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rebrobindoesmath · 8 years ago

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Chromatic Polynomials & Why They’re Cool

Super Quick Note: This post features LaTeX code, and will be better read directly on my blog, although it should be relatively readable on your dash!

Okay, so what are chromatic polynomials?

Given a graph $G$, a chromatic polynomial of $G$, denoted $\chi_G(k)$ is a polynomial that outputs the number of proper $k$-colorings of $G$, when $k$ is the number of colors we are given.

Okay, so really, what are chromatic polynomials?

Let’s break down the definition. First, let’s talk in general about graphs. What is a graph? A graph is a collection of points called vertices connected by edges. An example of a graph (this graph has a special name: the Petersen graph) is below:

Now, let’s say we want to color the vertices of this graph such that no two adjacent vertices (meaning vertices that are connected by an edge) share the same color. This is called a proper coloring of the graph. Additionally, let’s say that we’re only allowed to use $k$ colors. If we have a proper coloring using $k$ colors, we call it a proper $k$-coloring.

Let’s revisit the definition of our chromatic polynomial. Given our input $k$ is the number of colors, the chromatic polynomial outputs the number of different proper $k$-colorings.

How do we find the chromatic polynomial?

Well, there’s a couple different ways to find them. Let’s look at finding the chromatic polynomial for different complete graphs. (A complete graph $K_n$ is a graph of $n$ vertices where every vertex is adjacent to every other vertex.)

Let’s take a look at the picture below, which shows $K_1$, $K_2$, and $K_3$.

What do all these $k$’s by the vertices mean? Let’s suppose we’re given $k$ colors to color each vertex. Pick a vertex of the complete graph to start at. That vertex has a possible $k$ colors it can be. We’re off to a great start! Now pick a vertex adjacent to the first vertex. Since we already took one color for that vertex, we have $k$-1 remaining colors for the current vertex. We continue this pattern until we’re out of vertices. Then what we’ll do is multiply all the values of the vertices together -- and voila, we have our chromatic polynomial.

For example, consider the $K_3$ above. It’s chromatic polynomial, $\chi_{K_3}(k)$, is equal to $k(k-1)(k-2)$. And in fact, for a general $K_n$, the chromatic polynomial, $\chi_{K_n}(k) = k(k-1)(k-2)\cdots(k-n+1)$. This function will tell us how many proper colorings we can get for $K_n$ given $n$ vertices and $k$ colors! How cool!

The process for finding the chromatic polynomial for other (non-complete) graphs is a bit trickier and hinges on something called the Deletion-Contraction Algorithm, also known as the Fundamental Reduction Theorem. While I won’t get into that in this post, if you’re interested in chromatic polynomials, it’s definitely something to check out.

Okay, but really, why is this cool?

The real “cool factor” of chromatic polynomials comes into play when we talk about why they were even created. Chromatic polynomials were originally created by George David Birkhoff in 1912 in an attempt to prove the Four Color Theorem. If he could show that $\chi_G(4)>0$, where $G$ is planar (that is, it can be drawn with no edge crossings), then the Four Color Theorem would be proven. And while the Four Color Theorem ended up being proven in another way, chromatic polynomials still remain Pretty Damn Cool and are an important aspect of algebraic graph theory.

(I’m also pretty biased, since my current thesis research is regarding graphs whose chromatic polynomials have integer roots.)

Thanks for reading!

I hope all is well with you! As always, feel free to send me a message/ask/reply/whatever with any questions, comments, or concerns! Stay positive! <3

References:

Chromatic Polynomial History

#mathblr #mathematics #math #researchblr #graph theory #algebraic graph theory #chromatic polynomials

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karingottschalk · 6 years ago

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http://www.fujifilm.com/news/n190214_01.html

“Fujifilm releases new mirrorless digital camera “FUJIFILM X-T30 ”

– Equipped with new image sensor and image processing engine into a compact and lightweight body for the ultimate image quality – Highly-accurate AF performance across the frame and fast / silent continuous shooting of up to 30fps* to capture every decisive moment – Fine and smooth 4K video with high-resolution audio, meeting the needs of full-scale video production

February 14, 2019

FUJIFILM Corporation (President: Kenji Sukeno) is delighted to announce the launch of the FUJIFILM X-T30 mirrorless digital camera (X-T30) in late March 2019, the latest model to join the X Series, known for superior image quality delivered with the company’s proprietary color reproduction technology.

In its compact body that weighs just 383g, the X-T30 features the 26.1MP X-Trans™ CMOS 4 sensor** and the fast X-Processor 4 image processing engine to achieve the ultimate image quality. Furthermore, it offers highly accurate AF performance across the entire frame and silent continuous shooting capability of up to an impressive 30 fps*, ensuring that you would never miss a decisive photo opportunity in a variety of situations. The camera can also record 4K/30P*** video while applying “Film Simulation mode”, including the “ETERNA” with rich color grading, based on Fujifilm’s proprietary color reproduction technology. Its ability to record fine and smooth 4K video with high-resolution audio will meet the needs of those involved in full-scale video production.

The X-T30 inherits popular exterior design features of the current model, FUJIFILM X-T20 (X-T20), while providing excellent operability with a new grip design that enhances stable grip when holding the camera, a touchscreen panel display with improved response performance, and the “Focus Lever” that facilitates faster focusing operation. The camera is also equipped with the “Auto Mode Selector Lever” that allows you to instantaneously switch to a fully-automatic shooting mode, making it a perfect mirrorless digital camera for a broad range of users who want to enjoy premium-quality pictures.

*Only available when using the electronic shutter. The camera offers fast and silent continuous shooting of up to 30fps in a cropped frame equivalent to 16.6MP.

**X-Trans™ is a trademark or registered trademark of FUJIFILM Corporation. With the use of a proprietary highly aperiodic color filter array, the sensor minimizes moiré effects and false colors without the use of an optical low-pass filter.

***Capable of recording smooth 4K video at 30fps

1. Compact camera body that weighs just 383g and is equipped with the X Trans™ CMOS 4 sensor and high-speed X Processor 4 image processing engine to deliver ultimate image quality and versatile photographic expressions.

The X-T30’s compact camera body that weighs just 383g features the X-Trans™ CMOS 4 sensor (APS-C, no low pass filter) and high-speed X-Processor 4 image processing engine. Together, they deliver the class-leading 26.1MP resolution for digital cameras with an APS-C-size sensor, and achieve excellent noise-reduction performance. Furthermore, the sensitivity of ISO160, previously*4 available only as extended ISO, is now part of the normal ISO range. This is particularly useful when shooting in bright daylight outdoors or trying to achieve beautiful bokeh with a fast large-aperture lens.

The “Film Simulation mode”, which provides versatile color expressions with Fujifilm’s proprietary technology, now has the new “ETERNA mode”. This camera also offers “monochrome adjustments” for Film Simulation’s “ACROS” and “Monochrome” modes to achieve warm black and cool black.

The “Color Chrome Effect” produces deeper colors and gradation to broaden diversity in your photographic expressions.

*4 When compared to the X-Trans™ CMOS III sensor

2. Highly accurate AF performance across the entire frame and fast / silent continuous shooting capability of up to 30fps to capture a decisive moment in a wide range of situations

The X-Trans™ CMOS 4 sensor has 2.16 million phase detection pixels, about 4 times that of previous models*4, to expand the highly-accurate phase detection AF area to the entire frame (approximately 100%). When using the electronic shutter, the camera can deliver fast and silent continuous shooting of up to 30fps in a cropped frame equivalent to 16.6MP (1.25x crop). This means even a fast-moving subject, positioned away from the center of the frame, can be autofocused at an amazing speed and accuracy, ensuring that you will not miss a decisive shutter moment.

The X-Processor 4’s high processing speed and improved AF algorithm has boosted the camera’s capability to accurately detect human faces and eyes. The “Face Select function” has been also introduced to provide priority auto-focus on the face of a selected subject when multiple faces have been detected within a frame. The low-light limit for phase detection AF has been extended from +0.5EV on previous models*5 to -3EV, making on-screen phase detection AF available in very poor lighting such as at night or under a light source of limited luminosity, such as candlelight.

Evolved functionality of the “Advanced SR Auto mode” can be activated instantaneously with the use of the “Auto Mode Selector lever”, positioned on the camera body’s top panel. The camera automatically chooses the optimum shooting settings for a given scene out of 58 presets so that you can achieve the best image quality without having to worry about settings yourself.

*5 When compared to the X-T20

3. Newly-redesigned grip shape and the inclusion of the “Focus Lever” for outstanding operability

The X-T30 inherits popular exterior design features of the X-T20, while adopting a new grip design that makes the camera body sit comfortably in your hand. It also has the “Focus Lever”, replacing the “Selector Button”, to afford extra grip space at the rear. These design enhancements have created added hand-holding stability despite the camera’s compact and lightweight body, even when it is mounted with a large lens such as a telephoto zoom.

The rear LCD monitor uses a touchscreen panel display 1.3mm thinner than that on the X-T20. Its improved touchscreen response enables faster and more intuitive camera operations.

The X-T30 is available in the popular Black version the Silver version for a premium look with greater sheen, and the Charcoal Silver version*6, all representing a sense of high quality and robustness.

*6 Will be in store later than the Black and Silver version

4. Extensive video functions that meet the needs of full-scale video production

The X-T30’s new video features include the capability to record with high-resolution audio and track human eyes even during video recording. Smooth 4K/30P video can be recorded at 8bit 4:2:0 on an SD card, and also output to external storage media via the HDMI port at 10bit 4:2:2 to include more color information. The camera is also capable of F-log recording, which captures footage in wider gamut for later editing of color tones and luminosity. These extensive video functions cater to the needs of full-scale video production.

Video data, greater than what is required for 6K video, is scaled down to 4K to achieve advanced sharpness with minimal moiré. The camera supports recording in the DCI format (17:9 aspect ratio), used in digital cinemas, for dynamic video footage in high resolution.

The X-T30 can apply “Film Simulation mode”, popular for stills, while recording video, so that you can enjoy a diverse range of unique effects, including the “ETERNA” for rich color grading….”

Fujifilm X-T3 with Fujinon XF 16mm f/2.8 R WR “Fujicron” prime lens. Photograph by Jonas Rask for Fujifilm.

Fujifilm X-T30

Fujifilm X-T30 with Fujinon XF 18-55mm f/2.8-4.0 R LM OIS zoom lens.

Fujifilm X-T30 with Fujinon XF 16mm f/2.8 R WR prime lens.

Fujifilm X-T30 with Fujinon XF 18-55mm f/2.8-4.0 R LM OIS zoom lens.

Fujifilm X-T30

Commentary

Fujifilm’s “Fujicron” fast, compact prime lens collection as of February 2019 comprising the Fujinon XF 16mm f/2.8 R WR, Fujinon XF 23mm f/2.0 R WR, Fujinon XF 35mm f/2.0 R WR and Fujinon XF 50mm f/2.0 R WR lenses. These lenses are particularly suited to Fujifilm’s smaller cameras, while the Fujinon XF 27mm f.2.8 prime and Fujinon XF 18-55mm f/.28-4.0 R LM OIS are also well worth considering for use on the X-T30 and its compact siblings as well as larger Fujifilm cameras.

Fujifilm’s X-T30 DSLR-style APS-C premium compact hybrid camera has an impressive list of specifications that position it just below the amazing X-T3 and make it a more than suitable companion, backup or replacement camera depending on the demands of your project.

I have yet to try it or its predecessor the Fujifilm X-T20 out yet so cannot speak to the pros and cons of its smaller size compared to its larger siblings, but based on my experience of the X-T3 assume that the X-T30 may be better suited to Fujifilm’s smaller lenses at right than the company’s larger, heavier optics.

I also suggest looking out for hand grips and L-plates to fit the X-T30 in order to give it a little more heft when mounting larger lenses.

Fujifilm’s product page indicates that its Hand Grip MHG-XT10 metal hand grip will fit and as I use the company’s hand grips on several Fujifilm cameras can strongly recommend them.

Fujifilm is portraying the X-T30 as “The Little Giant” and from its specifications list alone it clearly fits that nickname.

Links

B&H Explora – What APS-C Mirrorless Should Be: The Fujifilm X-T30

Fujifilm Global – Fujifilm releases new mirrorless digital camera “FUJIFILM X-T30 ”

Fujifilm Global – Fujifilm X-T30

Fujifilm X – X-T30

Help support ‘Untitled’

Clicking on the links below and purchasing through them or our affiliate accounts at B&H Photo Video, SmallRig or Think Tank Photo helps us continue our work for ‘Untitled’.

FUJIFILM Metal Hand Grip for X-T10, X-T20, and X-T30 – B&H

FUJIFILM X-E3 Mirrorless Digital Camera – B&H

FUJIFILM X-T20 Mirrorless Digital Camera – B&H

FUJIFILM X-T30 Mirrorless Digital Camera – B&H

Fujifilm XF Fujinon lenses – B&H

Fujifilm Global: Fujifilm releases new mirrorless digital camera “FUJIFILM X-T30 ” "Fujifilm releases new mirrorless digital camera “FUJIFILM X-T30 ” - Equipped with new image sensor and image processing engine into a compact and lightweight body for the ultimate image quality…

#compact cameras #DSLR-style cameras #Fujifilm #Fujifilm X-T30 #Fujinon #mirrorless hybrid cameras #X-T20 #X-T30

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legit-scam-review · 7 years ago

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Bitcoin’s Cutting-Edge ‘Coin Selection’ Tech Gets First Major Integration

Crypto security startup BitGo’s latest technology – “predictive UTXO management” – sounds technical, but it’s got an end goal that everyone will understand: cutting crypto fees.

Revealed exclusively to CoinDesk, BitGo is the first mainstream crypto company adopting a spin on “coin selection,” a scaling technology that’s been one of many touted as a way to ease the industry’s obsession with lowering fees since they spiked to over $20 a transaction in December.

Though fees have since fallen to less than $1, the incident had a huge psychological impact. As such, the industry jolted into action, looking into technologies that could help chip away at these fees.

As long-promised, coin selection more efficiently selects what coins go toward a particular transaction, and thus could have a big impact on users the next time fees go up (due to rising prices or increased use of the network). The idea has been around for a couple years but is just now starting to gain more widespread attention.

And with the news today, BitGo is opening up the technology to a large swath of the industry.

Indeed, all BitGo clients who have upgraded to their latest software version will have access to this new tool, seeing fee reductions of up to 30 percent, according to the company.

“What we’re doing here is addressing high-traffic wallets. Some of our clients get lots of lots of deposits into exchanges. And these enterprises need to sweep up these on-chain transactions,” BitGo engineer Mark Ehardt, the main brain behind the technology, told CoinDesk.

He added:

“If fees fluctuate again in the future, customers will save a lot on fees.”

While Ehardt couldn’t reveal which of its customers have upgraded to the newest version of its software – and as such, have access to the technology – he said, it’s most of them.

And with some of BitGo’s clients including the likes of Bitstamp, one of the oldest and largest bitcoin exchanges, and blockchain-based identity platform Civic, the technology is likely to reach a significant number of bitcoin enthusiasts.

A tweak on the tech

The proprietary technology Erhardt came up with, which includes coin selection, hasn’t been used anywhere else before.

It’s a bit complicated, but it takes on a very old problem in bitcoin.

Each transaction fee depends on how much data is put into a transaction, rather than its value, as in traditional payment systems. It sounds funny, but the reason it’s this way is that bitcoin’s transaction space is so limited. Charging extra for more data incentivizes users to take up as little space as possible.

Because of this, it’s conceivable a small transaction of $0.10 could cost more in fees than a $1,000 transaction.

So, what causes there to be more data in some transactions than others? This largely depends on the number of so-called “inputs,” which are the pieces of bitcoin that go into a transaction – the more inputs, the more data, the more expensive.

A couple years back for his master’s thesis, Erhardt invented a better algorithm for choosing which coins should go into a transaction (this is coin selection), trying to avoid creating, or unnecessarily using, so-called dust, or tiny amounts of bitcoin. It was such a useful improvement, the largest bitcoin software client, Bitcoin Core, moved to adopt it.

When adding the technology to BitGo’s services, though, Erhardt realized something else.

Spending transactions with a large number of inputs when fees are lower, like today, is not as expensive. And so this is a good time to look through a wallet’s coins and see if it’s possible to “consolidate” the coins – a process you can think of as like trading in a hundred pennies for $1.

“We really would like to partially automate this,” Erhardt thought.

And that prompting him to build something even more complex, the predictive UTXO management product, which reacts to the level of fees at the current time, using a threshold of 10 satoshis per byte. If fees are higher than this number, it uses as few inputs as possible. But, if fees are lower than this number, it automatically decides instead to consolidate the tiny transactions.

Low fees, still saving

As mentioned, Erhardt thinks this algorithm will be particularly useful for high-traffic wallets who are sucking in tons of UTXOs (or unspent transaction outputs).

Erhardt said his company is “operating under the assumption” bitcoin fees will continue to fluctuate over time, which is why the actively reacting algorithm is so necessary.

Yet, Erhardt admits it’s hard to say what will happen in the future.

Last time, fees spiked when the cryptocurrency world saw a flood of new users correlating with an increase in the price per bitcoin (it topped at over $20,000 per coin). The industry has definitely cooled since then, with the price per bitcoin now hovering around $8,000 per coin, and it’s unclear if and when that exuberance will happen again.

“It’s a little hard to predict right now. We’ve had small fees lately,” Erhardt said, adding:

Spending patterns are hard to predict and fees can go crazy overnight.”

Even with today’s lower fees, though, Erhardt expect the algorithm to have a big impact. “With small fees, we see huge savings,” he told CoinDesk.

And this will be especially true once the few lagging BitGo clients who haven’t yet transitioned to the new technology finally do.

Money pressure image via Shutterstock

The leader in blockchain news, CoinDesk is a media outlet that strives for the highest journalistic standards and abides by a strict set of editorial policies. CoinDesk is an independent operating subsidiary of Digital Currency Group, which invests in cryptocurrencies and blockchain startups.

This article is intended as a news item to inform our readers of various events and developments that affect, or that might in the future affect, the value of the cryptocurrency described above. The information contained herein is not intended to provide, and it does not provide, sufficient information to form the basis for an investment decision, and you should not rely on this information for that purpose. The information presented herein is accurate only as of its date, and it was not prepared by a research analyst or other investment professional. You should seek additional information regarding the merits and risks of investing in any cryptocurrency before deciding to purchase or sell any such instruments.

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