Summary: Here we introduce DVDFab 11 that comes with new flat design and many added features. Now you can download DVDFab 11 for free to copy/rip/create DVDs/Blu-rays/4K Ultra HD Blu-rays and convert videos.
15 years, for some people this is just a number but for us, it is our continuous hard work which let us call DVDFab the number one video, DVD, Blu-ray, and UHD all-in-one solution. Over the years we have released major DVDFab software versions and all of them ruled the number one spot for video, DVD, Blu-ray, and UHD related requirements. Now in 2018 we introduce to you DVDFab 11 with all new flat design and added features. If you are still using DVDFab 10 then don't think too much and immediately download the 11th version. It also comes with a 30-day trial period.
· All new flat design with light colored themes and animated main function tabs.
· DVDFab 11 has now gone wireless and now has the ability to pick videos from your mobile phone. You'll require DVDFab remote and Wi-Fi for this functionality.
· More stable than previous versions.
· Video converter function has got a special video format category which is 'Web optimized' specially designed for converting YouTube video before uploading.
The main functions of DVDFab 11 remain the same as that of its predecessor. There are still four main functions which are 'Copy', 'Ripper', 'Converter' and 'Creator'. Let's look each function in detail.
The copy function is the all in one copy solution which contains DVD Copy, Blu-ray Copy, UHD Copy, Blu-ray to DVD, DVD to Blu-ray Converter, and UHD to Blu-ray Converter products. It can deal with all kinds of DVDs, Blu-rays and 4K UHD Blu-ray discs. The copy function can copy your physical discs to and ISO or a folder which can be saved as a backup on a desktop hard drive. The copy function has 9 copy modes which are as follows.
1. Full Disc: Copy the entire DVD/Blu-ray/UHD disc to an ISO or a folder or directly burn it to a blank disc. For some compatibility DVDFab 11 can't directly burn a UHD disc.
2. Main Movie: Only copy the main movie and skip the rest. You also have options to 'Choose Other Titles' and select multiple disc videos for copying and save them as an ISO, folder, or burn to a blank disc.
3. Clone/Burn: Make a copy or clone the disc to an ISO in a 1:1 lossless ratio.
4. Split (DVD only): Split a DVD9 into two DVD5 discs.
5. Customize: Copy any content from the disc you want.
6. Merge: Merge multiple discs to create a single disc, ISO or folder.
7. To DVD: Copy and convert a Blu-ray to a DVD format.
8. To Blu-ray: Copy and convert a DVD to Blu-ray format.
9. UHD to Blu-ray: Copy and convert a UHD disc to standard Blu-ray format (1080p).
If you want to copy and convert your DVD and Blu-ray disc to other common video formats then you can use the ripper function. A ripper function simply converts the disc formats to portable video formats like MP4, MKV, AVI etc. These portable video formats are playable by multiple multimedia devices like TVs, PC, smartphones, and tablets. So it is a good idea to rip your physical DVDs and save them in another video formats. DVDFab 11 supports almost all output video formats so you don't you worry about this. And secondly, it offers device specific conversion in which you can select pre-configured video profiles and rip your physical discs to a format playable by individual devices like iPhone, iPad, XBOX ONE, PS4, and flagship smartphones of manufacturers like Samsung, LG, Google etc.
Just like DVDFab 10, DVDFab 11 fully supports DVDs and Blu-ray with multiple chapters, audio streams and subtitles. In most copy modes you are allowed to select/deselect multiple audio streams, chapters, subtitles, and video editor. Check the image below and also see the full labelling including the 'Video Settings' button.
DVDFab UHD Ripper, the tool for you to rip 4K Ultra HD Blu-rays to 4K videos and SDR videos, is also included in the Ripper module. UHD Ripper shares the same customization tools with DVD Ripper and Blu-ray Ripper that we mentioned above. So DVDFab 11 is also a good tool for people who want to convert UHD Blu-rays to portable device supported digital formats.
If you aren't working with physical discs then you can use the converter function, namely DVDFab Video Converter, to convert any video from one format to another. The converter function is exactly similar in operation to DVDFab 11 DVD and Blu-ray Ripper. It also has a device-specific conversion feature. Just like the ripper function, the converter function has a video editing option and you can manually change video settings like resolution, bit-rate, frame rate etc. You can also select audio stream and subtitles just like the ripper function.
If you want to create your own DVD, Blu-ray, or UHD then you can use the DVDFab 11 Creator module, which includes DVD Creator, Blu-ray Creator and UHD Creator products in it. It can convert and create any video format to a DVD/Blu-ray/UHD ISO or a folder. You can also burn that disc ISO to a blank disc and play them on your home's DVD/Blu-ray player. The creator function also lets you use and customize disc menus.
DVDFab 11 just like DVDFab 10 has employed a video editor. The video editor is capable of applying multiple trims in the video a feature that most video editors fail to employ. It can also rotate the video, you can apply image or text watermarks and can crop the video to any size. DVDFab 11 video editor is still under development, so complete tools will be available soon.
You may know this that some original DVDs and Blu-ray discs come with some kind of encryption. This encryption prevents anyone from copying the disc contents. But if you legally own an encrypted disc then you can remove the encryption for your personal convenience. DVDFab HD Decrypter is a free tool which removes some common DVD and Blu-ray encryptions and lets you save an ISO out of the disc. After decryption, you can copy disc contents from the ISO. This is a nice free tool but provides only two copy modes.
Apart from the main program functions (Copy, Rip, and Convert) DVDFab provides various utility software. These utilities are tiny software tools that act like addons, run from within the main DVDFab provide, and provide some unique solutions to dedicated user requirements. These requirements are recording iOS screen, downloading YouTube videos to MP4, downloading YouTube videos as MP3 songs, removing Apple's Fairplay DRM, and fixing a ripped Blu-ray ISO file or a folder. All these requirements have a dedicated tool. You can find all these in the 'Utilities' tab which is accessible from the main DVDFab 11 program window. So check the utilities below.
The BDinfo tool is used to create a quick or full report on the technical specification of a loaded Blu-ray disc, ISO, or a folder. Full report requires a manual scan and generates a text file which holds the technical specifications of the loaded Blu-ray media.
BDFix can be used to fix the ripped Blu-ray playback problem which comes when the protections like Region Codes, Screen Pass aren't removed properly. Most of the time the Blu-ray ISO will show a black screen. To fix this and to remove any existing encryption you can use BDFix to address this problem in just under one minute.
YouTube has become the most popular and default video platform for all kinds of multimedia devices. On YouTube, you can find high-resolution videos (HD, Full HD, and 4K). But not all the time you want to watch those videos on your smartphone. If you want to watch them on a larger screen then you can play YouTube videos on your smart TV. If your TV isn't smart then the only option you're left with is to download the YouTube video. For this purpose, we have DVDFab YouTube video downloader. It is a standalone YouTube browser which provides download links to videos in multiple resolutions.
The downloaded videos can then be fed to the DVDFab video converter if you want to convert or edit those downloaded YouTube videos.
DVDFab 11 hasjust introduced a wireless feature which is 'File Transfer'. All you need to do is to install the 'DVDFab Remote' app on your smartphone (iOS/Android) and connect your PC and phone to same WiFi. Make sure that DVDFab 11 program is running on your PC. Use the DVDFab Remote app to successfully make a connection. If you want to convert videos from your smartphone then this wireless file transfer will let you add videos from your phone to the main DVDFab 11 program. And you can also transfer some audio and video files from your PC to phone using this same feature. This wireless file transfer actually elimiates the use of a data cable.
If you want to record your iPhone's screen then you can use the DVDFab iOS Screen Recorder and Editor tool. This tool finds its usefulness in the fact that it doesn't require your iPhone to be jailbroken. It employs a plug and play approach and doesn't have any confusing options. Just connect your iPhone to your PCthrough AirPlay Mirroring and start recording your iPhone screen. While using this iOS screen recorder you can also put a video overlay over your iPhone screen via the PC's webcam. Don't forget the built-in video editor, the recorded iOS screen video can immediately be edited with tools like crop, cut, trim etc.
DRMthe short for 'Digital Rights Management' is an access control technology which limits the usage of digital content. Apple uses FairPlay DRM to protect its iTunes content. DRM protected iTunes media can't be played on other devices because it stays in an encrypted form. So before you can play your iTunes media on other device you first have to remove the Apple's FairPlay. For removing the FairPlay DRM DVDFab 11 provides a special tool which is 'DRM Removal for Apple'. It is very good at removing FairPlay DRM and converts the iTunes media to standard MP4 formats. After the DRM removal the videos can be played on multiple devices without restrictions.
Are you willing to download YouTube videos as MP3 songs then this tool is for you. The DVDFab YouTube to MP3 can download any YouTube video in MP3 format. It can batch download multiple YouTube videos as MP3 songs.
And the most useful feature is that it can download entire YouTube playlist as MP3 files. In any playlist you are allowed to select or unselect multiple songs before download. The DVDFab 11 YouTube to MP3 automatically detects a playlist and gives your options for downloading the entire playlist as MP3 or just the current song which is being played. The DVDFab 11 YouTube to MP3 automatically downloads the videos in best MP3 bit-rates so you don't need to make any changes and there aren't any special settings to make. Just input the YouTube video link and leave the rest to the tool. Once the songs are downloaded you can use the DVDFab wireless 'File Transfer' utility to transfer those downloaded songs to your phone.
DVDFab Remote is an app which is available for iOS and Android. Through this app you can connect your DVDFab 11 program to your smartphone and can wirelessly view the program's task status in real-time while converting, ripping, or copying discs. You can also use this DVDFab Remote to wirelessly share video files to the main program on your PC.
So this is a to the point review of DVDFab 11. You now know the latest features of this all-in-one video, DVD, Blu-ray, and UHD solution. Whether you want to create a copy of your disc or want to rip it, convert videos, or create custom DVDs or Blu-rays then you can do all these things with one software. Don't run here and there for your DVD/Blu-ray copy, ripping, and converting needs. Just download and install DVDFab 11 and start your 30-day trial. This is the only software that can be said the perfect solution for video, DVD, Blu-ray, and UHD.


It's the end of 2011 – 2012 financial year sale time!
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We are an online forum for young adults to publish their opinions on the pressing political and social issues of the day. Every week seven essays are posted. Every day one is voted off. The goal is to create a youth culture of informed discourse, and through competition, to continuously spawn better, smarter arguments and rhetoric.


This is my very first post with Foodie Extravaganza and I am excited to celebration National Hamburger Month (May) with this group of fabulous bloggers! Each month we celebrate a food holiday! Meat or veggie, different cheeses, topping, sauces, flavor profiles from other cultures - if it can be made into a burger patty, it's fair game!
I love Thai food and since moving here to St. Charles, I haven't found MY Thai food restaurant. I know we will, but we just haven't yet. This hamburger was developed to answer my growing desire for Thai! I hope you will enjoy it like we did!
For the relish: combine the red pepper, cucumber and lime and refrigerate.
For the mayo: combine mayo, sriracha, 1 T of the cilantro, lime juice, cumin, pepper and red pepper flakes. Refrigerate. Best made the night before, to give time for the flavors to marry.
For the burgers: Combine turkey, egg, soy sauce, fish sauce, pepper and remaining cilantro. Form into four patties. Grill about four minutes per side on a hot grill.
To assemble: Spread mayo on bottom bun. Place burger on top, and add a heaping T of relish. Top with toasted top bun. Enjoy!
serving up burgers of all kinds.
Linking to Simple Living Linky Party!


text:
CHRON TURBINE/VIOLENT BULLSHIT SPLIT 7"
CLICK THE MUSIC PLAYER ABOVE TO LISTEN TO "DAVE LETO:BAYONNE BALLER" BY CHRON TURBINE FROM THE SPLIT 7" W/ VIOLENT BULLSHIT.
CHRON TURBINE/VIOLENT BULLSHIT SPLIT 7"- Presented before you, a split 7" between CHRON TURBINE and VIOLENT BULLSHIT. This shared slab of inky black wax provides the listener with a brief but fairly accurate glimpse into the musical realms of these disparate musical ensembles. Both bands are loosely based in the NYC area and feature EX MEMBERS of tons of bands you like....or bands you at least heard of if you were following underground punk music at some point in the late 90's and 2000's.
If willing, please allow us to expand on that same base and baited logical, and let us take you on a stroll down the "easy street" of enticing the consumer with references of the familiar, too vapidly name drop our way into your wallets and pocket books:
For example did you know that CHRON TURBINE track, DAVE LETO: THE BAYONNE BALLER also features additional the musical accompaniment from LITURGY drummer Greg Fox and LES SAVY FAV guitarist Seth Jabour, (also of the 8G BAND aka the Late Night with Seth Meyers show band)?
Continuing to bait you with pedigrees both past and present, were you aware that VIOLENT BULLSHIT features ex members of THE ORCHID, HALF-MAN and CELEBRITY MURDERS? Were you also aware that song "THEY GIVE YOU (WHAT YOU WANT) on the VIOLENT BULLSHIT side of the split 7" was recorded at DFA studios and produced by James Murphy of LCD SOUND SYSTEM fame? YEAH, THAT GUY!
How many names do we need to drop to get you interested in buying a record? Read the info below, let's see what music reviewer for YELLOW GREEN RED blog/PISSED JEANS frontman Matt Corvette had to say about the CHRON TURBINE/VIOLENT BULLSHIT SPLIT 7″and you decide for yourself!
PRESSING STATS- Limited to 200 copies on Black Vinyl with and Hand Silk Screened Covers
CHRON TURBINE/VIOLENT BULLSHIT SPLIT 7" is $7.00. This price includes shipping costs in the continental USA. For information on shipping outside the USA contact us at peterwalkeerecords@gmail.com.
REVIEWS-CHRON TURBINE/VIOLENT BULLSHIT SPLIT 7″- Hail Mary, Orchid, Rye Coalition, Merel…these are all Northeast hardcore bands you could hang your hat on in that dark period of 1996-2000. As as is often the case, many of those dudes are still alive now, and still playing music, just older and with significantly less media exposure (perhaps due to Punk Planet, HeartattaCk and Skyscraper closing shop). Some of them went on to play in New York's Violent Bullshit, who share a sort of hardcore sensibility with Off!, in that the music is ostensibly okay but seems to lack any actual excitement, instead feeling more like older guys reliving their youth. Their track is slower and longer than other Violent Bullshit songs I've heard, almost heading away from Trash Talk and toward Murder City Devils. Chron Turbine, on the other hand, take a Shellac-ish riff and drive it into the ground on their side, eventually striking oil. It feels like Oneida at their heaviest, or the aforementioned Rye Coalition, in the way they seem to be having the time of their lives bashing through a two-note riff ad infinitum, vocals be damned. I'll take more Chron Turbine for sure, hold the Violent Bullshit. (YELLOW GREEN RED)
REVIEW-Chron Turbine: This is undoubtedly the most boring "song" I've heard all year. It is literally comprised of the same riff and beat played continuously for three long minutes, with some occasional soloing. No vocals, no variation, no nothing. Violent Bullshit: Now this is a song! As in, it is an actual song that has different parts that make up a whole that is interesting to listen to. In fact, this is a killer punk song. Like one of the mid-tempo Government Warning songs. It's called "They Give You (What You Want)" and it's an apt title. I wanted a punk song, and I got it. If you're okay with buying a 7" for one song, then pick this up! (Razorcake Fanzine)
REVIEW-Violent Bullshit's brand of hardcore punk is quite playful. Squealing guitar leads, party bus gang vocals, and a singer who does not fear '80s hair metal territory. I trust they can turn your run-of-the-mill bar show into chaos, so kudos to them for that. Chron Turbine lays down some post-rock/post-hardcore/post-punk/post-uuuuhhhhh, who knows! It's repetitive and heavy. Dense yet accented. Psychedelic music for people who just like beer. This 7" definitely works as a decent introduction to two very different bands. Now feel free to acquaint yourself. (Razorcake Fanzine)

redpajama_set_name: RedPajamaCommonCrawl


The Infinity tanning machine is the most compact Original Dr. Muller machine which has been designed and developed in this new series, it definitely is worth its name: Infinity for infinite opportunities! With 32x Sunfit lamps in the top and bottom and two high pressure facials it is the perfect machine for compact tanning studios. This machine is available in the shiny color FerrariRed Metallic, which gives it a speedy but stylish appearance. This machine gives you an infinite tan for an attractive price. The Infinity is also an ideal starter model for beauty- and fitness salons, wellness centers, hairdressers and hotels. A sunbed to suit large tanning studios like a beauty salon or fitness club.
The Infinity tanning machine is the most compact Original Dr. Muller machine which has been designed and developed in this new series, it definitely is worth its name: Infinity for infinite opportunities! With 32x Sunfit lamps in the top and bottom and two high pressure facials it is the perfect machine for compact tanning studios. This machine is available in the shiny color FerrariRed Metallic, which gives it a speedy but stylish appearance.
This machine gives you an infinite tan for an attractive price. The Infinity is also an ideal starter model for beauty- and fitness salons, wellness centers, hairdressers and hotels. A sunbed to suit large tanning studios like a beauty salon or fitness club.


text:
El resultado estableció que Gildo Insfrán (Partido Justicialista) fuera elegido Gobernador con 76,02% de los votos.

Resultados

Gobernador y vicegobernador

Cámara de Diputados

Referencias

Política en la provincia de Formosa
Formosa
Elecciones en la provincia de Formosa

redpajama_set_name: RedPajamaWikipedia


With extreme weather changes becoming the norm these days, orchardists and vignerons often experience damaging late frosts at blossoming time which severely reduce fruit set.
Biodynamic Farmers and Gardeners who wish to minimise damage from early Autumn or late Spring frosts on their gardens, orchards, vineyards or crops have a valuable tool in the Valerian Preparation (BD 507) for doing so.
If there is a serious risk of night frosts, rhythmically stir 10ml of Valerian Prep/ha for 15 minutes in up to 125litres warmed water. Spray above the crops with a mist spray of the valerian solution late in the evening. As a rule there will be no damage from temperatures dropping to -3 or -4°C. If the frost is expected to be more serious than this you can use up to 60ml Valerian Prep/ha.
If frost remains on the crop in the morning or the evening spray has been neglected, the crop can still be rescued by applying another spray of Valerian Preparation in the early morning.
Valerian Preparation can be mixed with Seaweed Concentrate to spray out. An added benefit of spraying your crop with Valerian Preparation is that it enhances flowering in the crop.
There is also the protection offered by their annual application of diluted Biodynamic Tree Paste over the whole orchard to consider.


Since I was a boy, I have lived by the sea. Growing up on the south shore of Long Island, NY meant that the coast was only 15 minutes away but to me it always felt like too long of a drive. It could be the 'sweet air' as mountaineers call the oxygen rich air at sea level, regardless the song of the ocean siren is an undeniably powerful influence on me. In this gallery you'll experience why my love runs so deep - the calming, intense, vibrant and rugged stories as told by the place that the ocean meets the land.
Taken from a private estate on the Pacific coast - the varying cloud structures tell differing tales of coastal marine layering, temperature differences, moisture diffusion and atmospheric pressure gradients. Summarized, really cool clouds.


My 6th novel, a narrative of friendship, obsessive love, intellectual strivings and untimely death, was published by 4th Estate in 2009.
It tells the story of four friends who meet at Melbourne University in the late 1970s. They go on to Oxford for post-graduate studies, after which they go separate ways forging their careers. Twenty years later the friends are reunited in Melbourne, brought together as inaugural fellows of NOGA – the Network of Global Australians.
Ava, long married to Harry, is a successful novelist who carries a lifetime of secrets. At the time of her return to Melbourne she is struggling with her 7th novel. Conrad – Connie – now a celebrity philosopher living in the US, is questioning the seriousness of his work. A serial philanderer, he has also started to question his lifestyle. Helen is a molecular biologist, whose research combatting diarrhoea in some of the world's poorest locations can also be applied to bioterrorism. And lastly there is Jack, the brightest of them all, whose work in comparative religion has stalled owing to his long and futile obsession with Ava.
They arrive back in Melbourne each with their own problems and concerns. The friendships which they have long taken for granted are put under serious strain. When Ava finds herself caught in the worst of dilemmas it is not her old friends that come to her aid, but rather a secret lover from the past.
Writers have favourites among their works (although their next is always their best). My favourites are FACING THE MUSIC, THE PROSPEROUS THIEF and, yes, REUNION. REUNION is fair but unsentimental towards the characters – Ava, Jack, Connie and Helen – portraying them with all their attractions and all their flaws. And despite serious short-comings, it reveals friendship as sturdier and more forgiving than marriage. I'm rather drawn to that idea. And it plunges into fraught moral territory. There's a mutually beneficial sexual relationship between a teenager and a much older man, and one of the characters struggles against current laws and restrictions when they believe their life is no longer worth living.
The novel was well-received and there's plenty to be found online.
Slow TV recorded an on-stage interview conducted by Drusilla Modjeska at the 2009 Sydney Writers' Festival.
I have just read Reunion, having never heard of this writer before. There is plenty of philosophy and science basic references to show that the writer is indeed well read, but why the use of 'cell-phones'? Why the desperation to distinguish this is a writer from 'inner melbourne'? Was hoping this new decade would leave all the pretensions of the hipster generation behind, but this is pointless wallowing in the 'in crowd'. I am 46 if that makes any difference. From outer melbourne, because living in inner melbourne 20 years ago put me off the experience. For life.
Hi Natalie. 'Reunion' is a character-based novel. The story emerges from who the characters are, rather than who the author is. Connie would welcome being described as cool (hipster), but the others certainly wouldn't. Helen, the scientist, is quite daggy, and happy to be so.
When cell-phone rather than mobile is used it is because it comes from the point of view of a character who has spent much of his/her life in the US. Ava, for example, a long-time resident of the UK, would never use the term cell-phone. Re the location of the author, it is literary convention (certainly not desperation) that dictates how this is usually written.
I write about characters and situations that interest me. Novels take 3-4 years to write – a lovely length of time to look into what I don't know. I'm currently working on a novel about exile, the central character of which was born in the Soviet Union in 1962. My 5th novel opened in Berlin before the Great War.
One of the great attractions of fiction is its variety. Of course I would have preferred that you liked 'Reunion', but what matters far more is that fiction matters to YOU, that it connects with what you value. And it clearly does.


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text:
Webinar: Determining the Tg for Polymers and Polymeric Surface Coatings using Inverse Gas ChromatographyJohn Dowd2020-10-19T21:24:42+00:00
This webinar has already passed. For future webinars, please visit Webinars
Webinar: Determining the Tg for Polymers and Polymeric Surface Coatings
using Inverse Gas Chromatography (iGC)
Date: Wednesday, 27 May 2020
Inverse Gas Chromatography (iGC) can be used to determine the surface properties of many materials. In some cases, as well as surface adsorption, there is also bulk sorption of the iGC probe molecules, allowing bulk materials properties to also be determined.
The webinar will discus the use of iGC technique for:
Determining the Tg of polymeric materials and
Tg of polymer coatings when applied to highly crystalline fibres, such as fibre sizing and coatings
Prof. Daryl Williams Prof. Daryl R. Williams graduated with a B.Sc. (Hons) in Physical Chemistry from University of Melbourne, Australia and a M.Sc. in Polymer Science from Lehigh University, USA before coming to Imperial College London complete his PhD. He is founder and Managing Director of Surface Measurement Systems and the Professor of Particle Science in the Department of Chemical Engineering at Imperial College.
Prof. Williams has published over 100 papers in refereed journals and been granted international 5 patents. His research interests include the surface and bulk characterisation of complex organic solids, including especially biopharmaceutics, foods, pharmaceuticals and polymers as well as their manufacture using spray drying, crystallisation, freeze drying, milling and granulation. The Surface and Particle Engineering Laboratory at Imperial College he leads has 4 Postdoctoral research fellows and 11 PhD students.
He has invented one and has led the commercialisation of two standard techniques for materials characterisation, the Dynamics Vapour Sorption (DVS) and the Inverse Gas Chromatography (IGC) methods via Surface Measurement Systems. His work has been recognised by Imperial College with the President's Medal in 2017, the EEF's Future Manufacturing Award in Innovation for 2018 and the Institution of Chemical Engineers' Geldart Medal in 2018/19 for his "major contribution to particle technology".
Request for application notes


text:
\section{Introduction (temporary heading)}
The notion of a magic square has existed for thousands of years, with surviving written examples dating back to at least $650$ BC.  One of the oldest written examples is the so-called ``Lo Shu'' magic square -- a $3\times 3$ magic square from Chinese legends.  According to the legend, the Lo Shu square was observed as a pattern on the shell of a tortoise by Emperor Yu sometime between 2200 and 2100 BC.  As a simple idea that is often fiendishly difficult to implement, it is no wonder that magic squares have been studied and marveled at throughout history.  The idea is to fill in numbers into a square so that the sum along each row, column and diagonal are all equal to the same number -- often called the magic constant.  For instance the square in Figure~\ref{fig:square} is a representation of the ``Lo Shu'' magic square from the legends.  Here the magic constant is 15.

\begin{figure}[!htbp]
\captionsetup[subfigure]{labelformat=simple}
\centering
\begin{tikzpicture}
    \draw[thick] (0,0) -- (2,0);
    \draw[thick] (0,.667) -- (2,.667);
    \draw[thick] (0,1.333) -- (2,1.333);
    \draw[thick] (2,2) -- (0,2);
    \draw[thick] (0,0) -- (0,2);
    \draw[thick] (.667,0) -- (.667,2);
    \draw[thick] (1.333,0) -- (1.333,2);
    \draw[thick] (2,0) -- (2,2);
    \draw (.333,.333) node {$8$};
    \draw (1,.333) node {$1$};
    \draw (1.667,.333) node {$6$};
    \draw (.333,1) node {$3$};
    \draw (1,1) node {$5$};
    \draw (1.667,1) node {$7$};
    \draw (.333,1.667) node {$4$};
    \draw (1,1.667) node {$9$};
    \draw (1.667,1.667) node {$2$};
\end{tikzpicture}
\caption{The ``Lo Shu'' magic square.}
\label{fig:square}
\end{figure}

There have been numerous generalizations of magic squares to other shapes.  Ely introduced the idea of magic designs \cite{Ely}.  A design is an incidence structure consisting of ``points'' and ``lines,'' with each line being a subset of points.   A \emph{magic design} is then an injective function from the points to the natural numbers where the sum along any line is constant.  While Ely focused primarily on triangles and hexagons, other designs have since been studied (see e.g.\ \cite{Trotter}, \cite[Ch.17]{Gardner}, \cite{Hendricks}, and \cite{MN}).  A particulary nice family of designs comes from (combinatorial) configurations.  These are designs where every line has the same number of points, every point has the same number of lines through it, no two lines can intersect more than once, and no two points can appear on more than one line together.  For example, Trenkler \cite{Trenkler:3} studied magic stars which are configurations with two lines through every point.  More recently, Raney \cite{Raney} studied magic trilateral free $n_3$-configurations where three lines pass through every point, and each line contains three points.  In addition, Nash and Needleman \cite{NN2} studied magic finite projective planes which are configurations which contain a quadrilateral and further require all lines to intersect, and all pairs of points to be connected by a line.

On a different, but related front, Sedl$\acute{\text{a}}\check{\text{c}}$ek was the first to suggest studying magic labelings of graphs in \cite{Sedlacek}.  He suggested labeling the edges of a graph with real numbers and then defined an edge-labeling to be magic if the sum of the labels on all edges incident to a given vertex was a constant independent of the choice of vertex.  Stewart \cite{Stewart} studied some special cases of these labelings and called an edge-labeling \emph{supermagic} if the labels were consecutive integers starting from 1.  Note that the classical $n\times n$ magic square (ignoring diagonals) corresponds to a supermagic labeling of the complete bipartite graph $K_{n,n}$.  Many, many papers (see e.g.\ \cite{Doob2}, \cite{Jeurissen}, \cite{SGL}, \cite{Ivanco}, and \cite{LLSW} as examples) have since been devoted to determining whether and when various families of graphs can admit a supermagic edge-labeling (although many authors use the term ``magic" rather than supermagic).  For a more complete treatment of this history (and some of what follows) we suggest the ongoing survey by Gallian \cite{Gallian}.

In a departure from the above notion, Kotzig and Rosa \cite{KR1,KR2} defined a \emph{magic valuation} of a graph to be a labeling of both the vertices and the edges (now referred to as a \emph{total labeling} following Wallis \cite{Wallis}) with the consecutive integers starting from 1 such that the sum of any edge together with its two endpoints is a constant.  This idea was revived in 1996 when Ringel and Llad$\acute{\text{o}}$ \cite{RL} redefined these as \emph{edge-magic} (total) labelings.  Edge-magic total labelings of graphs have also been widely studied (see e.g.\ \cite{ELNR} and \cite{WBMS} for some important early examples).

More recently, MacDougall, Miller, Slamin, and Wallis \cite{MMSW} moved back towards Sedl$\acute{\text{a}}\check{\text{c}}$ek's original idea by introducing the notion of a \emph{vertex-magic} total labeling.  Such a labeling still labels both the vertices and the edges using the consecutive integers starting with 1, but instead requires that the sum of the labels on all edges incident to a given vertex plus the label on the vertex itself be a constant.  These magic graphs have also been extensively studied, with \cite{GM1}, \cite{GM2}, and \cite{AM} as more recent examples of work in this area.

Trenkler \cite{Trenkler:2} was the first to generalize supermagic edge-labelings to include \emph{hypergraphs}.  A hypergraph is a generalization of a graph which allows for multiple vertices to appear on each (hyper)edge.  In Trenkler's work a labeling of the complete $k$-partite hypergraph is supermagic if the sum of the labels on all edges incident to a fixed choice of $(k-1)$ vertices is independent of the choice of vertices.  Even more recently,  Boonklurb, Narissayaporn, and Singhun \cite{BNS} generalized super edge-magic labelings to hypergraphs as well.  Their generalization more closely parallels the original notion by requiring the sum of the label on an edge together with the labels on all vertices appearing on that edge to be a constant.  To be supermagic, all edge labels must be greater than any vertex labels in addition to the labels being consecutive integers starting from 1.

Generalizations have also been made to the labels over time.  While historically, $n \times n$ magic squares have most often been labeled by the integers 1 through $n^2$ -- and supermagic labelings are an attempt to hold on to this requirement -- there is no need for us to require such specificity.  To consider a function or labeling magic one simply needs a way to ``add'' the outputs or labels from each vertex and/or edge. Since the points in a graph, hypergraph, incidence structure, design, etc.\ do not appear in any fixed order, it is also necessary that the ``addition'' be commutative.  Thus, Doob \cite{Doob1} generalized magic edge-labelings to allow for labels coming from any abelian group.  He further added labels to the vertices and used those as restrictions on the sums of edge labels incident to each vertex while simultaneously relaxing the requirement that distinct edges have distinct labels.  In the special case where the sums are constant independent of vertex choice, these labelings have since been termed \emph{group-magic} edge-labelings (note, these are also sometimes referred to as $A$-magic edge-labelings, where $A$ denotes the (additive) abelian group used).  Some more recent work in this area can be seen in \cite{LSW,SLS,LL,Salehi,SL,JJ1,JJ2}.

One thing that nearly all magic shapes have in common (regardless of the type of magic labeling considered) is that they require each edge or line to contain the same number of points.  From the hypergraph perspective, such structures are called \emph{$k$-uniform} hypergraphs.  In what follows we consider the general case of $k$-uniform hypergraphs with vertices labeled by elements from an (additive) abelian group.  Note, we do not require our labels to be non-zero elements of the group (as in the $A$-magic situation) and we \emph{do} require that distinct vertices have distinct labels.

Observe, that we can incorporate the notion of an edge-magic total labeling (generalized to $k$-uniform hypergraphs) by imagining the label on each edge/line to be replaced by a label on an extra vertex created on that line that is not incident to any other line.  Our version of a magic labeling also serves to cover the cases of magic edge-labelings and vertex-magic total labelings by considering the dual (hyper)graphs where the sum along all edges incident to a fixed vertex becomes the sum of all vertices contained within a given edge.  Thus, we choose to follow \cite{NN2} in describing magic vertex-labelings of $k$-uniform hypergraphs over abelian groups.

In what follows, we demonstrate algorithms for determining both whether a given hypergraph is magic over some abelian group and also whether it is magic specifically over $\mathbb{Z}$.  The methods draw upon many results from number theory, and is an application of the local-global principle.

Much of the motivation for our discoveries stemmed from work specifically on $n_3$-configurations.  Thus, in the final section, we specialize to $n_3$-configurations and describe some results gained from the application of our algorithms to this type of hypergraph.

\subsection*{Labeling Hypergraphs}
While traditional graphs are collections of vertices with edges connecting them, they have been generalized to \emph{hypergraphs} by allowing edges to connect (or contain) more than two vertices.  A hypergraph can, therefore, be represented as a pair $\Gamma=(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is the set of \emph{points} and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ that we call the set of \emph{lines}.  A hypergraph is called \emph{$k$-uniform} if each line has exactly $k$ points on it.

A hypergraph is called a \emph{configuration} if, in addition:
\begin{itemize}
\item[(a)] Each pair of points $p_i \neq p_j \in \mathcal{P}$ is contained in at most one line.
\item[(b)] Each pair of lines $L_i \neq L_j \in \mathcal{L}$ intersects at at most one point.
\item[(c)] Each line is incident to the same number of points.
\item[(d)] Each point is incident to the same number of lines.
\end{itemize}
A configuration with $n$ points, each incident to $k$ lines, and $m$ lines, each incident to $l$ points is called an \emph{$(n_k, m_l)$-configuration}.  Note that such configurations can only occur if $nk=ml$.  A configuration is called \emph{symmetric} in the special case when $n=m$, in which case it follows that $k=l$, and these are often referred to as \emph{$n_k$-configurations}.  Figure \ref{fig:hypergraph} is an example of a $4$-uniform hypergraph which is not a configuration since the points $p_1$ and $p_2$ are simultaneously coincident to more than one line.

To study labelings on a  hypergraph $\Gamma=(\mathcal{P},\mathcal{L})$ we construct the $\mathbb{Z}$-modules
\[\mathbb{Z}\mathcal{P}=\left\{\left.\sum_{p\in \mathcal{P}}a_p p~\right|~a_p\in \mathbb{Z}\right\} \text{~and~} \mathbb{Z}\mathcal{L}=\left\{\left.\sum_{L\in \mathcal{L}}b_L L~\right|~b_L\in \mathbb{Z}\right\} \]
of the formal $\mathbb{Z}$-linear combinations of points and lines respectively.

 The hypergraph $\Gamma$ determines a relationship between $\mathbb{Z}\mathcal{P}$ and $\mathbb{Z}\mathcal{L}$ via a unique $\mathbb{Z}$-map $A:\mathcal{P}\rightarrow \mathbb{Z}\mathcal{L}$ known as the \emph{incidence map}.  For $p\in \mathcal{P}$, let
 \[Ap=\sum_{{L\in \mathcal{L}} \atop {p \in L}}L\]
 and then extend $\mathbb{Z}$-linearly.  This is the map that sends a point to the sum of lines through the given point. When  the points $p_1, \dots, p_n$ in $\mathcal{P}$ and the lines $L_1, \dots, L_m$ in $\mathcal{L}$ are enumerated, the map $A$ can be written as an $|\mathcal{L}| \times |\mathcal{P}|$ \emph{incidence matrix} whose $i,j$-entry is $1$ if $p_j \in L_i$ and is zero otherwise.
Figure~\ref{fig:incmatrix} is the incidence matrix from the hypergraph in Figure \ref{fig:hypergraph}.
We've chosen $L_1$, $L_2$, and $L_3$ to be the left, bottom, and right sides of the triangle, while taking $L_4$, $L_5$, and $L_6$, to be the blue, green, and red lines respectively.

\begin{figure}[ht!]
\captionsetup[subfigure]{}
\begin{subfigure}[b]{.55\linewidth}
\begin{tikzpicture}[scale=0.7, baseline={([yshift=-.8ex]current bounding box.center)}]]
    \draw[ultra thick] (1,0) -- (4,5.196);
    \draw[ultra thick] (4,5.196) -- (7,0);
    \draw[ultra thick] (7,0) -- (1,0);
    \draw[ultra thick,bend left,blue,dashed,rounded corners] (1,0) to (2,1.732) to (6,1.732) to (5,0);
    \draw[ultra thick,bend left,green,dotted,rounded corners] (4,5.196) to (5,3.464) to (3,0) to (2,1.732);
    \draw[ultra thick,bend left,red,dashdotted,rounded corners] (7,0) to (5,0) to (3,3.464) to (5,3.464);
    \draw[thick,fill=blue] (1,0) circle (0.1);
    \draw[thick,fill=blue] (2,1.732) circle (0.1);
    \draw[thick,fill=blue] (3,3.464) circle (0.1);
    \draw[thick,fill=blue] (4,5.196) circle (0.1);
    \draw[thick,fill=blue] (5,3.464) circle (0.1);
    \draw[thick,fill=blue] (6,1.732) circle (0.1);
    \draw[thick,fill=blue] (7,0) circle (0.1);
    \draw[thick,fill=blue] (3,0) circle (0.1);
    \draw[thick,fill=blue] (5,0) circle (0.1);
    \draw(0.1,0.2) node {\small $p_1$}
    \draw(1.1,1.932) node {\small $p_2$}
    \draw(2.1,3.664) node {\small $p_3$}
    \draw(4,5.496) node {\small $p_4$}
    \draw(5.9,3.664) node {\small $p_9$}
    \draw(6.9,1.932) node {\small $p_8$}
    \draw(7.9,0.2) node {\small $p_7$}
    \draw(5,-0.6) node {\small $p_6$}
    \draw(3,-0.6) node {\small $p_5$}
\end{tikzpicture}
\caption{A $4$-uniform hypergraph.}
\label{fig:hypergraph}
\end{subfigure}
\begin{subfigure}[b]{.4\linewidth}
$\kbordermatrix{
&\bm{p_1}&\bm{p_2}&\bm{p_3}&\bm{p_4}&\bm{p_5}&\bm{p_6}&\bm{p_7}&\bm{p_8}&\bm{p_9}\\
\bm{L_1}&1&1&1&1&0&0&0&0&0\\
\bm{L_2}&1&0&0&0&1&1&1&0&0\\
\bm{L_3}&0&0&0&1&0&0&1&1&1\\
{\color{blue}\bm{L_4}}&1&1&0&0&0&1&0&1&0\\
{\color{green}\bm{L_5}}&0&1&0&1&1&0&0&0&1\\
{\color{red}\bm{L_6}}&0&0&1&0&0&1&1&0&1
}$
\subcaption{An incidence matrix.}
\label{fig:incmatrix}
\end{subfigure}
\caption{}
\label{fig:hypergraphmatrix}
\end{figure}
Since every line $L\in \mathcal{L}$ is a collection of points from $\mathcal{P}$ we can view $L$ as being in $\mathbb{Z}\mathcal{P}$ by summing the points that make up $L$.  This is just the transpose of the incidence map, $A^\top:\mathbb{Z}\mathcal{L}\rightarrow \mathbb{Z}\mathcal{P}$ which is $A^\top L=\sum_{p\in L} p$.  It turns out, determining whether or not $\Gamma$ can be made magic over some group is determined by  the image of $A^\top$ or, equivalently, the $\mathbb{Z}$-row space of $A$.

Magic labelings of $\Gamma$ are functions on $\mathcal{P}$, so it will be useful to consider $\mathbb{Z}^\mathcal{P}$ (\textit{resp.} $\mathbb{Z}^\mathcal{L}$), the space of $\mathbb{Z}$-valued functions on $\mathcal{P}$ (\textit{resp.} $\mathcal{L}$).  It will be useful to extend $\mathbb{Z}$-valued functions $f$ on $\mathcal{P}$ to functions on $\mathbb{Z}\mathcal{P}$. We do this by
\[f\left(\sum_{p\in P}a_p p\right)=\sum_{p\in P}a_p f(p).\]
These functions can naturally be identified with $\mathbb{Z}\mathcal{P}$ (\textit{resp.} $\mathbb{Z}\mathcal{L}$) by the map
 \begin{eqnarray*}
 [\cdot]:\mathbb{Z}^\mathcal{P}&\rightarrow& \mathbb{Z}\mathcal{P}\\
 f&\mapsto & \sum_{p\in \mathcal{P}} f(p) p
 \end{eqnarray*}
 and similarly for $[\cdot]:\mathbb{Z}^\mathcal{L}\rightarrow \mathbb{Z}\mathcal{L}$.  Depending on the context, it may be more useful to consider functions and at other times consider combinations of points and lines.

We usually will not be working over $\mathbb{Z}$, but rather over abelian groups.  Since every abelian group $G$ is a $\mathbb{Z}$-module, the notions of $\mathbb{Z}\mathcal{P}$ and $\mathbb{Z}\mathcal{L}$ extend naturally to $G\mathcal{P}$ and $G\mathcal{L}$, and we can view the incidence map as a $G$-map.  When we do this, we will denote the incidence map/matrix as $A_G$.  In the special case where $G=\mathbb{Z}/r\mathbb{Z}$ for some $r\in \mathbb{N}$ we will just write $A_r$.

A $G$-labeling of $\Gamma$ can be represented as a function $f:\mathcal{P} \to G$.    Define $h\in \mathbb{Z}^\mathcal{L}$ by  \[[h]=A_G[f] = \sum_{{1\leq i \leq m}\atop {p \in L_i}} f(p)L_i\] as a function on the lines in $\mathcal{L}$.  Then $h(L_i)=\sum_{p\in L_i} f(p)$.   From this perspective, we consider $f$ as a function on the lines by $f(A^\top L)=\sum_{p \in L} f(p)$ instead.

We now introduce some terms  in order to get at the idea of ``magicness''.  A function $f:\mathcal{P} \to G$ is called \emph{line-invariant} if there exists a constant $c \in G$ such that $f(A^\top L) = c$ for all $L \in \mathcal{L}$.  Such a constant is called the \emph{magic constant} for $f$.  If, furthermore, we have that $f(p) \neq f(q)$ for all $p\neq q \in \mathcal{P}$, then we say that $f$ is a \emph{magic} function.  A $k$-uniform hypergraph $\Gamma=(\mathcal{P}, \mathcal{L})$ is said to be \emph{magic over} an abelian group $G$
if there exists a magic function $f:\mathcal{P} \to G$.  Observe that since $k$-uniform hypergraphs have the same number of points on each line, every constant function $f(p)=c$, for any abelian group $G$ and any $c \in G$, is line-invariant.  A non-constant example is the 4-uniform hypergraph from Figure~\ref{fig:hypergraph}, which is magic over $(\mathbb{Z}/2\mathbb{Z})^4$ with magic labeling given in Figure~\ref{fig:magicHG}.

\begin{figure}[ht!]
\captionsetup[subfigure]{labelformat=simple}
\centering
\begin{tikzpicture}[scale=0.7]
    \draw[ultra thick] (1,0) -- (4,5.196);
    \draw[ultra thick] (4,5.196) -- (7,0);
    \draw[ultra thick] (7,0) -- (1,0);
    \draw[ultra thick,bend left,blue,dashed,rounded corners] (1,0) to (2,1.732) to (6,1.732) to (5,0);
    \draw[ultra thick,bend left,green,dotted,rounded corners] (4,5.196) to (5,3.464) to (3,0) to (2,1.732);
    \draw[ultra thick,bend left,red,dashdotted,rounded corners] (7,0) to (5,0) to (3,3.464) to (5,3.464);
    \draw[thick,fill=blue] (1,0) circle (0.1);
    \draw[thick,fill=blue] (2,1.732) circle (0.1);
    \draw[thick,fill=blue] (3,3.464) circle (0.1);
    \draw[thick,fill=blue] (4,5.196) circle (0.1);
    \draw[thick,fill=blue] (5,3.464) circle (0.1);
    \draw[thick,fill=blue] (6,1.732) circle (0.1);
    \draw[thick,fill=blue] (7,0) circle (0.1);
    \draw[thick,fill=blue] (3,0) circle (0.1);
    \draw[thick,fill=blue] (5,0) circle (0.1);
    \draw(-0.1,0.2) node {\small $(1,1,1,1)$};
    \draw(0.9,1.932) node {\small $(1,1,0,1)$};
    \draw(1.9,3.664) node {\small $(1,1,1,0)$};
    \draw(4,5.596) node {\small $(1,1,0,0)$};
    \draw(6.1,3.664) node {\small $(0,1,1,0)$};
    \draw(7.1,1.932) node {\small $(1,0,0,1)$};
    \draw(8.1,0.2) node {\small $(0,0,1,1)$};
    \draw(5,-0.6) node {\small $(1,0,1,1)$};
    \draw(3,-0.6) node {\small $(0,1,1,1)$};
\end{tikzpicture}
\caption{A magic labeling over $(\mathbb{Z}/2\mathbb{Z})^4$.}
\label{fig:magicHG}
\end{figure}
Line-invariant functions are closely tied to $\text{Ker} A_G$.  If $[h]=A_G[f]=0$ for some $f:\mathcal{P}\rightarrow G$ then $f(A^\top L)=h(L)=0$ for all $L\in\mathcal{L}$.  Therefore, $f$ is line-invariant with magic constant $0$.  The converse also holds, and so $\text{Ker} A_G$ is the space of all line-invariant functions with magic constant $0$.  In the case of $k$-uniform hypergraphs where $gcd(k, |G|)=1$ one can obtain all line invariant functions by adding the constant functions to $\text{Ker} A_G$.

\subsection*{Substructures}
The existence of certain substructures or subconfigurations within a hypergraph places restrictions on the potential line-invariant labelings.  When considering the unique $7_3$-configuration -- the Fano configuration --  removing any point and the three lines incident to that point results in what is called a complete quadrilateral (see Figure~\ref{fig:fano}).  A \emph{complete quadrilateral}, $\mathcal{Q}=(\mathcal{P}_\mathcal{Q}, \mathcal{L}_\mathcal{Q})$ is an arrangement of points and lines consisting of four lines, no three of which pass through the same point, and the six points of intersection of these lines. So $\mathcal{Q}$ is a $(6_2,4_3)$-configuration in its own right.  Containing this substructure rules out many possible groups over which a magic labeling could exist due to the following  observation.

\begin{figure}[ht!]
\captionsetup[subfigure]{}
\centering
\begin{subfigure}[b]{.25\linewidth}
\begin{tikzpicture}[scale=0.8
    \draw[ultra thick] (1,0) -- (5,0);
    \draw[ultra thick] (5,0) -- (3,3.464);
    \draw[ultra thick] (1,0) -- (3,3.464);
    \draw[ultra thick] (3,0) -- (3,3.464);
    \draw[ultra thick] (2,1.732) -- (5,0);
    \draw[ultra thick] (1,0) -- (4,1.732);
    \draw[ultra thick] (3,1.155) circle(1.155);
    \draw[thick,fill=blue] (1,0) circle (0.1);
    \draw[thick,fill=blue] (2,1.732) circle (0.1);
    \draw[thick,fill=blue] (3,0) circle (0.1);
    \draw[thick,fill=blue] (3,1.155) circle (0.1);
    \draw[thick,fill=blue] (3,3.464) circle (0.1);
    \draw[thick,fill=blue] (4,1.732) circle (0.1);
    \draw[thick,fill=blue] (5,0) circle (0.1);

\end{tikzpicture}
\subcaption{The Fano $7_3$-configuration.\\}
\end{subfigure}
\hskip 1.5cm
\begin{subfigure}[b]{.4\linewidth}
\begin{tikzpicture}[scale=0.8
    \draw[ultra thick] (5,0) -- (3,3.464);
    \draw[ultra thick] (1,0) -- (3,3.464);
    \draw[ultra thick] (2,1.732) -- (5,0);
    \draw[ultra thick] (1,0) -- (4,1.732);
    \draw[thick,fill=blue] (1,0) circle (0.1);
    \draw[thick,fill=blue] (2,1.732) circle (0.1);
    \draw[thick,fill=blue] (3,1.155) circle (0.1);
    \draw[thick,fill=blue] (3,3.464) circle (0.1);
    \draw[thick,fill=blue] (4,1.732) circle (0.1);
    \draw[thick,fill=blue] (5,0) circle (0.1);
    \draw(0.5,0) node {$p$};
    \draw(5.5,0) node {$q$};
\end{tikzpicture}
\subcaption{The complete quadrilateral obtained by removing a point and its three incident lines.}
\end{subfigure}
\begin{comment}
$\kbordermatrix{
&p_1&p_2&p_3&p_4&p_5&p_6&p_7\\
\ell_1&0&1&1&1&0&0&0\\
\ell_2&1&0&1&0&1&0&0\\
\ell_3&1&1&0&0&0&1&0\\
\ell_4&1&0&0&1&0&0&1\\
\ell_5&0&1&0&0&1&0&1\\
\ell_6&0&0&1&0&0&1&1\\
\ell_7&0&0&0&1&1&1&0
}$
\end{comment}
\caption{
\label{fig:fano}
\end{figure}

\begin{prop}\label{prop:quad}
Let $f:\mathcal{P} \to G$ be any line-invariant function and let $\mathcal{Q}$ be a complete quadrilateral in $\Gamma$.  If $p$ and $q$ are the two points in $\mathcal{Q}$ that are not incident to a common line, then $2f(p)=2f(q)$.
\end{prop}
\begin{proof}
Let $L_1$, $L_2$ be the lines incident to $p$ and let $L_3$, $L_4$ be the lines incident to $q$.  Applying $f$ to the linear combination $L_1 + L_2 - L_3 - L_4$ we have
$$0 = f(L_1+L_2-L_3-L_4) = f(2p-2q) = 2f(p)-2f(q),$$
hence the result follows.
\end{proof}

For the Fano $7_3$-configuration specifically, we see as a corollary that, $2f(p)=2f(q)$ for all pairs of points $p,q \in \mathcal{P}$ and all line-invariant functions $f$.  This is because given any pair of points $p$ and $q$ there exists a line connecting them and thus, by removing the third point on that line, we are left with a complete quadrilateral with $p$ and $q$ in the appropriate positions.  More generally though, any time such a structure exists within our hypergraph $\Gamma$, it follows that $\Gamma$ is not magic over any group which does not have 2 as a zero diviso
.  For the Fano $7_3$-configuration, it is known \cite{NN2} that the smallest group over which it admits a magic labeling is $(\mathbb{Z}/2\mathbb{Z})^3$.

There are also substructures which forbid all line-invariant functions $f$ from being injective which we call \emph{forbidden substructures}.  Containing a forbidden substructure makes it impossible for a hypergraph to be magic over any abelian group.  More specifically, forbidden substructures create restrictions by admitting integer linear combinations of their lines which result in a difference of two points $p-q$, i.e.\ $p-q$ is in the image of $[\cdot]$.  For example, the smallest $n_3$-configuration which is not magic over any abelian group $G$ is the $9_3$-configuration which is denoted as configuration $(9_3)_3$ in \cite{G1} and is given in Figure~\ref{fig:(9_3)_3}.

\begin{figure}[ht!]
\captionsetup[subfigure]{labelformat=simple}
\centering
\begin{tikzpicture}[scale=0.9]
\draw[ultra thick, red] (0,0) -- (2,3.46);
\draw[ultra thick, red] (0,0) -- (4, 0);
\draw[ultra thick, green] (0,0) -- (2.81, 2.06);
\draw[ultra thick, green] (0.81, 1.4) -- (4,0);
\draw[ultra thick] (0.81, 1.4) -- (2.24, 1.26);
\draw[ultra thick] (1.79, 1.31) -- (2.38, 0);
\draw[ultra thick, green] (2,3.46) -- (2.38, 0);
\draw[ultra thick, red] (1.97, 0.89) -- (2.81, 2.06);
\draw[ultra thick] (2, 3.46) -- (4, 0);
\draw[thick,fill=blue] (0,0) circle (0.1);
\draw[thick,fill=blue] (4,0) circle (0.1);
\draw[thick,fill=blue] (2, 3.46) circle (0.1);
\draw[thick,fill=blue] (2.38, 0) circle (0.1);
\draw[thick,fill=blue] (2.81, 2.06) circle (0.1);
\draw[thick,fill=blue] (0.81, 1.4) circle (0.1);
\draw[thick,fill=blue] (2.24, 1.26) circle (0.1);
\draw[thick,fill=blue] (2.81, 2.06) circle (0.1);
\draw[thick,fill=blue] (1.79, 1.31) circle (0.1);
\draw[thick,fill=blue] (1.97, 0.89) circle (0.1);
\draw(-0.2,-0.2) node {\small $q$};
\draw(1.64,1.61) node {\small $p$};
\end{tikzpicture}
\caption{The smallest configuration which is not magic over any abelian group.}
\label{fig:(9_3)_3}
\end{figure}

Observe, that if we take the sum of the green lines and subtract the three red lines then we are left with exactly $p-q$.  Hence, if $f \colon \mathcal{P} \to G$ is any line-invariant function, then applying $f$ to this linear combination shows us that $f(p)=f(q)$.  We reproduce this subconfiguration more clearly (replacing green lines with black ones) in Figure~\ref{fig:subconfig1} together with another in Figure~\ref{fig:subconfig2}.  We have found that the majority of $n_3$-configurations for $n=11$, 12, 13, and 14 contain at least one of these two small examples of forbidden substructures.
In each case, we get the difference $p-q$ by adding the black lines and subtracting the red ones.

\begin{figure}[ht!]
\captionsetup[subfigure]{}
\centering
\begin{subfigure}[b]{.28\linewidth}
\begin{tikzpicture
    \draw[ultra thick,red] (1,0) -- (3,0);
    \draw[ultra thick,red] (1,1) -- (3,1);
    \draw[ultra thick,red] (1,2) -- (3,2);
    \draw[ultra thick] (1,0) -- (1,2);
    \draw[ultra thick,bend right] (1,0) to (2,1);
    \draw[ultra thick] (2,1) -- (2,2);
    \draw[ultra thick] (3,0) -- (3,2);
    \draw[thick,fill=blue] (1,0) circle (0.1);
    \draw[thick,fill=blue] (2,0) circle (0.1);
    \draw[thick,fill=blue] (3,0) circle (0.1);
    \draw[thick,fill=blue] (1,1) circle (0.1);
    \draw[thick,fill=blue] (2,1) circle (0.1);
    \draw[thick,fill=blue] (3,1) circle (0.1);
    \draw[thick,fill=blue] (1,2) circle (0.1);
    \draw[thick,fill=blue] (2,2) circle (0.1);
    \draw[thick,fill=blue] (3,2) circle (0.1);
    \draw(1,-0.3) node {\small $p$};
    \draw(2,-0.3) node {\small $q$};
    \draw[white] (0.5,0) node{a};
\end{tikzpicture}
\subcaption{A forbidden substructure derived from a $9_3$-configuration.}
\label{fig:subconfig1}
\end{subfigure}
\hskip 1.5cm
\begin{subfigure}[b]{.28\linewidth}
\begin{tikzpicture
    \draw[ultra thick,red] (2,0) -- (4,0);
    \draw[ultra thick,red] (1,1) -- (3,1);
    \draw[ultra thick,red] (1,2) -- (3,2);
    \draw[ultra thick] (1,0) -- (1,2);
    \draw[ultra thick] (2,0) -- (2,2);
    \draw[ultra thick] (3,0) -- (3,2);
    \draw[thick,fill=blue] (1,0) circle (0.1);
    \draw[thick,fill=blue] (2,0) circle (0.1);
    \draw[thick,fill=blue] (3,0) circle (0.1);
    \draw[thick,fill=blue] (1,1) circle (0.1);
    \draw[thick,fill=blue] (2,1) circle (0.1);
    \draw[thick,fill=blue] (3,1) circle (0.1);
    \draw[thick,fill=blue] (1,2) circle (0.1);
    \draw[thick,fill=blue] (2,2) circle (0.1);
    \draw[thick,fill=blue] (3,2) circle (0.1);
    \draw[thick,fill=blue] (4,0) circle (0.1);
    \draw(1,-0.3) node {\small $p$};
    \draw(4,-0.3) node {\small $q$};
\end{tikzpicture}
\subcaption{Another forbidden substructure forcing $f(p)=f(q)$.}
\label{fig:subconfig2}
\end{subfigure}
\caption{Forbidden substructures appearing commonly in $n_3$-configurations.}
\label{fig:subconfigs}
\end{figure}

In the setting of $k$-uniform hypergraphs, these types of forbidden substructures can be generalized and there are many other types of forbidden substructures as well.  For example, any pair of lines sharing $k-1$ common points forces the two additional points to be equal.  See Figure~\ref{fig:substructure1} for this example and Figures~\ref{fig:substructure2} and \ref{fig:substructure3} generalizations of the previous two.

\begin{figure}[ht!]
\captionsetup[subfigure]{}
\centering
\begin{subfigure}[b]{.28\linewidth}
\begin{tikzpicture}[scale=0.75,baseline={([yshift=-2ex]current bounding box.center)}]
    \draw[ultra thick,red,bend left] (0,0) to (1,1);
    \draw[ultra thick,red,bend right] (1,1) to (2,2);

    \draw[ultra thick,bend right] (0,0) to (1,1);
    \draw[ultra thick,bend left] (1,1) to (2,2);

    \draw[dotted,thick] (2,2) to (3,3);
    \draw[ultra thick,red] (3,3) -- (4,3);
    \draw[ultra thick] (3,3) -- (3,4);
    \draw[thick,fill=blue] (0,0) circle (0.133);
    \draw[thick,fill=blue] (1,1) circle (0.133);
    \draw[thick,fill=blue] (2,2) circle (0.133);
    \draw[thick,fill=blue] (3,3) circle (0.133);

    \draw[thick,fill=blue] (3,4) circle (0.133);
    \draw[thick,fill=blue] (4,3) circle (0.133);
    \draw(2.6,4) node {\small $p$};
    \draw(4,2.6) node {\small $q$};
\end{tikzpicture}
\subcaption{A forbidden substructure.}
\label{fig:substructure1}
\end{subfigure}
\qquad
\begin{subfigure}[b]{.28\linewidth}
\begin{tikzpicture}[scale=1.5,baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[ultra thick,red] (0,2) -- (0.67,2);
    \draw[thick,red,dotted] (0.67,2) -- (1.5,2);
    \draw[ultra thick,red] (1.33,2) -- (1.67,2);
    \draw[ultra thick,red] (0,1.67) -- (0.67,1.67);
    \draw[thick,red,dotted] (0.67,1.67) -- (1.5,1.67);
    \draw[ultra thick,red] (1.33,1.67) -- (1.67,1.67);
    \draw[ultra thick,red] (0,1.33) -- (0.67,1.33);
    \draw[thick,red,dotted] (0.67,1.33) -- (1.5,1.33);
    \draw[ultra thick,red] (1.33,1.33) -- (1.67,1.33);
    \draw[ultra thick,red] (0,0.5) -- (0.67,0.5);
    \draw[thick,red,dotted] (0.67,0.5) -- (1.5,0.5);
    \draw[ultra thick,red] (1.33,0.5) -- (1.67,0.5);
    \draw[ultra thick,red] (0,0) -- (0.67,0);
    \draw[thick,red,dotted] (0.67,0) -- (1.5,0);
    \draw[ultra thick,red] (1.33,0) -- (1.67,0);
    \draw[ultra thick] (0,1.33) -- (0,2);
    \draw[dotted,thick] (0,0.5) -- (0,1.3);
    \draw[ultra thick] (0,0) -- (0,0.5);
    \draw[thick,fill=blue] (0,0) circle (0.0667);
    \draw[thick,fill=blue] (0,0.5) circle (0.0667);
    \draw[thick,fill=blue] (0,1.33) circle (0.0667);
    \draw[thick,fill=blue] (0,1.67) circle (0.0667);
    \draw[thick,fill=blue] (0,2) circle (0.0667);
    \draw[ultra thick] (0.33,1.33) -- (0.33,2);
    \draw[dotted,thick] (0.33,0.5) -- (0.33,1.33);
    \draw[ultra thick,bend right] (0,0) to (0.33,0.5);
    \draw[thick,fill=blue] (0.33,0) circle (0.0667);
    \draw[thick,fill=blue] (0.33,0.5) circle (0.0667);
    \draw[thick,fill=blue] (0.33,1.33) circle (0.0667);
    \draw[thick,fill=blue] (0.33,1.67) circle (0.0667);
    \draw[thick,fill=blue] (0.33,2) circle (0.0667);
    \draw[ultra thick] (0.67,1.3) -- (0.67,2);
    \draw[dotted,thick] (0.67,0.5) -- (0.67,1.33);
    \draw[ultra thick] (0.67,0) -- (0.67,0.5);
    \draw[thick,fill=blue] (0.67,0) circle (0.0667);
    \draw[thick,fill=blue] (0.67,0.5) circle (0.0667);
    \draw[thick,fill=blue] (0.67,1.33) circle (0.0667);
    \draw[thick,fill=blue] (0.67,1.67) circle (0.0667);
    \draw[thick,fill=blue] (0.67,2) circle (0.0667);
    \draw[ultra thick] (1.33,1.33) -- (1.33,2);
    \draw[dotted,thick] (1.33,0.5) -- (1.33,1.3);
    \draw[ultra thick] (1.33,0) -- (1.33,0.5);
    \draw[thick,fill=blue] (1.33,0) circle (0.0667);
    \draw[thick,fill=blue] (1.33,0.5) circle (0.0667);
    \draw[thick,fill=blue] (1.33,1.33) circle (0.0667);
    \draw[thick,fill=blue] (1.33,1.67) circle (0.0667);
    \draw[thick,fill=blue] (1.33,2) circle (0.0667);
    \draw[ultra thick] (1.67,1.33) -- (1.67,2);
    \draw[dotted,thick] (1.67,0.5) -- (1.67,1.3);
    \draw[ultra thick] (1.67,0) -- (1.67,0.5);
    \draw[thick,fill=blue] (1.67,0) circle (0.0667);
    \draw[thick,fill=blue] (1.67,0.5) circle (0.0667);
    \draw[thick,fill=blue] (1.67,1.33) circle (0.0667);
    \draw[thick,fill=blue] (1.67,1.67) circle (0.0667);
    \draw[thick,fill=blue] (1.67,2) circle (0.0667);
    \draw(0,-0.3) node {\small $p$};
    \draw(0.33,-0.3) node {\small $q$};
    \draw[white] (-0.2,0) node {a};
\end{tikzpicture}
\subcaption{A generalization of Figure~\ref{fig:subconfig1}.}
\label{fig:substructure2}
\end{subfigure}
\qquad
\begin{subfigure}[b]{.28\linewidth}
\begin{tikzpicture}[scale=1.5,baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[ultra thick,red] (0,2) -- (0.67,2);
    \draw[thick,red,dotted] (0.67,2) -- (1.5,2);
    \draw[ultra thick,red] (1.33,2) -- (1.67,2);
    \draw[ultra thick,red] (0,1.67) -- (0.67,1.67);
    \draw[thick,red,dotted] (0.67,1.67) -- (1.5,1.67);
    \draw[ultra thick,red] (1.33,1.67) -- (1.67,1.67);
    \draw[ultra thick,red] (0,1.33) -- (0.67,1.33);
    \draw[thick,red,dotted] (0.67,1.33) -- (1.5,1.33);
    \draw[ultra thick,red] (1.33,1.33) -- (1.67,1.33);
    \draw[ultra thick,red] (0,0.5) -- (0.67,0.5);
    \draw[thick,red,dotted] (0.67,0.5) -- (1.5,0.5);
    \draw[ultra thick,red] (1.33,0.5) -- (1.67,0.5);
    \draw[ultra thick,red] (0.33,0) -- (0.67,0);
    \draw[thick,red,dotted] (0.67,0) -- (1.5,0);
    \draw[ultra thick,red] (1.33,0) -- (2,0);
    \draw[ultra thick] (0,1.33) -- (0,2);
    \draw[dotted,thick] (0,0.5) -- (0,1.3);
    \draw[ultra thick] (0,0) -- (0,0.5);
    \draw[thick,fill=blue] (0,0) circle (0.0667);
    \draw[thick,fill=blue] (0,0.5) circle (0.0667);
    \draw[thick,fill=blue] (0,1.33) circle (0.0667);
    \draw[thick,fill=blue] (0,1.67) circle (0.0667);
    \draw[thick,fill=blue] (0,2) circle (0.0667);
    \draw[ultra thick] (0.33,1.33) -- (0.33,2);
    \draw[dotted,thick] (0.33,0.5) -- (0.33,1.33);
    \draw[ultra thick] (0.33,0) to (0.33,0.5);
    \draw[thick,fill=blue] (0.33,0) circle (0.0667);
    \draw[thick,fill=blue] (0.33,0.5) circle (0.0667);
    \draw[thick,fill=blue] (0.33,1.33) circle (0.0667);
    \draw[thick,fill=blue] (0.33,1.67) circle (0.0667);
    \draw[thick,fill=blue] (0.33,2) circle (0.0667);
    \draw[ultra thick] (0.67,1.3) -- (0.67,2);
    \draw[dotted,thick] (0.67,0.5) -- (0.67,1.33);
    \draw[ultra thick] (0.67,0) -- (0.67,0.5);
    \draw[thick,fill=blue] (0.67,0) circle (0.0667);
    \draw[thick,fill=blue] (0.67,0.5) circle (0.0667);
    \draw[thick,fill=blue] (0.67,1.33) circle (0.0667);
    \draw[thick,fill=blue] (0.67,1.67) circle (0.0667);
    \draw[thick,fill=blue] (0.67,2) circle (0.0667);
    \draw[ultra thick] (1.33,1.33) -- (1.33,2);
    \draw[dotted,thick] (1.33,0.5) -- (1.33,1.3);
    \draw[ultra thick] (1.33,0) -- (1.33,0.5);
    \draw[thick,fill=blue] (1.33,0) circle (0.0667);
    \draw[thick,fill=blue] (1.33,0.5) circle (0.0667);
    \draw[thick,fill=blue] (1.33,1.33) circle (0.0667);
    \draw[thick,fill=blue] (1.33,1.67) circle (0.0667);
    \draw[thick,fill=blue] (1.33,2) circle (0.0667);
    \draw[ultra thick] (1.67,1.33) -- (1.67,2);
    \draw[dotted,thick] (1.67,0.5) -- (1.67,1.3);
    \draw[ultra thick] (1.67,0) -- (1.67,0.5);
    \draw[thick,fill=blue] (1.67,0) circle (0.0667);
    \draw[thick,fill=blue] (1.67,0.5) circle (0.0667);
    \draw[thick,fill=blue] (1.67,1.33) circle (0.0667);
    \draw[thick,fill=blue] (1.67,1.67) circle (0.0667);
    \draw[thick,fill=blue] (1.67,2) circle (0.0667);
    \draw[thick,fill=blue] (2,0) circle (0.0667);
    \draw(0,-0.3) node {\small $p$};
    \draw(2,-0.3) node {\small $q$};
\end{tikzpicture}
\subcaption{A generalization of Figure~\ref{fig:subconfig2}.}
\label{fig:substructure3}
\end{subfigure}
\caption{
\label{fig:substructures}
\end{figure}

Searching for particular substructures within complicated hypergraphs is computationally time consuming.  Furthermore, it may not provide definitive information as
it is unclear whether there is a finite list of forbidden substructures which accounts for all $k$-uniform hypergraphs which are never magic.  In fact, we do not even know if there is such a finite list in the special case of $n_3$-configurations.
With that in mind, it would be helpful to have a definitive, computational way to determine whether a given $k$-uniform hypergraph can be magically labeled over some abelian group.

\subsection*{When $\bm{\Gamma}$ is Not Magic}

If we can find a pair of points $p$ and $q$ that have equal values for all line-invariant functions, then our $k$-uniform hypergraph cannot be magic over any group.  The somewhat surprising result is that the converse is also true, i.e.\ if a $k$-uniform hypergraph cannot be magic over any group, then there must exist a fixed pair of points $p$ and $q$ that have equal values under \emph{every} line-invariant function.  To see this, suppose that no such pair of points exists.  Then for every pair of points $p,q \in \mathcal{P}$ there exists some abelian group $G_{p,q}$ and some line-invariant function $f_{p,q}:\mathcal{P} \to G_{p,q}$ such that $f_{p,q}(p)  \neq f_{p,q}(q)$.  Thus, the function
$$f=\bigoplus_{(p,q)\in\mathcal{P}} f_{p,q}:\mathcal{P} \to \bigoplus_{(p,q) \in \mathcal{P}} G_{p,q}$$
is both line-invariant and distinguishes all points.  Hence, we have found a group over which the configuration $\Gamma$ is magic.

\begin{theorem}\label{thm:equalpair} A $k$-uniform hypergraph $\Gamma=(\mathcal{P},\mathcal{L})$ is not magic over any abelian group if and only if there exists a pair of points $p,q \in \mathcal{P}$ such that, for all abelian groups $G$ and all line-invariant functions $f:\mathcal{P} \to G$, we have $f(p)=f(q)$.
\end{theorem}
This result is theoretical in nature, but it is the basis for a result  which allows one to computationally determine whether or not a $k$-uniform hypergraph can be made magic.  This is done by showing  $f(p)=f(q)$ for every line invariant function if and only if  $p-q\in \operatorname{Im} A^\top$.

Showing that a hypergraph cannot be made magic is relatively straight forward. Assume $p, q\in \mathcal{P}$ with $p-q\in \operatorname{Im} A^\top$.  Then there exists a $\mathbb{Z}$-linear combination of lines, $S$, such that $A^\top S=A^\top\sum_{i=1}^m b_iL_i=p-q$.  Next, let $f$ be a line invariant function to some group $G$ with magic constant $c\in G$.  It follows that $f(A^\top S)=f(p-q)=f(p)-f(q)$.  However, \[f(A^\top S)=\sum_{i=1}^m b_if(A^\top L_i)=c\sum_{i=1}^m b_i.\]
So, $c\sum_{i=1}^m b_i=f(p)-f(q)$.  If we can show $\sum_{i=1}^m b_i=0$ then we will have $f(p)=f(q)$ and the function cannot be magic.

Consider the $\mathbb{Z}$-linear map $\ast:\mathbb{Z}\mathcal{P}\rightarrow \mathbb{Z}$, where \[\left(\sum_{i=1}^n a_i p_i\right)^\ast=\sum_{i=1}^n a_i.\]
This function is counting, with multiplicity, how many points are in a given linear combination.  For a $k$-uniform hypergraph and any line $L$ we have $[A^\top L]^\ast=k$  since it is just counting how many points are on the line.  Therefore,
\[0=(p-q)^*=(A^\top S)^*=\sum_{i=1}^m b_i(A^\top L_i)^*=k \sum_{i=1}^m b_i,\]
and so $\sum_{i=1}^m b_i=0$ as desired.  This is summarized in the following:

\begin{prop}\label{prop:notmagic}
 A $k$-uniform hypergraph $\Gamma=(\mathcal{P}, \mathcal{L})$ is not magic over any Ableian group $G$ if there are $p, q\in \mathcal{P}$ with $p-q \in \operatorname{Im} A^\top$.
\end{prop}

Since $\operatorname{Im} A^\top$ is the $\mathbb{Z}$-row space of $A$, this is equivalent to having $e_i-e_j$ in the row space of the incidence matrix for some $i \neq j$, where $e_i$ and $e_j$ are standard basis vectors.  It is more difficult to show that if $\Gamma$ is not magic over any abelian group then there must exist a $p, q\in \mathcal{P}$ with $p-q\in \operatorname{Im} A^\top$.  To get at this result we solve the problem modulo $m$ for each $m\in \mathbb{N}$ ($m>1$) and then show a solution exists over $\mathbb{Z}$.  This technique is known as a local-global principle.

\subsection*{Local-Global Principles}
Local-global principles play an important role in number theory.  The basic idea is that if one wants to guarantee a solution to an equation defined over $\mathbb{Z}$, one only needs to show a solution modulo $r$ for all $r\in \mathbb{N} $ with $r>1$.  It should be noted that this concept often extends to other rings, but we focus here on the results over $\mathbb{Z}$.

For a linear equation in one variable this result follows from the Chinese Remainder Theorem, but the result is less obvious in other situations.  The local-global principle holds for any single variable polynomial equation over $\mathbb{Z}$.  However, it does not always hold for multi-variable equations.  For quadratic equations in multiple variables, this result is known as the Hasse-Minkowski theorem, and is the first major result of this kind. For a treatment of this result in English see \cite{Cohen}.  Unfortunately, a local-global principle is often too much to ask for in many settings and breaks down even for cubic equations in multiple variables.  Selmer \cite{Selmer} showed that the equation $3x^3+4y^3+5z^3=0$ has non-zero solutions $\text{mod } r$ and over $\mathbb{R}$, but no non-zero solutions over $\mathbb{Z}$.

In this paper we will be applying a local-global principle for systems of linear equations.  We provide a simplified version of the result below.

\begin{theorem}[\cite{LQ}, Section 2.3]
	For a matrix $M$ and column vector $y$ defined over $\mathbb{Z}$, if the system $Mx=y$ has a solution $\text{mod } r$ for all $r\in \mathbb{N}$, $r>1$, then $Mx=y$ has a solution over $\mathbb{Z}$.
\end{theorem}

\subsection*{Some Module Algebra}

The local-global principle states that if we want to know if $p-q\in\operatorname{Im}(A^\top)$ we can study the same question for $\operatorname{Im}(A_r^\top)$ for $r>1$ instead. A classic result from linear algebra (see e.g.\ \cite{Axler}, Proposition~6.46) states that $\operatorname{Im} M^\top = (\ker M)^\perp$ for any matrix defined over a field. However, this result breaks down for matrices defined over a ring, even one as nice as $\mathbb{Z}$.

For example, consider the matrix
\[M=\left(\begin{array}{cc}
2 & 0\\
0 & 0
\end{array}\right).\]
For this matrix, $\operatorname{Im} M^\top$ is spanned by $[2,0]$, $\ker M$ is spanned by $[0,1]$, and $(\ker M)^\perp$ is spanned by $[1,0]$ and hence  $\operatorname{Im} M^\top\neq  (\ker M)^\perp$ with respect to the standard inner product.

The classic result that $\operatorname{Im} M^\top=  (\ker M)^\perp$ over a field relies on the fact that for a subspace $W$ of a finite dimensional vector space $V$ we have $(W^\perp)^\perp = W$ with respect to an inner product.  This does not hold over most rings.  However, it does hold for $\mathbb{Z}/r\mathbb{Z}$.  We have not been able to find a proof in the literature, so we provide one here using the characters of the group $\mathbb{Z}/r\mathbb{Z}$.\footnote{The idea for this argument was suggested in response to a question on MathOverflow, see \cite{MO}.

Working over $\mathbb{Z}/r\mathbb{Z}$ for any integer $r>1$, observe that for each vector $\vec{v} = (v_1, v_2, \dots, v_s) \in (\mathbb{Z}/r\mathbb{Z})^s$ we have a cyclotomic character $\chi_{\vec{v}}:(\mathbb{Z}/r\mathbb{Z})^s \to \mathbb{C}^\times$ defined by setting \[\chi_{\vec{v}}(x_1, \dots, x_s) = e^{\frac{2\pi i}{n}(v_1x_1 + v_2x_2 + \dots + v_sx_s)}.\]  It follows that two vectors $\vec{v}$ and $\vec{x}$ are orthogonal in $(\mathbb{Z}/r\mathbb{Z})^s$ if and only if $\chi_{\vec{v}}(\vec{x}) = 1$.  For any vector $\vec{x} \in (\mathbb{Z}/r\mathbb{Z})^s$ and any character $\chi$ there is also a natural pairing $(\vec{x},\chi) = \chi(\vec{x})$.  Given any subspace $H$, we define $H^\downfree = \{\chi \mid \chi(\vec{x})=1 \text{ for all }\vec{x} \in H\}$.  Under this pairing, it has been shown (see \cite[Proposition 3.4]{Washington}) that $(H^\downfree)^\downfree = H$ for all subspaces $H \subseteq (\mathbb{Z}/r\mathbb{Z})^s$.  With a one-to-one correspondence between cyclotomic characters and vectors and the fact that $\langle \vec{v},\vec{x} \rangle = 0$ if and only if $\langle \vec{x}, \chi_{\vec{v}} \rangle=1$, it follows that $(V^\perp)^\perp=V$ for all subspaces $V$ when using the standard inner product as well.  This proves the following proposition:

\begin{prop}\label{prop:mperp}
	For a $\mathbb{Z}/r\mathbb{Z}$-map $M$, $\operatorname{Im} M^\top=  (\ker M)^\perp$.
\end{prop}

\subsection*{When $\bm{\Gamma}$ \emph{is} Magic}

We have already seen that a $k$-uniform hypergraph $\Gamma$ is not magic over any group if $p-q\in \operatorname{Im} A^\top$.  We now consider the situation where we know $\Gamma$ is not magic over any group and determine what we can say about $\operatorname{Im} A^\top$.

Theorem~\ref{thm:equalpair} implies that there exists some pair of points $p, q \in \mathcal{P}$ such that $f(p)=f(q)$ for all line-invariant functions for all groups $G$.   Since $\ker A_G$ contains all line-invariant functions with magic constant $0$, it follows specifically that $f(p)=f(q)$ for every function $f \in \ker A_G$.

The standard inner product on $\mathbb{Z}\mathcal{P}^2$ (\textit{resp. $(\mathbb{Z}/r\mathbb{Z})\mathcal{P}^2$}) with respect to the basis of points, yields the pairing
\[\langle p-q, [f]\rangle = f(p-q)= f(p)-f(q)=0.\]
So $p-q\in (\ker A)^\perp$ (\textit{resp.} $(\ker A_r)^\perp$).  When working over $\mathbb{Z}$, as we have noted, it is not necessarily true that $(\ker A)^\perp=\operatorname{Im} A^\top$.  However, it is true that $(\ker A_r)^\perp=\operatorname{Im} A_r^\top$, and $p-q\in \operatorname{Im} A_r^\top$ for all $r\in \mathbb{N}, r>1$.  Therefore, by the local-global principle for systems of linear equations $p-q\in \operatorname{Im} A^\top$.  This gives our main result.

\begin{main} A $k$-uniform hypergraph $\Gamma=(\mathcal{P}, \mathcal{L})$ with incidence matrix $A$ is not magic over any group $G$ if and only if there are $p, q\in \mathcal{P}$ with $p-q \in \operatorname{Im} A^\top$.
\end{main}

This result has a geometric interpretation as well. Assume $p-q$ is in the $\mathbb{Z}$-row space of $A$.  Then $p-q$ can be written as a linear combination of lines (rows), and the lines with non-zero coefficients correspond to a forbidden substructure.
For example, in the case of the $14_3$-configuration with points labeled $1, \ldots, 14$ and lines described as in Figure \ref{fig:14_3lines},
\begin{figure}[h!]
\[\begin{array}{cccccccccccccc}
L_1&L_2&L_3&L_4&L_5&L_6&L_7&L_8&L_9&L_{10}&L_{11}&L_{12}&L_{13}&L_{14}\\
1~&1~&1~&2~&5~&2~&8~&5~&3~&7~&6~&3~&9~&11\\
2&4&6&4&6&7&9&10&11&8&12&4&12&13\\
3&5&7&8&9&10&10&11&12&13&13&14&14&14\end{array}\]
\caption{Lines of a $14_3$-configuration.}
\label{fig:14_3lines}
\end{figure}
there is only one $\mathbb{Z}$-linear combination of the lines that results in $\vec{e}_i-\vec{e}_j$ for some $i<j$, namely the combination
$$-L_1 + L_2 + L_6 + L_7 - 2L_8 + L_9 - L_{10} - L_{13} + L_{14} = \vec{e}_4 - \vec{e}_5.$$
The lines that appear here form the subconfiguration given in Figure~\ref{fig:newsubconfig}.  By taking the black lines to have coefficient 1, the red lines to have coefficient -1, and the green line to have coefficient -2, this subconfiguration forces $f(p)$ to equal $f(q)$ for all line-invariant functions.

\begin{figure}[ht!]
\captionsetup[subfigure]{labelformat=simple}
\centering
\begin{tikzpicture
    \draw[ultra thick] (1,0) -- (4,0);
    \draw[ultra thick] (1,1) -- (4,1);
    \draw[ultra thick] (0,2) -- (2,2);
    \draw[ultra thick] (1,1) -- (3,2);
    \draw[ultra thick] (1,0) -- (4,-1);
    \draw[ultra thick,green] (1,0) -- (1,2);
    \draw[ultra thick,red] (2,0) -- (2,2);
    \draw[ultra thick,red] (3,-0.667) -- (3,2);
    \draw[ultra thick,red] (4,-1) -- (4,1);
    \draw[thick,fill=blue] (1,0) circle (0.1);
    \draw[thick,fill=blue] (2,0) circle (0.1);
    \draw[thick,fill=blue] (3,-0.667) circle (0.1);
    \draw[thick,fill=blue] (1,1) circle (0.1);
    \draw[thick,fill=blue] (2,2) circle (0.1);
    \draw[thick,fill=blue] (3,1) circle (0.1);
    \draw[thick,fill=blue] (1,2) circle (0.1);
    \draw[thick,fill=blue] (2,1.5) circle (0.1);
    \draw[thick,fill=blue] (3,2) circle (0.1);
    \draw[thick,fill=blue] (0,2) circle (0.1);
    \draw[thick,fill=blue] (4,-1) circle (0.1);
    \draw[thick,fill=blue] (4,0) circle (0.1);
    \draw[thick,fill=blue] (4,1) circle (0.1);
    \draw(0,2.3) node {\small $p$};
    \draw(1,2.3) node {\small $q$};
\end{tikzpicture}
\caption{A forbidden substructure.}
\label{fig:newsubconfig}
\end{figure}

\subsection*{When $\bm{\Gamma}$ is Magic over $\bm{\mathbb{Z}}$}

A slight adjustment to the ideas behind our main theorem allows us to move back towards the more traditional magic setting and determine whether a $k$-uniform hypergraph is magic over $\mathbb{Z}$ as well.  The result we give in Theorem~\ref{thm:magicoverZ} might be known in the dual setting of edge labelings of graphs (see \cite{Doob2}), however it is stated in a much different way and the proof appears to be lacking crucial details.  Before we give this result, we first need an analogous version of Theorem~\ref{thm:equalpair} specifically for the group $\mathbb{Z}$.

The idea is similar to our previous proof, but instead of taking a direct sum of labelings with values in a direct product of groups, we scale labelings in $\mathbb{Z}$ so that we may take an internal sum of labelings that will end up being injective.

Consider a $k$-uniform hypergraph $\Gamma=(\mathcal{P},\mathcal{L})$ and suppose that for each $p, q\in \mathcal{P}$ there is a line invariant function $f_{p,q}: \mathcal{P} \to \mathbb{Z}$ such that $f_{p,q}(p) \neq f_{p,q}(q)$. Given the points $p_1, \dots, p_n \in \mathcal{P}$ we let $b=\max\{|f_{p,q}(p_i)| \mid p,q,p_i \in \mathcal{P}\}$.  Then we define a new line-invariant function
\[f=\sum_{1 \leq i < j \leq n} (2b+1)[n(i-1) + j]f_{p_i,p_j}\]
By construction $f$ is injective and $\Gamma$ is magic over $\mathbb{Z}$. In summary

\begin{lemma}\label{lemma:Zequalpair}
A $k$-uniform hypergraph $\Gamma=(\mathcal{P},\mathcal{L})$ is not magic over $\mathbb{Z}$ if and only if there exists a pair of points $p,q \in \mathcal{P}$ such that for all line-invariant functions $f:\mathcal{P} \to \mathbb{Z}$ we have $f(p)=f(q)$.
\end{lemma}
We can use this result in determining whether or not a hypergraph is magic over $\mathbb{Z}$.  The result is similar to our main theorem, except now we are interested in whether or not $p-q$ is in $\operatorname{Im} A_\mathbb{Q}^\top$.  If $p-q$ is in $\operatorname{Im} A_\mathbb{Q}^\top$ then we can write
\[p-q =\sum_{i=1}^m b_i A_\mathbb{Q}^\top L_i\]
with $b_i\in \mathbb{Q}$. Let $d$ be the common denominator of the $b_i$.  Then by multiplying through we see that $dp-dq$ is a $\mathbb{Z}$-linear combination of the lines.  If $f$ is a $\mathbb{Z}$-line-invariant function with magic constant $c$ then we have
\[ =f\left(\sum_{i=1}^m db_i A_\mathbb{Q}^\top L_i\right)=\sum_{i=1}^m db_i f\left(A_\mathbb{Q}^\top L_i\right)=\sum_{i=1}^m db_i c=c\sum_{i=1}^m db_i.\]
This sum is $0$ for the same reasons as in the main theorem.  Similarly $f(dp -dq)=df(p)-d(q)$, and combining these we obtain $df(p)=df(q)$.  Since $f$ takes values in $\mathbb{Z}$ this implies $f(p)=f(q)$ and the function is not magic.

Conversely, if $\Gamma$ is not magic over $\mathbb{Z}$, then by Lemma~\ref{lemma:Zequalpair} it follows that there exists some pair of points $p, q \in \mathcal{P}$ such that $f(p)=f(q)$ for all line-invariant functions $f:\mathcal{P} \to \mathbb{Z}$.  Just as in the proof of the Main Theorem, this puts $p-q \in (\ker A_\mathbb{Q})^\perp$.  Note, however, that since we are now working over the field $\mathbb{Q}$ we may immediately conclude that $(\ker A_\mathbb{Q})^\perp = \operatorname{Im} A_\mathbb{Q}^\top$.   We summarize the results below.

\begin{theorem}\label{thm:magicoverZ} A $k$-uniform hypergraph $\Gamma=(\mathcal{P},\mathcal{L})$ with incidence matrix $A$ is not magic over $\mathbb{Z}$ if and only if there exists some $\mathbb{Q}$-linear combination of the rows of $A$ that is equal to $\vec{e}_i - \vec{e}_j$ for some $i \neq j$.
\end{theorem}

\subsection*{Applications to $n_3$-configurations}
By applying our algorithms to the known complete lists of $n_3$-configurations for $n=7, 8, \dots, 14$, we are able to determine how many of these configurations are magic over some abelian group as well as how many are magic over $\mathbb{Z}$ specifically.  In addition, it is important to point out that, for $n_3$-configurations, the two forbidden substructures in Figure~\ref{fig:subconfigs} are the only forbidden substructures using six or fewer lines.  As mentioned previously, these substructures account for the majority of $n_3$-configurations, for $n=7, \dots, 14$, which are not magic.  In the table below, we also count the number of configurations containing at least one of these small substructures.

$$
\begin{tabular}{||c|c|c|c|c|c||}
\hline \hline
$n$ & \# Configs. & \# w/ Small & \# Magic & Magic/Total & \# Magic over $\mathbb{Z}$\\
\hline
7 & 1 & 0 & 1 &1& 0\\
\hline
8 & 1 & 0 & 1 &1& 0\\
\hline
9 & 3 & 1 & 2 &2/3& 0\\
\hline
10 & 10 & 5 & 5 &1/2& 0\\
\hline
11 & 31 & 22 & 9 &$\approx$0.290& 1\\
\hline
12 & 229 & 187 & 34 &$\approx$0.148& 1\\
\hline
13 & 2036 & 1727 & 198 & $\approx$0.097 & 22\\
\hline
14 & 21399 & 17933 & 1467 & $\approx$0.069 &  125\\
\hline \hline
\end{tabular}
$$

Based on our findings in the table, it appears as though the fraction of magic $n_3$-configurations within the total number of configurations is approaching zero.
This seems reasonable from the perspective of forbidden substructures.  Observe that the small forbidden substructures found commonly in $n_3$-configurations (see Figure~\ref{fig:subconfigs}) involve collections of parallel lines.  In addition, as $n$ grows larger it becomes easier to find parallel lines within $n_3$-configurations.  Furthermore, there is an increased chance that an $n_3$-configuration will contain larger forbidden substructures as well.

\begin{conj} $\displaystyle \lim_{n\to\infty} \frac{\text{\# of $n_3$-configs magic over some group}}{\text{\# of $n_3$-configs}} \to 0$.
\end{conj}

The smallest example of an $n_3$-configuration that is magic over $\mathbb{Z}$ is the following $11_3$-configuration:
\begin{center}
$\begin{array}{ccccccccccc} L_1 & L_2& L_3 & L_4 & L_5 & L_6 & L_7 & L_8 & L_9 & L_{10} & L_{11} \\1&1&1&2&2&2&3&3&3&4&4\\6&7&8&5&6&7&4&5&7&5&6\\11&10&9&11&10&9&11&10&8&9&8\end{array}$
\end{center}

A magic labeling of this configuration is provided in Figure~\ref{fig:smallest_Z}.  Note that this labeling is actually \emph{supermagic} as it uses the consecutive integers 1, $\dots$, 11.
In an upcoming paper, the authors plan to explore magic $n_3$-configurations further -- both supermagic labelings and magic labelings over other abelian groups.

\begin{figure}[ht!]
\captionsetup[subfigure]{labelformat=simple}
\centering
\begin{tikzpicture}[scale=0.9]
\draw[ultra thick] (-1.84,3.56) -- (0.77, -1.27);
\draw[ultra thick] (-1.84, 3.56) -- (4, 3.56);
\draw[ultra thick] (-1.84, 3.56) -- (0.4, 2.04);
\draw[ultra thick] (4.13,0.01) -- (0.4, 2.04);
\draw[ultra thick] (0.47, 0.01) -- (4.13, 0.01);
\draw[ultra thick] (0.47, 0.01) -- (4, 3.56);
\draw[ultra thick] (-0.34, 3.56) -- (1.09, 0.63);
\draw[ultra thick] (-1.02, 2.04) -- (4.05, 2.04);
\draw[ultra thick] (0.77, -1.27) -- (4.05, 2.04);
\draw[ultra thick] (-1.02, 2.04) -- (2.04, 0.01);
\draw[ultra thick] (4, 3.56) -- (4.13, 0.01);
\draw[ultra thick] (-0.34, 3.56) -- (0.77, -1.27);
\draw[thick,fill=blue] (-1.84, 3.56) circle (0.1);
\draw[thick,fill=blue] (4, 3.56) circle (0.1);
\draw[thick,fill=blue] (-1.02, 2.04) circle (0.1);
\draw[thick,fill=blue] (4.05, 2.04) circle (0.1);
\draw[thick,fill=blue] (4.13, 0.01) circle (0.1);
\draw[thick,fill=blue] (0.47, 0.01) circle (0.1);
\draw[thick,fill=blue] (0.77, -1.27) circle (0.1);
\draw[thick,fill=blue] (-0.34, 3.56) circle (0.1);
\draw[thick,fill=blue] (2.04, 0.01) circle (0.1);
\draw[thick,fill=blue] (1.09, 0.63) circle (0.1);
\draw[thick,fill=blue] (0.4, 2.04) circle (0.1);
\draw(-2.2,3.56) node {\small $10$};
\draw(-0.34,3.86) node {\small $7$};
\draw(4.3,3.56) node {\small $1$};
\draw(4.4,2.04) node {\small $11$};
\draw(4.13,-0.29) node {\small $6$};
\draw(0.72,-0.24) node {\small $8$};
\draw(0.47,-1.27) node {\small $3$};
\draw(-1.32,2.04) node {\small $5$};
\draw(2.24,-0.29) node {\small $4$};
\draw(0.6,2.34) node {\small $2$};
\draw(1.44,0.68) node {\small $9$};
\end{tikzpicture}
\caption{The smallest configuration which is magic over $\mathbb{Z}$.}
\label{fig:smallest_Z}
\end{figure}

\begin{comment}
For an $n_3$-configuration to have a supermagic labeling, it must be the case that $n$ is odd. Why? Let $c$ be the magic constant. Since there are $n$ lines and each point lies on 3 lines, we must have $nc = 3 \sum_{i=1}^n i = 3n(n+1)/2$, or $c=3(n+1)/2$. Hence $n+1$ must be even, so $n$ is odd.

$$
\begin{tabular}{||c|c||}
\hline \hline
$n$ & \# Magic over $\mathbb{Z}$ using $1,2,\ldots, n$\\
\hline
11 & 1 \\
\hline
13 & 2 \\
\hline
15 & 18 \\
\hline
17 & 115 \\
\hline
19 & 2338 \\
\hline \hline
\end{tabular}
$$
\end{comment}

redpajama_set_name: RedPajamaArXiv


Tomorrow, when my brain is fresh, I'm sure something will occur to me. Or maybe I'll have a marvelous dream tonight which will illuminate all those hidden crevices in my mind where the plot lurks.
LOL! No, thunder and lightning and sounds of DOOOOOM are definitely appropriate for this year's book. Pretty much, I think all my books will be like that. I'm just a DOOM kinda gal, I guess.
I haven't started yet... Hmmm.


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