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Malaga Festival & Traditions
Cosmopolitan, open, fun and passionate, Malaga is all of these to those who whish to sample the province's culture and folklore. There is something for every visitor, wherever their taste, sensibility and sense of fun.
The first of a long list of events in the Summer Fair. Its origins date back to the incorporation of Malaga into the Kingdom of Castile at the hands of the Catholic Monarchs, who entered in the city in the 19th August 1487. This immensely popular week sees locals and visitors alike join in an effusive celebration lasting from midday to well into the early hours. Bathed in warm summer sunshine, the city's historical centre is bedecked with fairy lights, fans and castanets, matching to perfection the gipsy costumes and smiles worn by the thousands of people who contribute to this explosion of local song and dance. The laughter is occasionally interrupted for a most just and nutritional cause: the obligatory sampling of tasty tapas, with cheese and ham the most popular items on the menu.
And so it goes on until nightfall ushers in a change of scenery, the festivities continuing by moonlight at the Cortijo de Torres fairground.
Here, a whole spectrum of rides and attractions wait young and old alike, while a variety of music ranging from verdiales and flamenco to rock is played loud in the many different casetas, custom-built premises to be found throughout the fairground. Both the Young People's caseta and the recently opened Municipal Auditorium play host to some of the world's most famous performers. The latter, which holds 13,000 people, was filled to capacity during the 2005 fair.
This unbridled joy turns to impassioned fervour in Holy Week.
The men of some 40 religious societies take to the streets, through which the y conduct a carefully-coordinated procession, carrying on their backs their societies' tronos or floats, which bear enormous images and manifestations of human worship of the divine.
This 500-year-old religious and social phenomenon attracts thousands of parishioners and penitents alike, who embark on a pilgrimage through the heart of the city that starts at each society's chapel or headquarters. The liberation of a prisoner by El Cautivo society, the blessing of the city offered by El Rico; the huge float of the Virgen de la Esperanza, the fluttering entourage of the doves released by the Virgen de la Paloma or the much-acclaimed darling of Malaga, the Virgen del Rocio; the perfect synchronization with which La Expiracion has conducted its ceremonial march since 1931 and the immense popularity of the Zamarrilla procession – moments and images which evoke intense fervour here in Malaga.
Each town and village lives and celebrates Holy Week in its own particular way.
Riogordo stages a theatrical representation of the passion and death of Christ on Good Friday and Easter Saturday, a historic and moving performance which has been attended by practically the whole of the village since 1951 and whose lines date back to the VXII century. The outskirts of Alozaina also host an outdoor religious play, though of a more modern ilk. In Alhaurin de la Torre, Alhaurin El Grande and Coin. Easter is characterised by the brotherly rivalry between two societies, "the greens" and "the purples". Alora is famous for the dancing movements of the heavy floats of the Virgen de los Dolores and the Cristo de las Torres societies, which are borne by just eight carriers on each pole during the famous "Farewell" on the morning of Good Friday.
Wholly different affairs are the antics of the young men of Alfarnatejo, who take to the streets on Easter Saturday and ring cowbells, and the Easter Sunday pilgrimage to the country to enjoy a hearty feast that forms part of Benomacarra's traditional "pava".
Country get-togethers to enjoy good food are typical in many areas of the province, taking good advent age of the excellent local fare. Boiled or roasted chestnuts accompanied by liqueurs are devoured at the Tostonazo or Roast Chestnut Festival in Istan. Another fruit, this time of a juicer variety, was imported to Periana from Argentina 150 years ago: the peach, which enjoyed a particular boom in the 1970s. Considered the best in Spain, as many as four million kilos have been picked in single season to be sold at the popular fruit auctions known as "corrias".
The orange has a whole day devoted to it in Coin, as does the almond in Almogia. All of these products serve to accompany a dish made from small pieces of bread in the town of Torrox: Migas. In fact, "Breadcrumb Day" has been officially declared an event of national tourist interest and is celebrated throughout the whole of the Axarquia region in which Torrox lies, an area boasting the finest grape harvest festival. Algarrobo, Competa and El Borge are three of the locations in which this celebration is particularly popular, especially so in the case of the latter, where thousands of bags of raisins are given out each year during the Raisin Festival.
However, it is not only our taste buds that will be stimulated at these celebrations –our ears will also delight to the sound of verdiales music.
Comares, Algarrobo, Almogia and Alora are just a few of the village whose inhabitants put their vocal chords through their paces on a day specially devoted to this ancient form of song, while in Torremolinos, the huge green expanse of Los Pinares not only provides a delightful backdrop for the performance of this most unusual of styles, it is also a meeting point for the thousands of participants in a religious pilgrimage that is second only to El Rocio, the romeria dedicated to St. Michael the Archangel.
Some fifty wagons pulled by tractors or oxen, along with numerous horses and their riders, make up the huge procession that wends its way through the street of Torremolinos in late September before finishing up in the aforementioned pine forest, where a sumptuous feast awaits.
Another culinary-based celebration takes place on St. Mark's Day in Villanueva de la Concepcion, where a cake baking contest sees bakers and villagers alike give free rein to their imaginations by creating figures of animals, plants and other objects from a mixture of dough and eggs. These creative efforts work up an appetite that both competitors and spectators then satiate by devouring the works of art on a display in a delicious afternoon tea in the country.
On the same day in a similar setting, though in a different location, the people of Cuevas de San Marcos ay homepage to their patron saint by "tying up the devil". Legend has it that a when a travelling band of Christian warriors lost their way, they came across Belda Cave, from which a strange light and blood-curdling howls and screams were known to emanate. The warriors alerted the village chaplain, who immediately headed for the cave, cross in hand, to catch the devil. When he had done so, he tied some yellow flowers to the base of the cross. Each year, young and old alike take part in the symbolic ritual of trying to interweave handfuls of grass without breaking them. This followed by the eating to the aforementioned cakes, which are usually decorated with kidney beans.
This same fare is eaten, though in the month of May, on what as known as "Kidney Bean Day", a festival held in conjunction with Cross Day in Colmenar. Other towns and villages in the province also observe this celebration, turning interior patios and even streets into humble but welcoming shrines in which a cross is decorated with flowers and surrounded with manila shawls, ancient containers, sea shells and a variety of rural farming tools. While in the majority of these localities the religious images paraded through the streets are showered with rose petals, in Mijas stones are thrown instead.
No offence is intended, however. Quite the contrary, in fact. It is all done by way of making a request to the Virgin's male companion, a handsome individual of high standing. The target of the stones is a figure of St. Anton, and, in accordance with a XV century legend, the 17th January sees hundreds of single women bombard the saint with three stones that they have previously collected in the sacred surroundings of the chapel in which he has laid to rest. An original way of seeking a fiancé. Or fiancée. The young men of Tolox once requested their girlfriends' hand in marriage by covering their loved one's face in flour. This is now know as Power Day, for with the passage of time, the flour has been replaced by talcum powder, some 3,000 kilos of which are thrown in the village on Carnival Tuesday.
This musical celebration laced with a heavy dose of sarcasm has been held in Malaga for over twenty years, each edition attracting and increasing number of participants. The biting lyrics of the songs and hilarious fantasy of the carnival costumes worn take over the city streets in the month of February.
A very different form of dress known as Goyesque costume takes centre stage at Ronda Bullring in September during the Pedro Romero Fair. This festival began in 1954 to mark the bicentennial of the birth of the man who invented the art of bullfighting on foot. The Goyesque bullfight is now famed throughout the world and its fans include ministers and members of the Spanish Royal Family, who, from the Royal Box, watch, along with huge numbers of famous and anonymous spectators alike, as a scene from the times of Francisco de Goya, the painter commissioned to immortalise the celebrated matador on canvas, is played out before them. Behind this spectacle, the ashes of the great maestro lie buried beneath one of the arena's safety barriers.
Also reduced to ashes are the effigies known as "juas" burned on the bonfires lit beneath the moonlight on St. John's Night. With the arrival if the witching hour at midnight, the sands of the Malaga coast are filled with footprints; some belong to the revellers drawn to the heat of the flames, where sardines and other tasty morsels are cooked, others to midnight bathers heading for the sea, and other still to those observing the ritual of seeking the path to good fortune traditionally associated with the magical night of the 24th of June.
The sun then replaces its nocturnal friend to illuminate the "marengos", men who transport a float bearing the statue of the Virgen del Carmen. The Malaga coats is now carried along on a tide of emotion as the patron saint of fishermen is brought out from her chapel to embark upon a precession characterised by the aroma of geraniums and the sea, creating a fervour that reaches fever pitch for the traditional mass on the beach. After this, the marengos carry their "Queen of the Seas" into the water before depositing her on her boat. A colourful backdrop of vessels and fireworks light up the waves like marine glow worms before the Virgin is returned to dry land well into the night.


2014 Seat Leon ST X-Perience review | What Car?
The Seat Leon X-Perience is intended as a new flagship offshoot model in the highly successful Leon range, and also lays the groundwork for the planned arrival of Seat's first-ever SUV, due in 2016.
As the name suggests, the Leon X-Perience gets an electronic four-wheel-drive system, and taller ground clearance compared with the Leon ST, plus exterior cladding for the bodywork, and a mild interior makeover.
The UK range comprises two trim levels (SE and SE Technology), a 2.0-litre diesel engine in two states of tune (148bhp and 181bhp) and either a six-speed manual or a twin-clutch DSG automatic.
What's the 2014 Seat Leon ST X-Perience 2.0 TDI 184 DSG like to drive?
On the road, the Leon X-Perience feels much like the standard Leon ST, which is a positive. The driving position remains very estate-like, so you don't look down on other traffic like you would in an SUV.
Despite the subtly raised ride height the car rides and handles comfortably on most surfaces, while retaining its poise through corners, even in tight and tricky bends. There's a smidgen of body roll during tight cornering, but nothing that feels uncomfortable. The onus is on sure-footedness rather than precision, but the X-Perience never feels ponderous.
A short drive off-road proved that there's some merit in Seat's claims of a modicum of countryside rambling ability, with the Leon X-Perience competently dealing with the usual obstacles of rutted tracks, slippery grass and steep hills, but it still isn't as adept in the rough as some SUVs; think of it as a versatile estate rather than anything more.
Regardless of whether it's in 148bhp or 181bhp tune, the 2.0-litre turbodiesel engine offers more than enough shove from low revs all the way through the range. There's a hint of diesel clatter on start-up, and under heavy acceleration, but for the most part the motor is acceptably refined, and never overly intrusive.
The six-speed DSG transmission has an occasional tendency to hold a gear for longer than you'd desire at low speeds, but is unobtrusive for the most part; the manual has a perfectly acceptable shift that's slick enough for you to keep up with the prodigious low-end torque.
The most potent Leon X-Perience is currently only available with the DSG auto gearbox, while the less powerful model is manual only, which might be a deciding factor for buyers looking to use theirs for hauling a caravan or a trailer. Incidentally, the 181bhp Leon can tow up to 1600kg, although a tow-bar preparation is an option, not standard.
Another useful selling point is that the Leon X-Perience is one of few four-wheel-drive models on the market to dip below the 130g/km of CO2 banding regardless of engine choice; it consequently carries a 21% benefit-in-kind rate for company car tax. Seat anticipates that this rugged variant will be popular with fleet buyers too, with a straight 50:50 split between retail and fleet sales.
What's the 2014 Seat Leon ST X-Perience 2.0 TDI 184 DSG like inside?
Most of the cabin is carried over from the standard Leon ST, so you get decent interior space for four adults, a well-laid-out if slightly plain dashboard and decent-quality materials throughout. The high transmission tunnel in the back does mean foot space is a bit limited for a third passenger though.
The addition of extra mechanicals for the four-wheel drive system hasn't impinged on boot space, which remains the same as the Leon ST at an impressive 587 litres with the rear seats in place, expanding to 1470 litres with the bench folded, which can be achieved with just one pull of a lever in the luggage bay.
Trouble is, its VW Group stablemate the Skoda Octavia Scout is usefully more practical, with a larger boot, and it's also cheaper when equipped with the higher-powered engine and DSG gearbox, with prices starting at £27,990.
To set this Leon apart from other STs, there are a selection of bespoke interior colour combinations, including some brown Alcantara seats with contrasting orange stitching (which works better than it sounds) and some bespoke badges that are befitting of the plushest model in the range.
Equipment levels are competitive: the top SE Technology specification includes Seat's enhanced media system with a 5.8in colour touch-screen that now has easier to read maps, DAB and satellite-navigation, plus auto headlights, wipers and dimming rear view mirror.
Lesser SE models have to do without the Alcantara seats, 18-inch wheels and nav system of the flagship but still come with all the essentials, including a decent infotainment system, cruise and climate control, and rear parking sensors.
The X-perience is a worthy addition to the Leon range, because unlike the Skoda Octavia, this is the only Leon ST you can buy with four-wheel drive.
That said, we wouldn't choose one with the range-topping engine and DSG gearbox combination; Seat itself thinks the smart money will go on the lower-powered SE Technology model, which strikes a sweeter balance between generous kit levels and an attractive sticker price.
In fact, we'd say the 148bhp SE Technology is worth four stars, one more than the 184bhp DSG auto. It beats the all-wheel-drive Vauxhall Insignia Country Tourer and the front-wheel-drive Volvo V40 Cross Country on price and practicality, and is more frugal.
Despite the lure of all-wheel-drive, it's also worth thinking hard about the compelling necessity for the extra traction. If you don't plan to ever venture off road, a regular Leon ST shod with winter tyres may do the job as winter-proof family transport just as well, offering better fuel economy and a lower price.


thumb·screw / ˈ[unvoicedth]əmˌskroō/ • n. 1. a screw with a protruding winged or flattened head for turning with the thumb and forefinger. 2. (usu. thumbscrews) an instrument of torture for crushing the thumbs.
"thumbscrew." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. 18 Apr. 2019 <https://www.encyclopedia.com>.


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Taj Dearob News
Taj Dearob Pharma / Health News / Doxorubicin and Bleomycin: The Antibiotics Fighting Against Cancer
Doxorubicin and Bleomycin: The Antibiotics Fighting Against Cancer
Following the discovery of penicillin as an antibiotic by Alexander Fleming in 1928, searches were carried out in order to discover more potent and specialist agents for specific bacterial infections.
Throughout this search, compounds synthesised from soil microbes proved to be a fantastic source of agents, of which many were shown to less of a selective bacterial toxicity, but have cytotoxic effects on tumours, a class called anthracyclines. They are believed to work by interfering with enzymes involved with DNA replication, leading to a reduced synthesis of DNA and cell death. Since, anthracyclines have been exclusively and widely used in the field of oncology, known as one of the most potent classes of drugs.
Much akin to the search for more potent and specific antibiotics, there has been a continuous search for stronger anthracyclines, with the aim of tackling drug resistance and cardiotoxic effects since the discovery of Doxorubicin as one of the first to be discovered in the 1960s. At Taj Accura, we offer two widely used anthracyclines, Doxorubicin and Bleomycin at the most reasonable cost. As advances in formulation and newer drugs are found in this field, we strive to work with our leading R & D team in order to provide the best and most cost efficient treatment to patients possible.
Conclusion: According to the British National Formulary, Bleomycin is indicated in the treatment of non-Hodgkin's lymphoma and metastatic germ cell cancer, causing little bone marrow suppression. Taj Accura offers Bleomycin as the generic equivalent of Blenoxane, being indicated for squamous cell carcinoma and testicular carcinoma, being available as an injection for reconstitution in 15 units or 30 units per vial.
Doxorubicin on the other hand treats acute leukaemia, Hodgkin and non-Hodgkin lymphomas as well as solid breast cancer tumours1. At Taj Accura Pharmaceuticals, Doxorubicin is formulated as an injection for reconstitution, available in 10 mg or 50 mg doses per vial to the generic equivalent of Oncodox.

redpajama_set_name: RedPajamaCommonCrawl


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Evaluation: Genting Aims To Upend Macau Casino Landscape In Bidding War
All of Macau's 700,000 residents have to take a fast antigen test for Covid-19 everyday from Oct 30 to Nov 1, Macau's government said in a statement that made no mention of the MGM closure. In contrast, China's 'big three' internet corporations – Baidu, Alibaba and Tencent – have seen market place share rise a lot more than 20% for streaming platforms iQiyi, Youku and Tencent Video. Japanese bankNomuraputs the number of Chinese beneath restrictions at about 232 million, up from 225 million last week. Some 31 cities there are under some form of lockdown as of Oct. 27, restricting one particular in six people in China and covering 24.five% of its gross domestic solution, the Nomura analysis found. Rather, China reported its highest tally of new Covid-19 situations in two weeks on Thursday.
They focus purely on non-gaming marketing – selling the location, experiences, packages, hotels, retail and entertainment. The typical occupancy rate of Macau's hotels and guest homes in July grew by 51.6 percentage points year-on-year and 18.4 percentage points compared with June to 63.7%, according to data from the Statistics and Census Service . Non-resident workers can not be casino dealers below policy arrangements maintained by successive Macau administrations. But numerous non-residents employed in the casino resort sector operate in functions such as safety, hotel housekeeping, restaurant services, and cleaning duties.
With the utmost sincerity, we welcome additional and additional strategic investors and businesses to do enterprise and invest right here in the hope of bolstering the industrial development in Macau and Hengqin," said Li Ziwei, acting director of the Cooperation Zone. The value of US goods exported to China in 2021 amounted to $149 billion, and exports of goods and services to China assistance 858,486 US jobs, according to the US-China Small business Council's 2022 report on US exports. Revenues could be back on the rise by the end of this year, helped by the opening of new casino resorts, stated analyst Aaron Fischer in an opening speech at the gaming exhibition. Regardless of insisting they will rule much more moderately this time about, the Taliban have already introduced rules for what girls can put on at university, and beaten and harassed several Afghan journalists regardless of promising to uphold press freedoms. The year-on-year inflation rate in Macau rose to 1.32 per cent in June, due to higher wages for domestic workers, rates of meals away from residence, gasoline and liquefied petroleum gas, information released currently shows. The newly-opened Mesa Portuguese fine-dining at the Grand Lisboa Palace encourages guests to suit up and welcome the new year in style.
Muhyiddin, who was backed by an Islamist party, told reporters Tuesday that since no bloc had enough numbers, the king had initially asked him and Anwar to kind a "unity government". Sands Gallery opened an exhibition featuring more than 30 artworks from artists in China and Italy, aiming to market exchange and cooperation among regional and international talents. Suncity worker Tam mentioned she took a 1-third pay reduce to land a new secretarial job, adding that similar openings were routinely advertised with a month-to-month wage of just $1,one hundred. Macau's government has extended been keen to diversify away from gaming into tourism and leisure.
Now nearly nonexistent, the opaque VIP market made up extra than two-thirds of Macau's gambling revenue till just a couple of years ago, official data shows. The arrests announced on Sunday come two months soon after Macau authorities arrested the high-profile head of Suncity junket, Alvin Chau. Junket operators have traditionally supplied quick credit for mainland Chinese high rollers who play in Beijing-ruled Macau's casinos and collect on their debts working with underground financing channels, executives say. "The 'one China' principle is created to destroy the Republic of China and force Taiwanese to recognize the People's Republic of China as the suzerain. Instead of waiting until the last day, the council desires to prepare for the feasible outcome in advance, for the reason that there is no way that Taiwanese officials can be stationed in Macau with out a visa," Wang said.
The third is in the 38th week of gestation and has been placed beneath healthcare observation in quarantine as a precautionary measure because her husband tested constructive and she has been listed as a close make contact with only. There are at present 3,656 folks carrying out health-related observations in hotels, two,506 persons in designated health-related observation hotels, and 1,150 folks in optional health-related observation hotels. Goldstein tried to downplay a reliance on company from China at the company's Singapore resort, saying Las Vegas Sands can still make as significantly as $1.5 billion "without Chinese output" and instead from travelers from other Asian markets like South Korea, Malaysia, and Japan. Each Macau and Singapore, another important Asian gaming market place for Las Vegas Sands, struggled for the duration of the pandemic due to tighter travel restrictions than Las Vegas.
This applies to shoppers or users of a premises, and excludes employees that might be present. On Friday, the city completed a nearly two-week shutdown of non-critical companies, which includes casinos. Most Macau casinos reopened on Saturday with restricted operations as portion of countermeasures against Covid-19, and with company volumes noticed depressed as travel restrictions constrain the quantity of visitors from mainland China.
On the other hand, Inside Asian Gaming is anticipating an incredibly aggressive method to the passage of the bill by way of the AL, resulting in the bill becoming law possibly even as early as the end of February. Any drawn-out method would make it tricky for the government to comprehensive re-tendering by 26 June and open the door for an extension of existing licenses. The government has already stated that any such extension would be for a"short term", which IAG interprets as probably which means six months. As the bridge is positioned inside Mainland Chinese waters, all cross-boundary autos are needed to drive on the right — in Hong Kong, men and women drive on the left . This may be disorienting for drivers — and possibly 1 reason only two,305 cars used the bridge on opening day. A single Bus Hong Kong Macau Ltd runs 19 every day return trips involving Kwun Tong, the eastern side of Hong Kong's Kowloon and a handful of hotels in Macau.
The new border arrangements are pretty optimistic, but I would say that right after such a huge pandemic, its still a slow recovery. Still, the CE hoped that by 2022 there could be an "orderly relaxation of travel restrictions" and a "gradual recovery" of the tourism industry, right after vaccination prices against COVID-19 boost and with the continuous reinforcement of the prevention of the epidemic. Tour group visas for travel for mainland vacationers to Macau have been suspended considering the fact that last year, with individual travel visas applications currently only permitted in-particular person, whilst previously they have been permitted by way of a quicker on line application system.
Nevertheless, it is normally encouraged that you dress smartly casual when going to a casino. This signifies avoiding any ripped or tattered clothing, as well as any overly revealing outfits. If you navigate to this website are hunting for a really luxurious expertise, then you really should verify out the Mandarin Oriental Hotel & Casino.
Anticipating that any possible recovery won't commence till 4Q22, the analysts suggest everyday GGR could reach MOP$406 million (US$50.2 million) in Q4. By 2024, mass GGR is tipped to recover to 98% of 2019 levels with VIP GGR remaining at roughly a third of 2019 levels. To reflect its most recent GGR forecasts, the investment bank has slashed concessionaires' EBITDA forecasts by 19% to 53% and target rates by 15% to 60%. Despite its junket concerns, Credit Suisse mentioned it remains upbeat on Macau's gaming sector on "potential recovery and undemanding valuation", though it is less bullish amid current COVID-19 developments in mainland China. "The Macau government's current clarifications on amendments to the gaming law signal that the government remains supportive of sustainable development of the mass segment."
In addition, restaurants and lounges will also function Asaya, Rosewood's integrative properly-being notion, with eight treatment rooms, indoor and outside swimming pools, and a fitness center. Opening in the Fall of 2023, the new Paradise Valley home functions 215 guest rooms and suites, a 7,400-square-foot six-bedroom Presidential Casita, and a 400-foot swimming pool. With 300 guest rooms and suites, Singapore's very first Mondrian hotel will be positioned in the prominent Duxton Hill neighborhood.
Nevertheless, it's uncertain how lots of Chinese will take trips through the period, with authorities urging residents to stay put. The 5 Chinese regions that would kick-get started the visa resumption policy accounted for 40% of Macau's guests in 2019, according to Goldman Sachs. That could potentially bring casino operators back into the black, just after suffering losses in the past couple of years.
This entry was posted in casino and tagged border, holiday, macauzhuhai, measures, minute, october, quarantine, retained, selection on November 24, 2022 by totosite.in.


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Why Abbey Road
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5 Things in Your Room Right Now That Could Help You Learn a Language
by Staff, Jessica A. Scott
If you have dreams of studying abroad with a program like Abbey Road, you're going to want to at least have a basic grip on the language of the country you'll be staying in. You don't have to speak like a native to get by, but it is a lot more fun if you can understand what locals are saying to each other as they pass by you on the street!
Luckily, in this day in age, there are tons of ways to make language learning easier. In fact, there are five things in your room right now that could help you master your target language in no time, like:
Image Credit: Adriana Calvo/Pexels.com
1. Your Smartphone
Smartphones keep us connected to our friends and family, but they also connect us to some great language learning resources. There are many free apps available (Duolingo and Memrise, for example), that help you learn a language by playing games, making flashcards, and listening to native speakers pronounce words.
Image credit: OpenClipart-Vectors/Pixabay
2. Your Laptop
Movies are a fantastic way to immerse yourself in a language. If you want to learn Italian, try downloading a movie like La vita è bella on your computer, and watch it with the original Italian audio track. If you're not quite fluent enough for that yet, turning on the Italian subtitles could help you to follow along, and to see which words you need to study.
Image Credit: kinkate/Pixabay
3. Headphones or Earbuds
There are hundreds of websites that let you listen to music or audiobooks in your target language. Plugging in your headphones while you listen can help to block out outside sound, so you can focus on the words without distractions.
Image Credit: Tolbxela/ Flickr
4. Ebook Reader
Whether you prefer ebooks or paperbacks, reading is one of the best ways to learn and retain new words. Websites like Project Gutenberg have a huge supply of free foreign language stories, novels, and textbooks that you can download for free.
Image Credit: qimono/Pixabay.com
The number one most important thing you need to do when learning a language is to immerse yourself in that language. For an hour or so a day, try to shut out everything and everybody else in your house that speaks your native language, while you focus on using the above tips to read, listen, practice, and (most importantly) think in your target language.
Jessica A. Scott graduated from the University of Louisville with degrees in English and Humanities with concentrations in Literature, Linguistics, and Classical and Modern Languages. She has a passion for languages and has made many trips to Italy, where she works on perfecting her own Italian language skills as she visits some of the most beautiful places in the world.
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The Cynefin framework helps leaders and managers to identify how they perceive situations and make sense of what actions are appropriate for their given context. Essentially it is a sense-making device that asks what context are you in and builds on the premise that different situations require different ways of navigating.
The Cynefin framework identifies three domains: Order, Unorder and Disorder.
The domain of Order is characterized by cause and relationships that can be understood and perceived over time.
The Domain Unorder is more complex and here only patterns can be perceived, mostly in retroperspective, while keeping in mind that once those patterns emerged and can be described, they are probably already outdated. Moreover, our observation and intervention also affected and changed the pattern. The Unorder domain is a constant flux, constantly changing.
The domain of Disorder is mainly arising when different stakeholders or agents that need to make sense of a given situation cannot agree on one domain to categorize the situation in. This is the space where you do not know where you are and feel lost. The action to take, also called the decision model, is to gather information, identify one of the other four domains, and move on.
Let's have a closer look at the four domains in the Order and Unorder category and what are the appropriate decision models for each context.
The first domain is the simple or obvious domain. "Simple contexts are characterized by stability and clear cause-and-effect relationships that are easily discernible by everyone." The right answer is 'known' by everyone involved and "decisions are unquestioned because all parties share an understanding". It is the realm of "known knowns".
Typical examples of the obvious domain are order processing and fulfillment. For example, an invoice is not paid after delivery. The person handling the invoices realizes the unpaid invoice, looks up the conditions in the contract and takes actions according to a standard handbook, manual or laws. Usually there is one best practice present and known to everyone involved, e.g. written down in a contract. Single-point forecasting, field manuals, and operational procedures are legitimate and effective practices in this domain.
The action mode for the obvious domain is to sense the situation, categorize the situation into a known bucket and respond with a well-known solution. This seems straightforward, which is why this domain was also named simple in the past versions of the Cynefin framework.
Another risk is path dependency, or named entrained thinking by Dave Snowden. This refers to a conditioned response, where people are not open to new ways of thinking. Leaders and managers become blinded by the perspectives they acquired through their past experience, training and success. The personal bias limits their ability to see new possibilities, new paths.
Another risk is complacency, which can occur when things appear to be going smoothly. A sudden change in the context is likely to be missed, and any reaction that follows is too late.
It is no surprise that within the Cynefin framework the simple domain lies next to the chaotic one. A catastrophic failure, meaning a shift into chaos, occurs most frequently because success has bred complacency. Just think of the many dynamic technology inventions that 'suddenly' disrupted whole industries.
The complicated or knowable domain contains multiple good practices that can achieve you the desirable outcome. This is based on that there is a clear relationship between cause and effect, but not everyone can see it. It is basically separated over time and space. It is the realm of "known unkowns". Even if I do not know the solution or the path leading me to solving a challenge, I probably have a sense of who would know the answer. It is the call for expert opinions, which could differ, because they have their own unique interpretation and solution for the cause and effect relationship they observe. Often a "complicated analysis and understanding of consequences at multiple levels" is required.
The action mode is to analyze the problem and roadmaps, sense the problem and respond with a plan (often provided by an expert who has a good practice available).
As in the obvious contexts, entrained thinking is a danger in the complicated domain as well. However, here it is the experts (rather than the leaders) who are prone to it. Experts that have invested in building their knowledge are unlikely to tolerate controversial ideas, thereby dismissing or overlooking innovative suggestions by nonexperts. Additionally, there is a key dependency on trust between expert advisor and decision maker.
The complex domain is where we face wicked problems. The truth is out there somewhere and through trusting the process and cultivating our ability to play the emerging practice will present itself to us. The path appears while we are walking on it. Most platform technology designs are emerging in this way, starting with a safe enough to try idea and building the necessary features on the way, perhaps through input from its users.
To be very clear, what we are observing in a complex context is not a direct cause and effect relationship, rather a pattern that emerges. However, we need to keep in mind that "once a pattern has stabilized, its path appears logical, but it is only one of many that could have stabilized, each of which also would have appeared logical in retrospect." Moreover, it could be that a pattern stabilizes and repeats itself, but it is not sure if that pattern continues to repeat, "because the underlying sources of the patterns are not open to inspection." In addition observing of the objective of interest may in itself influence the pattern and change what we observe.
"The decision model in this space is to create probes to make the patterns or potential patterns more visible before we take any action. We can then sense those patterns and respond by stabilizing those patterns that we find desirable, by destabilizing those we do not want, and by seeding the space so that patterns we want are more likely to emerge. Understanding this space requires us to gain multiple perspectives on the nature of the system. This is the time to "stand still" (but pay attention) and gain new perspective on the situation." Narrative techniques are particularly powerful in this space.
The primary challenge that leaders face in the complex domain "is the temptation to fall back into traditional command-and-control management styles—to demand fail-safe business plans with defined outcomes. Leaders who don't recognize that a complex domain requires a more experimental mode of management may become impatient when they don't seem to be achieving the results they were aiming for. They may also find it difficult to tolerate failure, which is an essential aspect of experimental understanding." If leaders try to overcontrol the organization, they shut down the opportunity for informative patterns to emerge.
In a chaotic context, searching for right answers is pointless, because the relationships between cause and effect are impossible to determine. They are shifting constantly, thus no manageable patterns exist—only turbulence. This is the realm of unknowables.
It is possible to change domains in the Cynefin framework. As knowledge increases there is a clockwise drift from chaotic through complex and complicated to obvious.
There can also be counter-clockwise movement as people die and knowledge is forgotten, or as new generations question the rules; and a counter-clockwise push from chaotic to obvious can occur when a lack of order causes rules to be imposed suddenly.
It is essential for decision-makers to know the domain their current challenge is in, so that you can apply the appropriate action modes.
It becomes obvious that the transformation towards agile, human-centric, self-managed, socially sustainable work environments or teal, which is at this point in time an unordered path, creates and brings forward novel practices like for example Holacracy, etc. At the same time there are emerging practices and observable patterns that are pointing to principles that can be beneficial for organizational transformation, like focusing on the human aspect instead of only procedural changes or engagement and involvement over top-down directives. In complex contexts the solutions need to emerge from the community instead of being imposed from the leader in a top-down way.
As we are moving to an increasingly complex future, or even chaotic one, where we cannot predict the future outcomes by past behaviors, forecasting or planning, it also emerges and becomes obvious that there is a novel quality and practice required for decision-makers and leaders, not only in organizational contexts. That is to lead, oneself and others, from a space of intuitive insight. We need to pause and reflect to make sense of our context, act from a place of increased consciousness and often go against what our rationality or instinct tells us would be the right thing to do.
We also need to question our own way of thinking and beliefs that we hold, question everything we know. Are we stuck with what we have done in the past or are we open to novel practices and thoughts and ideas from non-experts, or places we normally would not look for inspiration? Are we flexible enough to look at each situation and context separately and make a new, individual assessment? Are we willing to respond to each context differently?
"Business schools and organizations equip leaders to operate in ordered domains (simple and complicated), but most leaders usually must rely on their natural capabilities when operating in unordered contexts (complex and chaotic). In the face of greater complexity today, however, intuition, intellect, and charisma are no longer enough. Leaders need tools and approaches to guide their firms through less familiar waters.
David Snowden created the Cynefin framework in 1999 then working for IBM Global services. Cynefin is Welsh for habitat, a word "that signifies the multiple factors in our environment and our experience that influence us in ways we can never understand".
In 2013 he and Cynthia Kurtz, an IBM researcher, described the framework in detail in a paper, "The new dynamics of strategy: Sense-making in a complex and complicated world".
In 2007 Snowden and Mary E. Boone described the Cynefin framework in the Harvard Business Review, under the headline "A Leader's Framework for Decision Making" and were awarded "Outstanding Practitioner-Oriented Publication in OB" from the Academy of Management's Organizational Behavior division.
The Cynefin Centre – a network of members and partners from industry, government and academia – began operating independently of IBM in 2004.
What context are you, your organization or your current task in?


text:
Paclitaxel, 5-fluorouracil, and cisplatin combination chemotherapy for the treatment of advanced gastric carcinoma
Cancer. 1999 Jan 15;85(2):295-301.
Y H Kim 1 , S W Shin, B S Kim, J H Kim, J G Kim, Y J Mok, C S Kim, H S Rhyu, J H Hyun, J S Kim
1 Department of Hematooncology, Korea University, College of Medicine, Seoul.
Background: Although the clinical efficacy of paclitaxel in the treatment of gastric carcinoma has not been clearly defined, recent reports have suggested a possible role in the treatment of upper gastrointestinal carcinomas in vitro and in vivo. In this study, the authors evaluated the efficacy and toxicity of a combination chemotherapy that included paclitaxel, 5-fluorouracil (5-FU), and cisplatin in the treatment of patients with advanced gastric carcinoma.
Methods: Forty-one gastric carcinoma patients with metastatic disease, unresectable advanced disease, or relapsed disease were treated with the following regimen, administered every 28 days: paclitaxel 175 mg/m2 by 3-hour intravenous (i.v.) infusion on Day 1, 5-FU 750 mg/m2 by 24-hour continuous i.v. infusion on Days 1-5, and cisplatin 20 mg/m2 by 2-hour i.v. infusion on Days 1-5. Twenty-six patients had measurable disease, and 15 had evaluable disease. All patients were assessable for toxicity.
Results: Twenty-one of the 41 patients (51%; 95% confidence interval [CI], 36.5-65.7%) demonstrated an objective response, including 4 complete responses (10%; 95% CI, 3.9-22.5%). Sixty-five percent of the patients with measurable disease (17 of 26; 95% CI, 58-92.5%) and 27% of the patients with evaluable disease (4 of 15: 95% CI, 11.1-52.3%) achieved a complete response or a partial response. The median response duration was 17 weeks (range, 4-90 weeks), and the median survival duration for all patients was 26 weeks (range, 8 to 118+ weeks). The major toxicity of this treatment was myelosuppression with neutropenia of World Health Organization Grade 3 and 4 in 24% and 10% of the patients, respectively. Nonhematologic toxicity included mucositis, nausea/vomiting, diarrhea, neurotoxicity, and alopecia. Fluid retention occurred in two patients, and one patient had an anaphylatic reaction. Dose reduction was necessary for one patient, because Grade 4 neutropenia and mucositis occurred.
Conclusions: Paclitaxel, 5-FU, and cisplatin was an active combination regimen in the treatment of advanced gastric carcinoma. The toxicity of this regimen was tolerable. Based on these findings, this combination regimen could be an attractive treatment in the preoperative setting.
Antineoplastic Agents, Phytogenic / administration & dosage
Antineoplastic Agents, Phytogenic / adverse effects
Antineoplastic Combined Chemotherapy Protocols / administration & dosage
Antineoplastic Combined Chemotherapy Protocols / adverse effects
Antineoplastic Combined Chemotherapy Protocols / therapeutic use*
Cisplatin / administration & dosage
Cisplatin / adverse effects
Fluorouracil / administration & dosage
Fluorouracil / adverse effects
Paclitaxel / administration & dosage
Paclitaxel / adverse effects
Stomach Neoplasms / drug therapy*
Stomach Neoplasms / pathology
Antineoplastic Agents, Phytogenic
Fluorouracil


text:
\section{Introduction}\label{sec:intro}

\emph{Order diagrams}, also called \emph{line diagrams} or \emph{Hasse
  diagrams}, are a great tool for visualizing the underlying structure of
ordered sets. In particular they enable the reader to explore and interpret
complex information. In such diagrams every element is visualized by a point on
the plane. Each edge of the covering relation is visualized as an ascending line
connecting its points. These lines are not allowed to touch other points. These
strong requirements are often complemented with further soft conditions to
improve the readability of diagrams. For example, minimizing the number of
crossing lines or the number of different slopes. Another desirable condition is
to draw as many chains as possible on straight lines. Lastly, the distance of
points to (non-incident) lines should be maximized.

Experience shows that in order to obtain (human) readable drawings one has to
balance those criteria. Based on this notion, there are algorithms that produce
drawings of order diagrams optimizing towards some of the criteria mentioned
above. Drawings produced by such algorithms are sufficient to some
extent. However, they may not compete with those created manually by an
experienced human. However, such an expert is often not available, too
expensive, or not efficient enough to create a large number of order
diagrams. Hence, finding efficient algorithms that draw diagrams at a suitable
quality is still an open task. An exemplary requiring such algorithms is Formal
Concept Analysis (FCA) \cite{fca-book}, a theory that can be used to analyze and
cluster data through ordering it.

In this work we present a novel approach which does not employ the optimization
techniques as described above. For this we make use of the structure and its
properties that are already encapsulated in the ordered set. We base our idea on
the observation that ordered sets of order dimension two can be embedded into
the plane in a natural way. Building up on this we show a procedure to embed the
ordered sets of order dimension three and above by reducing them to the
two-dimensional case. To this end we prove an essential fact about
inclusion-maximal bipartite induced subgraphs in this realm. Based on this we
link the naturally emerging $\mathcal{NP}$-hard computation problem to a
formulation as a SAT problem instance. Our main contribution with respect to
this is \Cref{thm:main}.

We investigate our theoretical result on different real-world data sets using
the just introduced algorithm \texttt{DimDraw}. Furthermore, we note how to
incorporate heuristical approaches replacing the SAT solver for faster
computations. Finally, we discuss in part surprising observations and formulate
open questions.

\section{Related Work}
\label{sec:relatedwork}

Order diagrams can be considered as acyclic (intransitive) digraphs that are
drawn upward in the plane, i.e., every arc is a curve monotonically increasing
in $y$-direction. A lot of research has been conducted for such upward
drawings. A frequently employed algorithm-framework to draw such graphs is known
as Sugiyama Framework \cite{Sug81}. This algorithm first divides the set of
vertices of a graph into different layers, then embeds each layer on the same
$y$-coordinate and minimizes crossings between consecutive layers. Crossing
minimization can be a fundamental aesthetic for upward drawings. However the
underlying decision problem is known to be $\mathcal{NP}$-hard even for the case
of two-layered graphs \cite{eades1986edge}. A heuristic for crossing reduction
can be found in~\cite{eades1986median}. The special case for drawing rooted
trees can be solved using divide-and-conquer algorithms
\cite{journals/tse/ReingoldT81}. Such divide-and-conquer strategies can also be
used for non-trees as shown in~\cite{journals/tsmc/MessingerRH91}.  Several
algorithms were developed to work directly on order diagrams. Relevant for our
work is the dominance drawing approach. There, comparable elements of the order
relation are placed such that both Cartesian coordinates of one element are
greater than the ones of the other \cite{kelly1975planar}. Weak dominance
drawings allow a certain number of elements that are placed as if they were
comparable \cite{journals/corr/abs-1108-1439} even when they aren't. Our
approach is based on this idea. Previous attempts to develop heuristics are
described in~\cite{conf/gd/KornaropoulosT12a}.  If an ordered set is a lattice
there are algorithms that make use of the structure provided by this. The
authors in~\cite{stumme95geometrical} make use of geometrical representations
for drawings of lattices. In~\cite{freese2004automated} a force directed
approach is employed, together with a rank function to guarantee the ``upward
property'' is preserved. A focus on additive diagrams is laid out in
\cite{ganteradd}.

\section{Notations and Definitions}
\label{sec:notations}
We start by recollecting notations and notions from order theory
\cite{trotter1992combinatorics}. In this work we call a pair $(X,R)$ an
\emph{ordered set}, if $R\subset X \times X$ is an \emph{order relation} on a
set $X$, i.e., $R$ is reflexive, antisymmetric and transitive. In this setting
$X$ is called the \emph{ground set} of $(X,R)$. In some cases, we write
$(X,{\leq})$ instead of $(X,R)$ throughout this paper. We then use the
notations $(a,b)\in {\leq}$, $a \leq b$ and $b \geq a$ interchangeably. We write
$a<b$ iff $a \leq b$ and $a \neq b$. Alike, if $b \geq a$ and $b \neq a$, write
$b > a$. We say that a pair $(a,b)\in X \times X$ is \emph{comparable}, if
$a \leq b$ or $b \leq a$, otherwise it is \emph{incomparable}. An order relation
on $X$ is called \emph{linear} (or total) if all elements of $X$ are pairwise
comparable. For $(X,{\leq})$ the order relation $L$ on $X$ is called a
\emph{linear extension} of $\leq$, iff $L$ is a linear order and
${\leq} \subset L$. If $\mathcal{R}$ is a family of linear extensions of $\leq$
and ${\leq}=\bigcap_{ L \subset \mathcal{R}}L$, we call $\mathcal{R}$
\emph{realizer} of $\leq$. The minimal $d$ such that there is a realizer of
cardinality $d$ for  $(X,{\leq})$ is called its \emph{order
  dimension}. We use the denotation of order dimension for ordered sets and
order relations interchangeably. For a set $X$ and
$\mathcal{C}\subset X \times X$ we denote
$\mathcal{C}^{-1}:=\{(a,b) \mid (b,a)\in \mathcal{C}\}$. The \emph{transitive closure} of
$R$ is denoted by $R^+$.

For our work we consider simple graphs denoted by $(V,E)$, where
$E \subset\binom{V}{2}$. For an ordered set $(X,{\leq})$, its \emph{comparability
  graph} is defined as the graph $(X,E)$, such that $\{a,b\}\in E$, if and only
if $a,b\in X$ are comparable. Similary the \emph{cocomparability graph}
(sometimes called incomparability graph) is the graph on $X$ where $\{a,b\}$ is
an edge if and only if $a,b\in X$ are incomparable. Two order relations on the
same ground set are called \emph{conjugate} to each other, if the comparability
graph of one is the cocomparability graph of the other.  We refrain from a
formal definition of a \emph{drawing} of $(X,{\leq})$. However we need to
discuss the elements of such drawings used in our work. Each element of the
ground set is drawn as a point on the plane. The \emph{cover relation} is
defined as
${\cover(X,{\leq})\coloneqq\{(a,b)\in {\leq}\mid \nexists c \in X : a < c <
  b\}}$. Each element of the cover relation is drawn as a monotonically
increasing curve connecting the points.

From here on some definitions are less common. For $(X,{\leq})$ we denote the
set of incomparable elements by $\inc(X,{\leq})$. Two elements
$(a,b),(c,d)\in \inc(X,{\leq})$ are called \emph{incompatible}, if their addition
to $\leq$ creates a cycle in the emerging relation, i.e., if there is some
sequence of elements $c_1,\ldots ,c_n\in X$, such that each pair
${(c_i,c_{i+1})\in{\leq}\cup(a,b)\cup(c,d)}$ with $i \in\{1,\ldots,n\}$ and
$c_{n+1}=c_1$. We call the graph $((\inc(X,{\leq}),E)$ with
$\{(a,b),(c,d)\}\in E$, iff $(a,b)$ and $(c,d)$ are incompatible the
\emph{transitive incompatibility graph}. Denote this graph by $\tig(X,{\leq})$.
We say a pair $(a,b)\in\inc(X,{\leq})$ \emph{enforces} another pair
$(c,d)\in\inc(X,{\leq})$, iff $(c,d)\in ({\leq}\cup(a,b))^+$. If and $(a,b)$
enforces $(c,d)$ in $\leq$, we write $(a,b)\rightarrow(c,d)$.

\section{Drawing Ordered Sets of Dimension Two}
\label{sec:2d}
Ordered sets of order dimension 2 have a natural way to be visualized using a
realizer by their dominance drawings \cite{kelly1975planar}. Let $(X,{\leq})$ be
an ordered set of dimension two. First define the \emph{position} for each $x\in X$ in a linear extension $L$ as the number of vertices that are
smaller, i.e., $\pos_{L}(x)\coloneqq\{y\in X\mid y < x\}$.

Now let $\mathcal{R}=|\{{\leq_1},{\leq_2}\}|$ be a realizer consisting of two
linear extensions of $\leq$.  Each element is embedded into a two-dimensional
grid at the pair of coordinates $(\pos_{\leq_1}(x),\pos_{\leq_2}(x))$. Embedding
this grid into the plane is done using the generating vector $(-1,1)$ for $x_1$
and $(1,1)$ for $x_2$.
Each point now divides the plane into four quadrants using the two
lines that are parallel to $x_1$ and $x_2$. It holds that $a < b$, if and only
if the point $b$ is in the quadrant above the point $a$ by construction. Draw
the elements of the cover relation as straight lines. This guarantees that all
elements of the cover relation are drawn as monotonically increasing curves.
In order to compute such drawings, a preliminary check of the two-dimensionality
of the ordered set is required. If so computing a realizer in polynomial time is
possible as a result of the following theorem.

\begin{theorem}[Dushnik and Miller, 1941 \cite{dushnik1941partially}]
\label{thm:dushnikmiller}
The dimension of an ordered set $(X,{\leq})$ is at most 2, if and only if there is
a conjugate order $\leq_C$ on $X$. A realizer of $P$ is given by $\leq$ given by
$\mathcal{R}=\{{\leq}\cup{\leq_C},{\leq}\cup{\geq_C}\}$.
\end{theorem}

\noindent In 1977, Golumbic gave an algorithm \cite{golumbic1977complexity} to
check whether a graph is transitive orientable, i.e., whether there is an order
on its vertices, such that the graph is exactly the comparability graph of this
order. It computes such an order, if it exists. This algorithm runs in
$\mathcal{O}(n^3)$, with $n$ being the number of vertices of the
graph. Combining this with \Cref{thm:dushnikmiller} provides an algorithm to
compute whether an ordered set is two-dimensional. Furthermore, it also returns a
realizer, in the case of two-dimensionality. For the sake of completeness, note that
there are faster algorithms (as fast as linear)
\cite{Mcconnell97linear-timetransitive}) for computing transitive
orientations. However those only work if the graph is actually transitive
orientable and return erroneous results otherwise.  On a final note, deciding
dimension $3$ or larger is known to be $\mathcal{NP}$-complete as shown by
Yannakis in 1982 \cite{yannakakis1982complexity}. This is in contrast to the
just stated fact about two-dimensional orders.

\section{Drawing Ordered Sets of Higher Dimensions}
\label{sec:3d}

The idea of embedding two-dimensional orders does generalize in a natural way to
higher dimensions, i.e., gives us a nice way to embed $n$-dimensional ordered
sets into $n$-dimensional space (Euclidean space) by using an $n$-dimensional
realizer. However, projecting an ordered set from a higher dimension into the
plane turns out to be hard. See, for example, the projections of an ordered set
in example \Cref{fig:project}.  For this reason our algorithm makes use of a
different method to compute drawings of order diagrams for higher dimensions.

\begin{figure}[t]
  \centering
  \includegraphics[height=10em]{img0}\hfill
  \includegraphics[height=10em]{img1}\hfill
  \includegraphics[height=10em]{img2}\hfill
  \includegraphics[height=10em]{img3}\hfill
  \includegraphics[height=10em]{img4}\hfill
  \includegraphics[height=10em]{img5}\hfill
  \includegraphics[height=10em]{img6}\hfill
  \includegraphics[height=10em]{img7}
  \caption{A three-dimensional ordered set embedded -- based on its realizer -- into
    three dimensional Euclidean space and then projected into the plane using a
    parallel projection from multiple angles. Even though the structure of the
    ordered set is recognizable, the drawings are all not satisfactory.}
  \label{fig:project}
\end{figure}

In short, the main idea of this section is the following: for a given order
relation we want to insert some number of additional pairs in order to make it
two-dimensional. This allows the resulting order to be drawn using the algorithm
for the two-dimensional case described in the previous section. Afterwards we
once again remove all the inserted pairs. By construction, the property that if
$a < b$, the point $a$ is inserted below $b$ is still preserved.  Such drawings
are sometimes called \emph{weak dominance drawings}
\cite{journals/corr/abs-1108-1439}. However, for each inserted pair $(a,b)$, we
obtain two points in the drawing that are drawn as if $a$ and $b$ were
comparable. This poses the question for minimizing the number of inserted pairs.

\begin{definition}
  Let $(X,{\leq})$ be an ordered set. A set $\mathcal{C}\subset \inc(X)$ is called
  a \emph{two-dimension-extension} of $(X,{\leq})$, iff ${\leq}\cup\mathcal{C}$ is
  an order on $X$ and the ordered set $(X,{\leq}\cup\mathcal{C})$ is
  two-dimensional.
\end{definition}
Such an extension always exists: a linear extension of dimension one
always exists. If an order contains exactly one incomparable pair it has
dimension two.

It is known to be $\mathcal{NP}$-complete to decide whether an ordered set can
be altered to be two-dimensional by inserting $k$ pairs \cite{Brightwell_2013}.
Hence, we propose an algorithm that tackles the problem for approximating the
corresponding optimization problem. The idea of the algorithm is based on the
following theorem.

\begin{theorem}[Doignon et.al., 1984 \cite{doignon1984realizable}]\label{thm:doignon}
  The ordered set $(X,{\leq})$ has order-dimension two if and only if
  $\tig(X,{\leq})$ is bipartite.
\end{theorem}

\noindent Thus, for $(X,{\leq})$ of dimension greater than two,
$\tig(X,{\leq})$ is non-bipartite. We want to find a maximal induced
bipartite subgraph of $\tig(X,{\leq})$.

\begin{lemma}\label{thm:ring}
  Let $(X,{\leq})$ be an ordered set and $(a,b),(c,d)\in\inc(X,{\leq})$. Then the
  following are equivalent:
  \begin{enumerate}[label=\roman*)]
  \item $d \leq a$ and $b \leq c$.
  \item $(a,b)\rightarrow(d,c)$.
  \item $(c,d)\rightarrow(b,a)$.
  \item $(a,b)$ and $(c,d)$ are incompatible.
  \item $(b,a)$ and $(d,c)$ are incompatible.
  \end{enumerate}
\end{lemma}

\begin{proof}
  $(i)\Rightarrow(iv)$. Consider the relation
  $\prec \coloneqq{\leq} \cup (a,b) \cup (c,d)$. This yields
  $d \prec a \prec b \prec c \prec d$, i.e, $\prec$ contains a
  cycle. Analogously $(i) \Rightarrow (v)$.\\%
  $(iv)\Rightarrow(ii)$. The assumption directly implies that
  $(d,c) \in ({\leq}\cup(a,b))^+$ due to the cycle generated by $(a,b)$ and
  $(c,d)$. By the same argument $(v)\Rightarrow (iii)$.\\%
  $(ii)\Rightarrow(i)$. Assume $d\not \leq a$ or $b \not \leq c$. Since
  $(d,c)\in({\leq}\cup(a,b))^+$ it follows $(d,c)\in {\leq}$ which contradicts
  $(c,d)\in \inc(X,\leq)$. Similarly follows $(iii)\Rightarrow (i)$.
  \null\hfill $\square$
\end{proof}

\noindent Recall the definition of $\tig(X,{\leq})$ defined on $\inc(X,{\leq})$
with incompatible pairs being connected. Call a cycle in
$\tig(X,{\leq})$  \emph{strict}, iff for each two adjacent pairs $(a,b)$ and
$(c,d)$ it holds that $d < a$ and $b < c$. A strict path is defined analogously.

\begin{lemma}[Doignon et al., \cite{doignon1984realizable}]
  \label{thm:strict}
  Let $(X,{\leq})$ be an ordered set and let the pair $v\in\inc(X,{\leq})$. Then the
  following statements are equivalent:
  \begin{enumerate}[label=\roman*)]
  \item $v$ is in contained in an odd cycle in $\tig(X,{\leq})$.
  \item $v$ is contained in a strict odd cycle in $\tig(X,{\leq})$.
  \end{enumerate}
\end{lemma}

\begin{remark}
  This is stated implicitly in their proof of Proposition 2, verifying the
  equality between the two chromatic numbers of a hypergraph corresponding to
  our cycles and a hypergraph corresponding to our strict cycles.
\end{remark}

\begin{theorem}\label{thm:main}
  Let $(X,{\leq})$ an ordered set. Let $\mathcal{C}\subset\inc(X,{\leq})$ be minimal with
  respect to set inclusion, such that $\tig(X,{\leq})\backslash\mathcal{C}^{-1}$ is
  bipartite. Then $(X,{\leq}\cup\mathcal{C})$ is an ordered set.
\end{theorem}

\begin{proof}
  Refer to the bipartition elements of $\tig(X,{\leq})\backslash\mathcal{C}^{-1}$ with
  $P_1$ and $P_2$.

  \begin{claim}
    The arrow relation is transitive, i.e., if $(a,b)\rightarrow(c,d)$ and
    $(c,d) \rightarrow (e,f)$ then $(a,b)\rightarrow (e,f)$. If
    $(a,b)\rightarrow (c,d)$ then $c\leq a$ and $b\leq d$ by
    definition. Similarly $(c,d)\rightarrow (e,f)$ implies $e \leq c$ and
    $d \leq f$. By transitivity of $\leq$ this yields that $e\leq a$ and
    $b \leq f$ which in turn implies that $(a,b)\rightarrow(d,f)$.
  \end{claim}

  \begin{claim}[$\star$]Let $(a,b),(c,d)\in\inc(X,{\leq})$ with
    $(a,b)\not \in \mathcal{C}^{-1}$ and $(a,b)\rightarrow (c,d)$, then
    $(c,d)\not \in \mathcal{C}^{-1}$. Assume the opposite, i.e., $(c,d) \in \mathcal{C}^{-1}$. Without
    loss of generality let $(a,b) \in P_1$. As $(c,d) \in \mathcal{C}^{-1}$ there has
    to be a pair $(e,f)\in P_1$ that is incompatible to $(c,d)$, i.e.,
    $(e,f)\rightarrow (d,c)$, otherwise $(c,d)$ can be added to $P_1$ without
    destroying the independet set. However $(a,b)\to (c,d)$ is
    equivalent to $(d,c)\rightarrow (b,a)$ and yields together with the transitivity
    of the arrow relation $(e,f)\rightarrow (b,a)$. But than $(e,f)$ and $(a,b)$
    are incompatible, a contradiction since both are in the independent set
    $P_1$.
  \end{claim}

  \begin{claim}[$\star \star$]
    If $(x,y)\in \mathcal{C}$, then $(y,x)\not\in\mathcal{C}$. As $(y,x)\in \mathcal{C}^{-1}$, there is
    a pair $(a,b)$ in $P_1$, such that $(a,b)$ and $(y,x)$ are incompatible,
    i.e., $(a,b)\rightarrow (x,y)$, otherwise $(y,x)$ can be added to
    $P_1$. However, since $(a,b)\not \in \mathcal{C}^{-1}$ follows $(x,y)\not \in \mathcal{C}^{-1}$ by
    Claim ($\star$).
  \end{claim}

  \noindent Reflexivity: $\forall x\in X$ we have
  $(x,x)\in {\leq}\subset {\leq}\cup\mathcal{C}$.

  \noindent Antisymmetry: assume $(x,y)\in {\leq}\cup\mathcal{C}$ and
  $(y,x)\in{\leq}\cup\mathcal{C}$. We have to consider three cases. First,
  $(x,y)\in{\leq}$ and $(y,x)\in{\leq}$. Then $x=y$, as $\leq$ is an order
  relation. %
  Secondly, $(x,y)\in{\leq}$ and $(y,x)\in\mathcal{C}$. If $(x,y)\in{\leq}$, then $x$
  and $y$ are comparable, i.e., the pair $(y,x)$ can't be in
  $\inc(P,{\leq})$. Then $(y,x)\not\in\mathcal{C}$, a contradiction. %
  Thirdly, $(x,y)\in \mathcal{C}$ and $(y,x)\in\mathcal{C}$. This may not occur by
  Claim~$(\star\star)$.

  \noindent Transitivity: let $(x,y)\in{\leq}\cup\mathcal{C}$ and
  $(y,z)\in{\leq}\cup\mathcal{C}$ we show $(y,z)\in{\leq}\cup\mathcal{C}$. We have to
  consider four cases. First, $(x,y)\in{\leq}$ and $(y,z)\in{\leq}$ implies
  $(x,z)\in{\leq}$. %
  Secondly, $(x,y)\in{\leq}$ and $(y,z)\in\mathcal{C}$ and assume that
  $(x,z)\not\in({\leq}\cup\mathcal{C})$. Then $(z,x)\not \in \mathcal{C}^{-1}$, but
  $(z,x)\rightarrow (z,y)$, as $(x,y)\in{\leq}$. From Claim $(\star)$ follows
  that $(z,y)\not\in\mathcal{C}^{-1}$, a contradiction to $(y,z)\in \mathcal{C}$. The case
  $(x,y)\in \mathcal{C}$ and $(y,z)\in{\leq}$ is treated analogously. %
  Lastly, $(x,y) \in \mathcal{C}$ and $(y,z) \in \mathcal{C}$ and assume
  $(x,z)\not \in \mathcal{C}$, i.e., $(z,x)\in \mathcal{C}^{-1}$. There has to be an odd cycle in
  $P_1\cup P_2$ together with $(y,z)$, otherwise $(y,z)$ can be added to
  $P_1\cup P_2$ to create a larger biparite graph. By \Cref{thm:strict}, there
  also hast to be a strict odd cycle. Let the neighbors of $(y,z)$ in
  $P_1\cup P_2$ be $(a,b)$ and $(c,d)$. Then $a < z$, $c < z$, $y < b$ and
  $y < d$, and the pairs $(a,b)$ and $(c,d)$ are connected by a strict path on
  an even number of vertices through the strict odd cycle. By the same argument
  there are pairs $(e,f)$ and $(g,h)$ with $e < y$, $g < y$, $x < f$ and $x < h$
  and $(e,f)$ and $(g,h)$ connected by a strict odd path. We now show, that
  $(z,x)$ is in an odd cycle with $P_1\cup P_2$ to yield a contradiction. For
  this consider the following paths $A=(c,f)(z,x)(h,a)$, $B=(d,h)(h,g)$ and
  $C=(f,e)(e,b)$. Each of those is a path in $\tig(P,{\leq})$ by definition.

  \begin{claim}
    Between $(h,a)$ and $(d,h)$ there is a path on an even number of vertices in
    $P_1\cup P_2$. To show this, let $(a_1,b_1),\ldots ,(a_{2k},b_{2k})$ be the
    strict path on an even number of vertices connecting $(a,b)$ and $(d,c)$
    such that $(a_1,b_1)$ = $(a,b)$ and $(a_{2k},b_{2k})=(d,c)$. This implies
    $a_{2i+1}< b_{2i+2}$, $a_{2i}< b_{2i+1}$, $b_{2i+1}> a_{2i+2}$ and
    $a_{2i}> b_{2i+1}$ and for each $i \in \{0,\ldots ,k-1\}$. However, this
    yields the path $(h,a)=(h,a_1)(b_2,h)(h,a_3)\cdots(b_{2k},h)=(d,h)$ which is
    even and connecting $(h,a)$ and $(d,h)$ in $P_1\cup P_2$, as required.
  \end{claim}

  \noindent Analogously we obtain a path between $(c,f)$ and $(b,f)$ on an even
  number of vertices. Moreover $(h,g)$ and $(f,e)$ are also connected by a path
  on an even number of vertices in $P_1\cup P_2$, since $(g,h)$ and $(e,f)$ are
  connected by an even path. Reversing all pairs of this path yields the
  required path. Combining the segments $A$, $B$ and $C$ with the paths
  connecting them yields an odd cycle in $P_1\cup P_2\cup \{(z,x)\}$, a
  contradiction.  \null\hfill $\square$
\end{proof}

\subsection{The Importance of Inclusion-Maximality}
Consider the standard example $S_3=(X,{\leq})$ where the ground set is defined as
$X=\{a_1,a_2,a_3,b_1,b_2,b_3\}$ and $a_i\not\leq a_j$ for $i \neq j$,
$b_i \not \leq b_j$ for $i \neq j$ and $a_i\leq a_j$ if and only if $i \neq
j$. This example is well known to be a three-dimensional ordered set. However,
it becomes two-dimensional by inserting a single pair $(a_i,b_i)$ into the order
relation $\leq$ for some index $i\in\{1,2,3\}$, i.e., the transitive
incomparability graph becomes bipartite if we remove for example the pair
$(b_1,a_1)$. Now assume we do not require to removing a set minimal with respect
to set inclusion, take for example both pairs $(a_1,b_1)$ and
$(b_1,a_1)$. However the set $(X,{\leq}\cup\{(a_1,b_1),(b_1,a_1)\})$ is not an
ordered set, as both pairs $(a_1,b_1)$ and $(b_1,a_1)$ are in
${\leq}\cup\{(a_1,b_1),(b_1,a_1)\}$. This is a conflict with $a_1\neq b_1$,
i.e., we do not preserve the order property as the resulting relation is not
antisymmetric.

\subsection{Bipartite Subgraph is not Sufficient}
\label{sec:insuffic}
From \Cref{thm:main} one might conjecture that finding an inclusion-minimal
bipartite subgraph of $\tig(X,{\leq})$ is sufficient to find a two-dimensional
extension of $(X,{\leq})$. However, it may occur that two pairs are not
incompatible in $\tig(X,{\leq})$ and are incompatible in
$\tig(X,{\leq}\cup\mathcal{C})$ with $\mathcal{C}$ being an inclusion-minimal set such that
$\tig(X,{\leq})\backslash\mathcal{C}^{-1}$ is bipartite. This can arise in particular, if
the following pattern occurs: the ordered set contains the elements $a,b,c$ and
$d$, such that all elements are pairwise incomparable, except $b < d$ and
$(c,a)\in \mathcal{C}$ exhibits this observation, see \Cref{fig:counterexample1}. Then
$(a,b)$ and $(c,d)$ are not incompatible in $\tig(X,{\leq})$, but they become
incompatible with the relation ${\leq}\cup\mathcal{C}$.  An example for this is provided by the
ordered set in \Cref{fig:counterexample2}.  The transitive incomparability graph
of this ordered set has 206 vertices. By removing pairs

\begin{figure}[t]\centering
  \begin{minipage}[t]{0.4\textwidth}
    \centering
    \includegraphics{counterexample1.pdf}
    \caption{An example how new incompatibilities can arise. $\leq$ is the
      continous line, $\mathcal{C}$ is the dashed line. $(a,b)$ and $(c,d)$ are not
      incompatible in $\leq$ and incompatible in ${\leq} \cup \mathcal{C}$.}
    \label{fig:counterexample1}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.55\textwidth}
    \centering
    \includegraphics{counterexample2.pdf}
    \caption{An example for an ordered set that has a transitive incompatibility
    graph with an inclusion-minimal bipartite subgraph of the transitive
    incompatibility graph that does not give rise to a two-dimension extension.}
    \label{fig:counterexample2}
  \end{minipage}
\end{figure}

\noindent \quad \texttt{\small (5,9), (3,13), (6,15), (17,15), (5,15), (5,13), (11,10),
  (9,6), (5,6), \\\null\quad(4,15), (1,8), (3,15), (11,12), (2,8), (11,7), (0,15), (1,16)},

\noindent the transitive incomparability graph of this ordered set becomes
bipartite. However, if we once again compute the transitive incomparability
graph of the new ordered set, we see that it is not bipartite, i.e., the new
ordered set is once again not two-dimensional. We have to add the additional
pair \texttt{(17,8)} to make the graph two-dimensional. It may be remarked at
this point that it is in fact possible to make the transitive incomparability
graph bipartite by adding only eleven pairs (in contrast to the seventeen added
in this particular example), see \Cref{fig:counterexample_alg}. Those pairs give
rise to a two-dimension extension.

\section{Algorithm}
\label{sec:alg}
Building up on the ideas and notions from the previous sections we propose the
Algorithm \texttt{DimDraw} as depicted in \Cref{alg:dimdraw}. Given an ordered
set $(X,{\leq})$, one calls \texttt{Compute\_Coordinates}. Until this procedure
identifies a conjugate order using the \texttt{Comupte\_Conjugate\_Order} (and
in turn the algorithm of Golumbic \cite{golumbic1977complexity}), it computes
bipartite subgraphs of the transitive incompatibility graphs. Furthermore it
adds the so-computed pairs to the $\leq$. For finite ground sets the algorithm
terminates after finitely many steps.

\begin{algorithm}[t]
  \caption{\texttt{DimDraw}}
  \label{alg:dimdraw}
  Execute \texttt{Compute\_Coordinates} on the ordered set that is to be drawn.
  \hrule
\begin{lstlisting}[mathescape=true]
Input: Ordered set ($P$,$\leq$)
Output: Conjugate order of ($P$,$\leq$)

def Compute_Conjugate_Order($P$,$\leq$):
    $C$ $\coloneqq$ Cocomparability_Graph($P$,$\leq$)
    if Has_Transitive_Orientation($C$):
        ($P$,$\leq_C$) $\coloneqq$ Transitive_Orientation($C$)
        return $\leq_C$
    else:
        return $\perp$
\end{lstlisting}
\hrule
\begin{lstlisting}[mathescape=true]
Input: Ordered set ($P$,$\leq$)
Output: Coordinates of the drawing of ($P$,$\leq$)

def Compute_Coordinates($P,\leq$):
    $\leq_C$ $\coloneqq$ Compute_Conjugate_Order($P,\leq)$
    $\mathcal{C}$ $\coloneqq$ $\emptyset$
    while $\leq_C$ $=$ $\perp$:
        $I$ $\coloneqq$ Transitive_Incomparability_Graph($P,{\leq}\cup\mathcal{C}$)
        $B$ $\coloneqq$ Maximum_Bipartite_Subgraph($I$)
        $\mathcal{C}$ $=$ $\mathcal{C}\cup V(I\backslash B)^{-1}$
        $\leq_C$ $\coloneqq$ Compute_Conjugate_Order($P$,${\leq} \cup \mathcal{C}$)
    $\leq_1$ $\coloneqq$ ${\leq}\cup{\leq_C}$
    $\leq_2$ $\coloneqq$ ${\leq}\cup{\geq_C}$
    for $x$ in $P$:
        Coord($x$,1) $\coloneqq$ $|\{k\mid k \leq_1 x\}|-1$
        Coord($x$,2) $\coloneqq$ $|\{k\mid k \leq_2 x\}|-1$
    return Coord
\end{lstlisting}
\end{algorithm}

\subsection{Postprocessing}
The algorithm does not prevent a point from being placed on top of lines
connecting two different points. This however is not allowed in order
diagrams. Possible strategies to deal with this problem are the following. One
strategy is to modify the coordinate system, such that the marks of different
integers are not equidistant. Another one is to perturb the points on lines
slightly. A third way is to use splines for drawing the line in order to avoid
crossing the point.

\section{Finding large induced bipartite subgraphs}
\label{sec:bip}
Our algorithm has to compute an inclusion-minimal set of vertices, such that
removing those vertices from the transitive incompatibility graph results in a bipartite
graph. Deciding for a graph whether it is possible to make it bipartite by
removing a set of cardinality $k$ is known to be $\mathcal{NP}$-complete
\cite{LEWIS1980219}. Even approximations are known to be in this complexity
class \cite{approx}. Therefore we propose different approaches.

\subsection{Exact solution using a reduction to SAT}

Even for small example, i.e., orders on less than 30 elements, a naive approach
is infeasible.  As we will see in \Cref{sec:evaluation} the question for
computing $\mathcal{C}$ results in $\binom{182}{5}$ tests for an example on $19$
elements (\cref{fig:fische}) and $\binom{294}{29}$ tests for example
(\cref{fig:car}) on 24 elements. Therefore we need a more sophisticated solution
for the problem. We reduce the problem for finding biparite subgraphs to a SAT
problem and then solve this problem with a SAT-Solver, in our case MiniSat
\cite{minisat} in version 2.2. In other words we want to know for some graph
$G=(V,E)$ on $n$ vertices and $m$ edges whether by deleting $k$ vertices we can
make the graph bipartite. Solving is done by finding a partition of $V$ into the
three sets $P_1,P_2,\mathcal{C}^{-1}$, such that $P_1$ and $P_2$ are independent sets and
$|\mathcal{C}^{-1}|\leq k$. For this we construct a conjugative normal form as follows: for
each vertex $v_i$ we have three variables, call them
$V_{i,1},V_{i,2},V_{i,3}$. The first two variables indicate, whether the vertex
is placed in $P_1$ or $P_2$, respectively and the third variable indicates,
whether the vertex is placed in $\mathcal{C}^{-1}$. For each vertex we have to guarantees,
that it is placed in at least one of $P_1$, $P_2$ or $\mathcal{C}^{-1}$, i.e., at least one
of $V_{i,1},V_{i,2},V_{i,3}$ is true for each $i$. We achieve this with the
clause $V_{i,1} \vee V_{i,2}\vee V_{i,3}$ for all $i \in \{1,\ldots,n\}$. Also
we want to guarantee that $P_1$ and $P_2$ are independent sets, i.e., no two
vertices in $P_1$ or $P_2$ are connected by an edge. We achive this by adding
the two clauses $\neg V_{i,1} \vee \neg V_{j,1}$ and
$\neg V_{i,2} \vee \neg V_{j,2}$ for each edge $\{v_i,v_j\}\in E$. This ensures
that no two vertices connected by an edge are placed in set $P_1$ or
$P_2$. Finally we have to guarantee, that there are at most $k$ vertices in
$P_3$, i.e., that at most $k$ of the variables
$\{V_{1,3},V_{2,3},\ldots,V_{n,3}\}$ are true. There are multiple ways to
achieve this. We employ the method as described in~\cite{conf/cp/Sinz05}. This
results in an additional $(n-1)\cdot k$ auxiliary variables and $2nk+n-3k-1$
clauses. Altogether our SAT instance has in total $(n-1)(k+3)+3$ variables and
$2m+2nk+2n-3k-1$ clauses. For this CNF the following holds by construction.
\begin{theorem}
  The SAT instance as constructed above is satisfiable if and only if $G$ has an
  induced bipartite subgraph on $n-k$ vertices.
\end{theorem}

\noindent Now we build this SAT instance for $k$ increasing from 1 until it is
satisfiable. Then the set $\{v_i \mid V_{i,3}=\text{true} \}$ is exactly the
subset of vertices we have to remove to make the graph bipartite. Obviously
methods like binary search may be applied here.  Furthermore, we may also plug
heuristic procedures into our algorithm to find an inclusion-minimal set
$\mathcal{C}$. We experimented on this with a greedy algorithm, a simulated annealing
approach and a genetic algorithm with promising results, especially for the
genetic algorithm. However, it is preferred to use the SAT algorithm as long as
there is enough computational power and the problem instances are not too large.

\section{Experimental evaluation}
\label{sec:evaluation}
\begin{figure}[t]
  \null\hfill
  \includegraphics[height=10em]{fische_hand2}\hfill
  \includegraphics[height=10em]{fische}\hfill\null
  \caption{Two drawings of the ``Living Beings and Water'' lattice, by hand
    (left) and with our algorithm (right).}
  \label{fig:fische}
\end{figure}

\begin{figure}[t]
  \null\hfill
  \includegraphics[height=10em]{cars_hand2} \hfill
  \includegraphics[height=10em]{car1} \hfill
  \includegraphics[height=10em]{car2} \hfill\null
  \caption{The ``Drive concepts for motorcars'' lattice. Drawn by an expert
    (left) and two drawings of our algorithm (middle and right).}
  \label{fig:car}
\end{figure}

The \texttt{DimDraw} algorithm was originally designed with the idea in mind to
draw the order diagram of lattices. Those are employed in a particular in Formal
Concept Analysis (FCA), a mathematical theory for analyzing data. Note that any
complete lattice can be represented by a concept lattice in FCA. We tested our
algorithm on all lattice examples from a standard literature book on FCA
\cite{fca-book}. In all those cases the quality of the produced drawings came
close to examples hand drawn by experts. For example, consider the lattice that
arises from ``Living Beings and Water'' \cite[p.18]{fca-book}. In
\Cref{fig:fische} we compare the hand-drawn example (left) to the result drawn
by our algorithm (right). For \Cref{fig:car} \cite[p.40]{fca-book} there are two
different solutions depicted, both having the minimal number of pairs
inserted, note that the algorithm stops after it finds a single solution.

Because of the importance of drawings in FCA we tested the algorithm on every
lattice with eleven or less vertices. The reader might want to have a look at the
document containing all 44994 drawings on 7499 pages
\cite{durrschnabel_dominik_2019_3242627}.

Concluding the experiments we want to present an example that our algorithm also
works on non-lattices. Consider the ordered set from
\Cref{fig:counterexample2}. While the hand-drawn version of this order diagram
makes use of splines, our algorithm-generated version
\Cref{fig:counterexample_alg} uses exclusively straight lines.

An interesting observation during the experiments was the following: for all
examples of that we are aware, even including those not presented in this work,
one pass of the SAT solver was sufficient for reducing the order dimension to
two. This is surprising, in particular in light of~\cref{fig:counterexample2}
from~\cref{sec:insuffic}.

\begin{figure}[t]
  \centering
  \includegraphics[height=10em]{counterexample_alg}
  \caption{A non-lattice example drawing produced by the algorithm. See
    \Cref{fig:counterexample2} for a hand-drawn version.}
  \label{fig:counterexample_alg}
\end{figure}

\section{Conclusion and Outlook}
\label{sec:outlook}
We presented in this work a novel approach for drawing diagrams of order
relations. To this end we employed an idea by Doignon et al.\ relating order
dimension and bipartiteness of graphs and proved an extension. Furthermore, we
linked the naturally emerging problem to SAT. Finally, we demonstrated various
drawings in an experimental evaluation. The drawings produced by the algorithm
were, in our opinion, satisfying.  We would have liked to compare our algorithm
(exact and heuristic type) to the heuristics developed
in~\cite{conf/gd/KornaropoulosT12a}. Unfortunately, we were not able to
reproduce their results based on the provided description.

A notable observation is the fact that in all our experiments the SAT-Solver
blend of \texttt{DimDraw} was able to produce a solution in the first pass,
i.e., the algorithm found a truly minimal two-dimension extension. This raises
the natural question, whether the maximal induced bipartite subgraph approach
does always result in a minimal two-dimension extension. Further open questions
are concerned with employing heuristics and to improve the postprocessing
stage. The SAT-solver version of \texttt{DimDraw} is included in the software
conexp-clj\footnote{\url{https://github.com/tomhanika/conexp-clj}}. At a later
time we also want to include heuristic versions.

\subsection*{Acknowledgement}
The authors would like to thank Torsten Ueckerdt for pointing out the research
about diametral pairs and Maximilian Stubbemann for helpful discussions.

\bibliographystyle{bib/splncs04}

redpajama_set_name: RedPajamaArXiv


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text:
Dog found abandoned on park bench with child's heartbreaking note
Footage shows a dog who was abandoned in a park by a young child who claimed that adults were abusing the animal and it was too "painful" to watch the pooch suffer
Sofie JacksonVideo News Reporter
Sign up to FREE email alerts from Daily Star - Weekly #TeamDog
A heartbreaking video shows a woman find a puppy tied to a park bench with a note explaining that it was no longer safe at home because it was being abused.
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The dog was found in the bench of the park with a tear-jerking note (Image: @MascotaCoyoacan/Newsflash)
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Dog's adorable routine of bringing owner presents while she works melts hearts
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The letter left by the dog in the park (Image: Newsflash / @MascotaCoyoacan)
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text:
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lizmfwd2019-12-31T15:17:55-07:00December 31st, 2019|


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"Steadfast love and faithfulness will meet; righteousness and peace will kiss each other." (Psalm 85:10 NRSV).
What does it mean to live as a people who believe in God"s mercy and justice, and that they actually make a difference in this world? In our lives? These were some of the questions on my mind as I went for my morning run and listened to an interview with renowned justice activist Bryan Stevenson.
The answer that God seemed to give through the words of Stevenson was both unexpectedly simple and at the same time complex—hope. "Hopelessness is the enemy of justice," Stevenson emphasized. "If we allow ourselves to become hopeless, we become part of the problem. . . . Injustice prevails where hopelessness persists."
As I write these words, the year is still new, a time that is often filled with renewed energy, hope, and possibility. But this last year felt different, to me at least. In the midst of a pandemic and deep social division, hope these days feels like a commodity dispensed in unusually small portions—just enough to survive, perhaps, but hardly thrive. We're all tired, desperately so.
But Stevenson's words made me realize that maybe fatigue and despair offers us an opportunity to realize how deeply we need to draw from a source of strength beyond ourselves. Vital hope, the kind described in Scripture, has always been rooted in something deeper than the ups and downs our emotions, than the amount of stress we're carrying in our bodies, and whether we have the strength to face what life puts in our path.
Hope is rooted in the goodness and justice of God—in the promise that the universe is held in love. And, as Martin Luther King, Jr. famously said, hope is rooted in the surety that, however grim the currently reality is, in the long run the moral arc of the universe is tilted towards justice. The kind of hope that sustains and carries us through the hard times is hope only found in the movement of God's Spirit—carrying, sustaining, and guiding us towards life.
Psalm 85:10 offers an intriguing picture of that kind of hope—one in which our longings for heaven and earth to be united in wholeness and justice are finally satisfied. "Righteousness"—tsadiq—and "peace"—shalom—united in a kiss, our world at last governed by the wholeness and flourishing God has always intended. The other intriguing thing about this verse is its one of those Hebrew verses that can be translated in either past or future tense—either fulfilled or future. Skim a few different translations, and you'll find many translations that choose both.
But of course, for New Testament believers, both ways of reading it are true now. Christ's death and resurrection has already secured this promised future, and it's something we experience now. But it's also something that, until Christ returns, lies tantalizingly just out-of-reach in its fullness.
"Righteousness" is a word we too often think of as denoting some kind of abstract moral purity, a bit intangible and out of reach. But the Hebrew word often translated "righteousness"—tsadiq—can be translated with equal accurateness as "justice," and it's a word with tangible on-the-ground impact. As Cornel West famously put it, "justice is what love looks like in public." And in Scripture, to be one of the righteous—the tsadiqim—is to be a part of a community that fiercely believes in and lives for God's justice to be seen on earth, not just in heaven. It's to be part of a community that believes that shalom—creation flourishing as it was meant to be, with no one harming the flourishing of another—is not only possible, but the future promised in Christ. The future we're called to live into now.
To live that way requires hope. Not blind optimism, but a hope rooted in the belief that God's grace, goodness, and resurrection power is big enough to reach deeper than the most broken parts of ourselves, our country, and our world.
That kind of life also requires hope that God's justice is actually good for us—that God's justice is not designed to tear down or destroy, but to invite us into something better. Hope gives us the courage to let God's Spirit show us the truth about our past and carry us a different future. Because, as Stevenson puts it, "We all need mercy, we all need justice, and — perhaps — we all need some measure of unmerited grace."
Monica LaRose
Source: https://onbeing.org/programs/bryan-stevenson-love-is-the-motive/
About Monica La Rose
Scandalous Savior
How Jesus Revealed the Heart of God
The Real Gift of Christmas
Changing Everything
The Amazing Names of the Messiah
The Light of The World
The Faithfulness of God – The Floor Beneath Our Feet
Is Jesus God?
God's Invitation into Wholeness
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