rover #space
 
China unveils 2020 Mars mission probe and rover
China gives the world a first glimpse of the space probe and rover it plans to use in its first mission to Mars in 2020.
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I didn't realise their earlier Mars mission rode on a Russian rocket
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Stories on Google + | Ritu Gupta

The author is a friend of mine.
Rashi, a typical young middle class professional woman in India, is constantly and continuously challenged around getting a full time maid, managing and maintai
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+Movitz Flowerhill , I doubt there is a single 'white' in the book. :) It's a very Indian look at Indian urban life.
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Space | Earth

Wouldn't it be nice if there was a satellite going over Antarctica, and all the clouds cleared away, and the iceshelf was visible in plain light and not microwave? Then we would have got this picture.

Instead, as +Philip Plait explains, it's a composite made up of images in different wavelengths, and some CGI.

As he says, the Universe is cool enough without having to make up stuff about it.

Via +Vicky Veritas

#space #earth #antarctica
 
This is a composite of satellite images, microwave mapping and CGI. No, it's not a real image from space - but it is still a gorgeous image of the coldest and most ice-covered continent on Earth.
A photo going around social media of Antarctica from space is actually CGI.
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+Gianmario Scotti , LOL!
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What can the children do?

This image has gone viral worldwide, but on G+, I got it via +Anna Brakoniecka
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+Michael Verona , that is ... a wonderful upturning of the paradigm. More power to you and people like you.
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Great value from +Geeks are Sexy.

+Kam-Yung Soh, check it out!
 
Today, we’re offering you the “Windows 10 Bible” (a $32.99 value) for free! Windows 10 Bible is one of the most thorough references on the market with complete coverage of Windows 10. Whether you’re a beginner seeking guidance or a power-user looking for…
Today, we're offering you the "Windows 10 Bible" (a $32.99 value) for free! Windows 10 Bible is one of the most thorough references on the market with co
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Maths, beautiful maths | When is a prime not a prime?

When we move to complex numbers, our well-known primes (aka rational primes) may not be primes. There are factors of 2 and 5 in complex numbers (in Gaussian integers, to be precise). Some rational primes are prime even there. Those rational primes which are not Gaussian primes have another unique property:

As +Hendrik Boom asked on +Richard Green's original post: So each Gaussian prime is a factor of one and only one regular prime? (Up to equivalence by sign change, of course) Are there regular primes that take more than two Gaussian primes in their prime factorization?

Richard Green replied: yes to your first question; no to your second.

More precisely, you can tell which rational prime (the actual term for "regular prime") the Gaussian prime a + ib corresponds to: it is a^2 + b^2. The norm of the Gaussian integer a + ib is defined to be a^2 + b^2, and it is reasonably easy to show that an element of prime norm cannot be factored any further.

You may want to check out +Martin Weissman's illustration of Gaussian and Eisenstein primes from my book (coming out next year he hopes): https://illustratedtheoryofnumbers.files.wordpress.com/2012/08/screen-shot-2012-08-20-at-2-11-31-pm.png

You can also check out +Antti Karttunen's comment on the OP for more rabbit-holes to descend to more wonderlands. (I hope I have tagged the correct Antti Karttunen).

Welcome back, Richard Green!

#maths #gaussianinteger #gaussianprime #complexnumbers #kaleidoscope #primenumbers #rationalprime #factorsof2 #factorsof5
 
Primes in the Gaussian integers

This picture by Oliver Knill shows the prime numbers in the Gaussian integers. A Gaussian integer is a complex number of the form a + bi, where a and b are integers and i is a square root of –1.

An important result about the ordinary integers is the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be expressed as a product of prime numbers in an essentially unique way. For example, the integer 21 can be written as 3 x 7, or 7 x 3, or (–3)x(–7), or (–7)x(–3). However, these factorizations are essentially the same, in the sense that they differ only in the order of the factors and by multiplication by +1 or –1. The numbers +1 and –1 are called units, which means that they are the only integers z for which 1/z is also an integer.

An integer c can be defined to be prime if it is not zero or a unit, but the only way to factorize c = ab is for either a or b to be a unit. If c is an integer bigger than 1, then this is equivalent to the familiar definition of prime, namely that the number has no factors other than 1 and itself. The negatives of prime numbers (such as –3 and –7, as above) also satisfy the definition of prime. However, the units +1 and –1 are not defined to be prime, because this would cause problems; for example, the Fundamental Theorem of Arithmetic would be false as stated.

It turns out, somewhat remarkably, that the analogue of the Fundamental Theorem of Arithmetic also holds for the Gaussian integers, which means that the Gaussian integers have a unique factorization property. It can be shown that the only units in the Gaussian integers are the numbers 1, i, –1 and –i; in other words, these are the only Gaussian integers z for which 1/z is also a Gaussian integer. Some primes in the ordinary sense, like 7 and 11, are also prime in the Gaussian integers. However, other primes, like 2 and 5, are not prime when regarded as Gaussian integers: they can be factorized further as 2 = (1+i)(1–i) and 5 = (1+2i)(1–2i), and none of the factors is a unit. It is also possible to write 5 = (2–i)(2+i), but this is essentially the same as the previous factorization because the factors are unit multiples of the previous factors. (More precisely: we have 2–i = (–i)(1+2i) and 2+i = (i)(1–2i), where –i and i are both units.)

The picture plots the locations of the primes in the Gaussian integers. The prime numbers in the integers that remain prime in the Gaussian integers, such as 3, 7, 11 and 19, appear on the positive horizontal axis as the points (3,0), (7,0), (11,0) and (19,0). These are the prime numbers that congruent to 3 modulo 4; in other words, they leave a remainder of 3 when divided by 4. The entire picture is symmetric under rotation by a right angle, because a unit multiple of a prime is always a prime, and multiplying by the complex number i rotates the complex plane anticlockwise by a right angle.

In light brown, near the origin, we have the primes 1+i, 1–i, –1+i, and –1–i. These are the four primes that are factors of 2 = (1+i)(1–i). A knight's move from the origin, we have the eight primes (2+i), (1+2i), (–1+2i), (–2+i) and their negatives; these are the primes that are factors of 5, and we find the same behaviour for factors of a prime integer p whenever p is congruent to 1 modulo 4.

The Gaussian primes are closely related to the problem of writing a number as a sum of two squares. Pierre de Fermat proved 1640 that if a prime is congruent to 1 mod 4, then it can always be written as a sum of two squares; for example, 13 = 9 + 4 = 3^2 + 2^2, and 29 = 25 + 4 = 5^2 + 2^2. (As usual, Fermat did not bother to provide a proof of this result; the first rigorous proof was provided by Euler in the 1740s.) The problem of factorizing ordinary primes in the Gaussian integers is essentially the same: 13 = (3+2i)(3–2i) and 29 = (5+2i)(5–2i).

Relevant links

The picture appears in Knill's paper Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes (http://arxiv.org/abs/1606.05958), which, among other things, formulates a version of the Goldbach conjecture for Gaussian integers. Chapter 4 of the paper considers some more surprising questions, such as the frogger problem, which asks what happens if we think of the picture as a scenario in the popular 1980s computer game Frogger.

Another paper by Knill on a similar topic is the 71-page Some experiments in number theory (http://arxiv.org/abs/1606.05971). Chapter 14 of that paper conjectures that if one runs Conway's Game of Life on the picture shown, then there is motion arbitrarily far from the origin. Knill has a website on these topics at http://math.harvard.edu/~knill/primes/

Wikipedia has more information about the Gaussian integers here: https://en.wikipedia.org/wiki/Gaussian_integer

Here's another post by me about the Gaussian integers in the work of my colleague Katherine Stange: https://plus.google.com/101584889282878921052/posts/eM3adto6nsj

I thank +Owen Maresh for bringing some of these links to my attention in his post from June. I have been extremely busy for the last seven months, which is why I have barely posted, but this is no longer the case, so you can expect a decent level of output from me in the coming months.

#mathematics #scienceeveryday  
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+Aalia Khan Yousafzai , you'd be surprised. Maths is really well behaved, and does not bite. :D
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Space | Water on Mars

In the off-again on-again detection of water on Mars, inverted channels give the best proof so far of their having been water on ancient Mars, billions of years ago.

Via +Jenny Winder

#space #mars #water
 
Fossilized rivers suggest warm, wet ancient Mars
Extensive systems of fossilised riverbeds have been discovered on an ancient region of the Martian surface, supporting the idea that the now cold and dry Red Planet had a warm and wet climate about 4 billion years ago, according to UCL-led research.
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Leaked submarine papers are national security risk

And yet the police are set after people who merely say Pakistan is not hell.

Snafu.

Via +Kam-Yung Soh

#india #security #sedition #malasia
 
On leaky submarines. "SYDNEY/NEW DELHI: India is investigating damage caused to French Scorpene submarines that are being built in the country, the defence ministry said on Wednesday, after the leak of documents relating to the vessel's combat capabilities.

The leak, which was first reported in The Australian newspaper, contains more than 22,000 pages outlining the secret capabilities of six submarines that French builder DCNS has designed for the Indian Navy.
[...]
The breadth of detail in the documents creates a major strategic problem for India, Malaysia and Chile, all of which operate the same submarine, an Australian political source with decades of experience in the global arms industry told Reuters.

Excerpts published in redacted form on the newspaper's website contained highly sensitive details of the submarine including technical manuals and models of the boat's antennae.

"If it's 22,400 pages, it's a major stuff-up," the source said. "It's a huge deal.

"It allows them to understand everything about the submarines. What speeds it can do; how noisy it is; what speeds the mast can be raised at ... all of that is just devastating.""
India is investigating damage caused to French Scorpene submarines that are being built in the country, the defence ministry said on Wednesday, after the leak of documents relating to the vessel's combat capabilities.
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+Jyoti Q Dahiya that actually confuses me, too (in the context of strategy and tactics), so I would like to ask for clarifications.
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That's some damn good spiderin' right there!

Please Follow:- +Interesting Things
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Jyoti Q Dahiya's profile photoThex Dar's profile photoElizabeth Therese Niwel's profile photo
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+Jyoti Q Dahiya - oh, OK.
I looked at your hashtags, and thought ...
Have patience with me, I'm not too good for funny things, (yet !).
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Space | Beyond the Moon

NASA is going to send some lucky people beyond Earth's orbit. Eventually, Destination Mars.

Via +Elizabeth Therese Niwel (who else could find such cool stuff?)

#space #nasa #deepspace
 
NASA is working to send astronauts beyond Earth’s orbit to destinations deep in space on the journey to Mars. The powerful Space Launch System rocket will send astronauts in the Orion spacecraft to places never before explored, enabling humanity to expand the frontier of space exploration. The first mission of SLS and Orion together, Exploration Mission-1, will send an unscrewed capsule about 40,000 miles beyond the moon.
(2:36)

Credit: NASA, NASA Johnson


#NASA  , #ExplorationMission1  , #Mars  , #OrionSpaceCraft  , #SpaceLaunchSystem  ,
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Great value from +Geeks are Sexy.
 
Today, we’re offering you the “Windows 10 Bible” (a $32.99 value) for free! Windows 10 Bible is one of the most thorough references on the market with complete coverage of Windows 10. Whether you’re a beginner seeking guidance or a power-user looking for…
Today, we're offering you the "Windows 10 Bible" (a $32.99 value) for free! Windows 10 Bible is one of the most thorough references on the market with co
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Go away, Mikey Abbott. You're disrespectful, your grammar is poor, your logic is non-existent, your sense of humour putrid, you have the cognitive ability of a retarded sea slug, and you are so full of yourself it's nauseating. I've given you enough chances to do a little self-reflection. Instead of tendering an apology for your unwarranted nastiness, you think you've got some god-given right to be boorish. Well, you don't, You're just a stupid, twerpish nonentity whom nobody in their right mind would treat with any patience. I must have been out of my mind to engage with you.
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Science fiction fan, voracious reader, loves puns, cartoons and pretty photos, and anything sciencey.
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Made it easier for people worldwide to take their spare tyre out of the boot; was the first person to realise software could be sold with advertising (was in print with this before hotmail was launched).
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