Guest post by Arpita Mitra, MSc student in Criminology and Criminal Justice, University of Oxford. Arpita has a BA in sociology from the University of Delhi. Her research interests include youth justice, transnational crimes and juvenile delinquency in the context of transitional justice, with special emphasis on South Asian politics. Follow Arpita on Twitter @msarpitamitra.
This article examines the situation of refugee and migrant children, who are increasingly becoming victims of human trafficking in their efforts to reach Europe. The concept of 'missing children' is examined, in order to emphasise the lack of adequate documentation of unaccompanied minors, the increasing control of refugee camps by traffickers and other gangs of organised crime, and the multi-faceted understanding of the conditions of criminality in Europe.
The recent trend of mass migration has been a common feature of media outlets and the topic of numerous reports. With over 1.2 million people entering the European continent since 2015, questions around the experiences of conflict, the threat of persecution, and gross human rights violations have resurfaced political and academic consciousness. The 'burden' of the refugee crisis has developed alongside the discourse of responsibilisation, which downplay the needs of individuals coming from countries such as Syria, Afghanistan, and Sudan, to name a few. Consequently, the news concerning a three-year old Syrian boy found dead on the shores of Bodrum, Turkey became an alarming call for political leaders to revisit the plight of refugee unaccompanied children.
An unaccompanied minor is defined as an individual 'who has been separated from both parents and other relatives and is not being cared for by an adult who, by law or custom, is responsible for doing so'. As the borders along regions such as the Balkans are increasingly shut and re-fortified to limit the in-flow of refugees and asylum seekers, security concerns and rights of unaccompanied children pose greater challenges. It has been reported that children currently account for more than one third of the total migrant population arriving in Europe. According to the European Commission, around 3,500 unaccompanied minors applied for asylum status in Europe in the month of January 2016 alone. Boys between the ages of 14 and 16 years constitute 90 percent of the total proportion of minors travelling in the absence of parents or guardians. They're typically sent off by their respective guardians based on the calculation that even the potentially life-threatening journey to Europe provides their children with a better chance at a good life than remaining in countries where bombs and bullets are part of their daily existence.
In this context, questions are emerging with respect to 'missing and disappearing children' and, more specifically, the conditions and situations to which they are disappearing. Such concerns place unaccompanied child refugees in a continuum wherein they not only leave one site of exploitation (i.e., the state of conflict in their home country), but are increasingly made vulnerable to other zones of exploitation through their migration. They also highlight the multiple, inter-related causes of deprivation that child refugees experience and the manner in which acts of criminality may occur as the by-product of coercive circumstances and 'alternative' techniques of survival.
An existing 'pan-European criminal infrastructure' with regards to organised crime groups is mushrooming, as gangs seek to exploit the increasing migration flow in order to smuggle children into human trafficking, under the pretence of easier entry into Europe. Unaccompanied refugee children are, thus, key targets for the purposes of sexual exploitation and child labour. Not only does this situation indicate the extent to which camps fall under the control of traffickers, it also problematises the notion that border control mechanisms exist at the cusp, demarcating binaries of legality-illegality.
Under the EU law specific to child refugees and children seeking asylum, key requirements are established to: (a) recognise enhanced vulnerability of unaccompanied minors to sexual abuse/exploitation and/or human trafficking; (b) appoint guardians or representatives for the child victim of human trafficking; and (c) ensure 'special protective provisions' such as free legal counselling, representation and child-sensitive conduct of investigations and trial hearings among others. The case of 'missing children' goes against the basic principle of providing 'international protection for unaccompanied minors' (also mentioned under the Directive 2013/33/EU).
Beyond the prevailing 'anti-migration sentiments,' a key limitation in addressing the refugee crisis relates to the lack of infrastructure in Europe to support unaccompanied minors. In fact, president of METAdrasi, Lora Pappa, acknowledges the delayed and complicated procedure of registering unaccompanied minors and cross-checking the identity of adults who present themselves as separated relatives of the child. With the number of unaccompanied children entering Europe increasing fourfold since 2014, individual nation-states present grim statistics on missing and disappearing children at the border.
For example, local authorities in Finland, including the Finnish reception units, have 'no clue' about the 2,500 asylum seekers who have 'disappeared' under their supervision. Similar trends are noted in the more 'favoured destinations' for unaccompanied under-age asylum seekers such as Germany, Austria, and Hungary. Approximately 5,835 refugee children have been reported missing in Germany alone, 555 of whom are under the age of 14. While some of them will have been reunited with members of their family, a significant number comprise of unaccompanied minors, who have come into contact with gangs that facilitate various forms of international organised crime. Therefore, the link between agents seeking to smuggle people inside the European Union and trafficking gangs exploiting the vulnerabilities of unaccompanied minors accounts for part of this disturbing reality.
The tightening of border control policies have heightened the 'fear' of getting caught, or being sent back to the country of origin. This fear urges some unaccompanied minors to 'invisibilise' themselves to the authorities as a means to continue their respective journeys in Europe. At the same time, the decision to be invisible becomes counterproductive to the protection of rights of unaccompanied children in the light of growing risks of trafficking. How do we then account for 'crimes of coercion' or criminality involving such children, which are clearly driven by force, threat, or other forms of coercion?
This statement by 16-year old Martha reveals the experiences of unaccompanied children on their way to the Calais refugee camp along the French coast. Her experiences echo several other reported crimes concerning young refugee boys who have been routinely raped in these camps. The inability to pay steep entry-fee subjects unaccompanied children to various forms of coercion, one of them being to commit an act otherwise deemed as 'criminal'. This is a phenomenon reported by UNICEF as 'helping migrants evade law enforcement, opening lorries and selling drugs or food donated by volunteers in trafficker-controlled night-time markets.' Thus, loopholes in the protection of unaccompanied refugee children under the European Law create a favourable environment for members of organised crime groups to take advantage of the vulnerabilities of these children in the context of migration.
Although they enter Europe in the hope of escaping the violence and uncertainty plaguing their home country, they are continuously pulled in the vicious cycle of deprivation that involves forced recruitment into acts of economic and sexual exploitation. The 'necessity of allowing oneself to be sexually abused' could become an everyday compromise for many children, transforming our understanding of what constitutes the causality for acts of crime. This, in turn, brings to fore the question of responsibilisation that not only problematises the enforcement of European law with regards to refugees, but also places children beyond a victim-perpetrator binary.
While political leaders and policy makers have been concerned about the likely 'risks' of incorporating immigrants and refugees in European nation-states, little attention has been paid to the conditions that not only deprive unaccompanied children of their rights but also coerce them into further acts of criminality. Who constitutes a greater risk and to whom? What does this refugee crisis comprise of? Why does the increasing victimisation of unaccompanied refugee children continue to remain sparsely represented? These questions call for greater elaboration in the academic and political discourse.


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Model aircrafts with turbojet engines compete in air in China's Shandong (1/7)
2018-10-14 10:20:37 Xinhua Editor :Mo Hong'e
Photo taken on Oct. 13, 2018 shows model aircrafts with turbojet engines perform in the air during a contest in Rongcheng, east China's Shandong Province. A total of 12 teams from around the world compete in the air. (Xinhua/Lin Haizhen)
Photo taken on Oct. 13, 2018 shows model aircrafts with turbojet engines perform in the air during a contest in Rongcheng, east China's Shandong Province. A total of 12 teams from around the world compete in the air. (Xinhua/Wang Fudong)
Photo taken on Oct. 13, 2018 shows model aircrafts with turbojet engines perform in the air during a contest in Rongcheng, east China's Shandong Province. A total of 12 teams from around the world compete in the air. (Xinhua/Yang Zhili)
Photo taken on Oct. 13, 2018 shows model aircrafts with turbojet engines perform in the air during a contest in Rongcheng, east China's Shandong Province. A total of 12 teams from around the world compete in the air. (Xinhua/Zhang Binbin)
A contestant controls his model aircraft with turbojet engine during a contest in Rongcheng, east China's Shandong Province, Oct. 13, 2018. A total of 12 teams from around the world compete in the air. (Xinhua/Yang Zhili)
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Popsugar Must Have October 2015 Theme Spoiler + Coupon!
The theme for Popsugar's October 2015 box was released today!
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All about headache types of headaches causes of headache headache symptoms diagnosis of headaches treatments for headaches cluster headache causes of cluster headache symptoms of cluster headache cluster headache diagnosis cluster headache relief alternative cluster headache remedies migraine headache causes of migraine headache migraine headache triggers symptoms of migraine headaches diagnosis of migraine headache migraine headache cure and remedy migraine medications prevention of migraine headaches migraine alternative treatments ocular migraine treatment for menstrual migraine migraine headache during pregnancy tension headache tension headache causes tension headache symptoms diagnosis of tension headache tension headache treatment prevention of tension headache rebound headaches analgesic rebound headache ergotamine rebound headache sinus headache sinus headache causes sinus headache symptoms diagnosis of sinus headache sinus headache cures
What causes a migraine headache?
The precise mechanisms of a migraine headache are not completely understood. There appears to be general agreement, however, that a key element is changes in the blood flow within the brain due to a variety of triggers. The most widely accepted hypothesis of migraine suggests that a migraine attack is precipitated when pain-sensing nerve cells in
the brain (called nociceptors) release chemicals called neuropeptides. At least one of the neurotransmitters, substance P, increases the pain sensitivity of other nearby nociceptors.
Other neuropeptides act on the smooth muscle surrounding cranial blood vessels. This smooth muscle regulates blood flow in the brain by relaxing or contracting, thus dilating (enlarging) or constricting the enclosed blood vessels. At the onset of a migraine headache, neuropeptides are thought to cause muscle relaxation, allowing vessel dilation and increased blood flow. Other neuropeptides increase the leakiness of cranial vessels, allowing fluid leak, and promote inflammation and tissue swelling. The pain of migraine is though to result from this combination of increased pain sensitivity, tissue and vessel swelling, and inflammation. The aura seen during a migraine may be related to constriction in the blood vessels that dilate in the headache phase.
Susceptibility to migraine may be inherited. A child of a migraine sufferer has as much as a 50% chance of developing migraine. If both parents are affected, the chance rises to 70%. However, the gene or genes responsible have not been identified, and many cases of migraine have no obvious familial basis. It is likely that whatever genes are involved set the stage for migraine, and that full development requires environmental influences as well.
The early warning signs of a migraine are thought to be due to a narrowing of the blood vessels within the brain, while the pain in the head is due to the subsequent expanding or dilating of the blood vessels. The dilation of the blood vessels results in the release of pain-producing substances called prostaglandins. Other chemicals that cause inflammation and swelling, along with some that increase sensitivity to pain, are also released. These various chemicals, along with the dilation of the scalp vessels, stimulates pain receptors and results in a headache.
More information on headache
What is a headache? - A headache is a condition of mild to severe pain in the head. Tension headaches and migraine headaches account for 90% for all headaches. Headaches can be single or recurrent in nature.
What types of headaches are there? - A wide range of headache types have been classified in detail by the International Headache Society. Including tension-type headache (TTH), migraine, cluster headache.
What causes headache? - Causes of headaches range from eyestrain to inflammation of the sinus cavities to life-threatening conditions. Most headaches are due to tension, migraine, or a combination of the two.
What're the symptoms of a headache? - Signs and symptoms of a headache involve pain in the head, visual impairment such as blurred vision, nausea and vomiting, hearing impairment, irritability, confusion, malaise.
How're headaches diagnosed? - A physical exam assesses general health and a neurologic exam evaluates the possibility of neurologic disease that is causing the headache.
What headache remedies are available? - Appropriate treatment for headaches is highly specialized depending on the type of headache, individual response, and any associated health conditions.
What's a cluster headache? - Cluster headaches are rare headaches that occur in groups or clusters. Cluster headaches are frequency associated with drooping eyelids, red, watery eyes, and nasal congestion.
What causes a cluster headache? - The immediate cause of pain is in the trigeminal nerve. Cluster headaches are due to an abnormality in the hypothalamus.
What're the symptoms of a cluster headache? - The primary cluster headache symptom is excruciating one-sided head pain centered behind an eye or near the temple.
How is a cluster headache diagnosed? - Cluster headache symptoms guide the diagnosis. A neurologic examination may help doctor detect physical signs of a cluster headache.
How to cure cluster headache? - Treatment for cluster headaches is composed of induction, maintenance, and symptomatic therapies. Medications to treat cluster headaches are classified as either abortives or prophylactics.
What alternative cluster headache remedies are available? - Since some cluster headaches are triggered by stress, stress reduction techniques, such as yoga, meditation, and regular exercise, may be effective.
What's a migraine headache? - Migraine headache is a primary headache disorder with a genetic basis. Adults with migraine describe episodic attacks with specific features.
What causes a migraine headache? - Migraine headache attack is precipitated when pain-sensing nerve cells in the brain (called nociceptors) release chemicals called neuropeptides.
What're migraine headache triggers? - A wide variety of foods, drugs, environmental cues, and personal events are known to trigger migraine headaches.
What're the symptoms of migraine headaches? - Symptoms of migraine headaches include hallucinations, partial obstruction of the visual field, numbness or tingling, or a feeling of heaviness.
How is a migraine headache diagnosed? - The diagnosis of migraine headache rests solely on what a patient describes to a doctor. A diary can help document headache characteristics.
What's the treatment for migraine headache? - Nonsteroidal anti-inflammatory drugs (NSAIDs) are helpful for early and mild migraine headache. Other drugs used for acute attacks include meperidine and metoclopramide.
What migraine medications are available? - Several nonprescription drugs and some relatively inexpensive prescription drugs are available for migraine headaches. Tricyclic antidepressants (TCAs may be effective migraine prevention.
How to prevent migraine headaches? - The occurrence of common migraine may be reduced in high-risk people who avoid factors that have been associated with previous migraine.
What alternative migraine treatments are available? - Alternative treatments are aimed at prevention of migraine. Migraine headaches are often linked with food allergies or intolerances.
What is ocular migraine? - Ocular migraine is a term which may have different meanings to different physicians. Ocular migraines usually affect only one eye at a time.
How to relieve menstrual migraine? - Menstrual migraine is a migraine headache that affects a woman each month between the 2nd day before the start of the menstrual period and the end of menstruation.
Migraine headache during pregnancy - Over half of women find that their migraines occur less often in the last few months of pregnancy. About 15 to 20 percent of pregnant women have migraines.
What is tension headache? - Tension headaches are the most common type of headache. The primary cause of the pain is sustained muscle tension. Tension headaches can be episodic or chronic.
What causes tension headache? - Tension headaches are caused by muscle tension around the head and neck. tension headaches seem to be more common in women than in men.
What're the symptoms of tension headache? - A tension-type headache may cause diffuse pressure or tightness. The pain of tension headaches usually is mild (not disabling) and bilateral.
How is tension headache diagnosed? - Most tension-type headaches are diagnosed based on a complete and comprehensive history and physical examination.
What's the treatment for tension headache? - Nonprescription painkillers such as aspirin, acetaminophen or ibuprofen most commonly relieve tension headaches.
How to prevent tension headache? - To prevent tension headaches, many different antidepressants may be effective for individuals with and without depressive symptoms.
What're rebound headaches? - Rebound headaches are those caused by medication taken to relieve regular headaches. Rebound headaches frequently occur daily and can be very painful.
What's analgesic rebound headache? - Analgesic-rebound headaches are caused by successively frequent use of pain relievers, both prescription and over-the-counter medications.
What's ergotamine rebound headache? - Throbbing pain of migraine, in conjunction with escalating doses of ergotamine compounds usually indicates ergotamine rebound headache.
What's a sinus headache? - A sinus headache is a headache that occurs because your sinuses are congested with mucus. Sinus headache pain is due to swelling in the sinus cavities.
What causes a sinus headache? - Sinus headaches can be caused by a sinus infection or allergies. They may become infected with bacteria, a virus, or even fungus.
What're the symptoms of a sinus headache? - Symptoms of a sinus headache include pain or pressure in the forehead or around the eyes and cheek(s), tender skin and tender bones over and under the eyes.
How is a sinus headache diagnosed? -To diagnose sinus headache, the doctor will ask about your symptoms and medical history, and perform a physical exam.
What's the treatment for sinus headache relief? - Treatment of sinus headache is by treating the underlying condition, which may be allergic or infectious, and by nasal vasoconstrictors and analgesic medication.


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What Do Bugs Do In the Winter?
By Sara LeDuc
It's not uncommon for us humans to be well aware of what common yard animals such as squirrels, deer, and even bears do during the colder months, but have you ever wondered what bugs do during the winter? The behavior of different bugs ranges widely during this season, and it helps to be aware of their cold weather habits so you can prepare your home accordingly.
Cockroaches During the Winter
Most cockroach species can't withstand temperatures under 15 degrees Fahrenheit. For this reason, these unwanted pests may end up retreating to different spaces in your home in pursuit of warmth. They most frequently inhabit dark spaces like basements or crawl spaces. The good news about these bugs during the winter, though, is that their development rate stalls. That means that they reproduce more slowly, and are thus easier to extinguish from your home.
To prevent them from entering in the first place, seal any openings from the outside of your house that would allow them easy access. If you find any roaches in your home, eradicate them by making your own natural bait. Mix three parts boric acid with one part confectioners or powdered sugar. Once they're lured into the sweet smelling mixture, the boric acid will make short work of them. Sprinkle this mixture wherever you suspect these roaches may gather, and soon your infestation will be a non-issue. Importantly, this mixture is not toxic to kids or pets.
Spiders During the Winter
Overall, spiders are less active and present during the winter months. For instance, brown recluse spiders tend to be inactive once temperatures dip below 44 degrees Fahrenheit. Other spider species, like adult black widows, tend to seek shelter for the winter months. This means that these spiders may make their way into your home. A good way to rid your home of spiders during the winter is to thoroughly vacuum your space to get rid of all cobwebs.
Mosquitoes During the Winter
One of the most irksome parts of summer in many areas of the country is the presence of mosquitoes. Not only are their bites irritating, these bugs can carry dangerous viruses and diseases. The good news about winter is that these tiny vampires typically retreat during the colder weather. During this time, they cease blood feeding and increase their fat reserves. These bugs will live in burrows, storm drains, or other underground areas over the winter.
Ants During the Winter
Ants frequently plague homes during warmer months, but they struggle to survive in the cold temperatures—many ant colonies perish during this season. Some species, such as fire ants, tunnel underground in the cold to avoid the plunging temperatures. Others, such as the Argentine Ant, become inactive during this season, hibernating in loblolly pine trees, the bark of which absorbs sunlight, providing enough warmth for these ants to wait out the frost.
Bees During the Winter
Honeybees are a hot topic in regards to Earth and its environment, so that may prompt you to wonder where these insects go during the winter. After temperatures drop below 50 degrees Fahrenheit, honeybees retreat to their hive. Inside the hive, they gather in a central area, forming a "winter cluster." The queen bee sits at the center, with all the surrounding bees working to keep her safe and warm for the duration of the season. The bees flutter their wings and shiver, using their constant motion to warm the inside of the hive, even as temperatures outside plummet.
Butterflies During the Winter
Perhaps the most picturesque species of insects, butterflies tend to migrate to warmer temperatures during the winter. Monarch butterflies make their way to the highlands of Michoacán in Mexico for the season. Other species of butterflies simply enter a state of dormancy during this season. They can do so in a variety of ways—as eggs, larva, or in a chrysalis. Most become dormant in a larval state. This is important as it keeps butterflies warm during the winter and allows for their awakening once food is available to them.
Luckily, bugs don't present much of a worry to your home during the winter months. Other winter pests are something to think about, though.
Most Common Cool Weather Pests and How t...
Most Common Cool Weather Pests and How to Get Rid of Them
By Perry Carpenter
How to Get Rid of Flying Ants
How to Get Rid of Bugs in the Kitchen
By Natasha Gilani
3 Cockroach Control Products Explained
3 Common Winter Pests to Guard Your Home...
3 Common Winter Pests to Guard Your Home Against


The ban will be made effective all over the state over the next two years, Patnaik said.
Moving towards its target of making Odisha plastic-free over the next two years, Chief Minister Naveen Patnaik today announced a complete ban on the use of plastic in several places from October 2, this year.
Patnaik made the announcement while participating in the "Ama Mukshya Mantri, Ama Katha" (Our Chief Minister, our talk) programme), a video conferencing initiative to connect with the people in every nook and cranny of the state.
Patnaik said the plastic ban would be imposed in municipal corporation areas and Puri town from Gandhi Jayanti on October 2. The major cities where the ban will be imposed in the first phase include Bhubaneswar, Cuttack, Berhampur, Samalpur and Rourkela.
The Housing & Urban Development department and Panchayati raj department have been mandated for executing the order in urban and rural areas, respectively. The Forest & Environment department will monitor the plastic pollution control work across the state.
Patnaik also appealed all to plant a sapling each during the monsoon and take care of it. If we all plant tree and take care of it, the environment will be protected and the human beings will live in good condition and we can work for the development of our family and society, he said.
Patnaik expressed satisfaction at the increase of green cover in the state which is more than the national average.


Thanks to the young George III, the Royal Collection contains the world's finest group of paintings, drawings and prints by Venice's most famous view-painter, Canaletto. This exhibition brings together the best 18th-century Venetian art in the Collection, with Canaletto's greatest works shown alongside paintings and drawings by his contemporaries, Sebastiano and Marco Ricci, Francesco Zuccarelli, Pietro Longhi and Giovanni Battista Piazzetta. Canaletto & the Art of Venice explores the many delights of the Italian city, from the splendours of the Grand Canal and St Mark's Square to its festivals, theatre and masked carnival.


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Home » Kikanetsu: The Art of Daisukerichard (Paperback)
Kikanetsu: The Art of Daisukerichard (Paperback)
By Daisukerichard (Illustrator)
The world of expressionless girls on the verge of adulthood living in rooms where reality and fantasy blend
The distinctive factor in His work is high school girls in languid poses that arise from the uncertainty that is peculiar to girls during adolescence. Girls in his work show expressionless faces and the artist intentionally makes it hard to gauge their emotions. But the way that the artist presents the subject - depicting the uncertainty felt in adolescence, while at the same time celebrating these youths through pale shades of color and unique composition - is definitely something that resonates with teenage girls. Pale colors give an impression of the fragility and sweetness of young girls, while each work's unique composition shows the girls dwelling alone in between the two incompatible worlds of adulthood and childhood.
This is His first artwork collection showcasing over 170 artworks, including never-before-seen original artworks, remakes of already published artworks, and rough sketches. This book also includes interesting features like the "motif collection", the "making of 170cm cover art illustrations". This book will surely inspire DaisukeRichard fans and those who are new to his distinctive illustrated works alike.
Publisher: Pie International
Publication Date: June 1st, 2019
Individual Artists - Artists' Books
Manga - School Life


Perspective view of the Mars north polar ice cap and its distinctive dark troughs forming a spiral-like pattern. The view is based on images taken by ESA's Mars Express and generated using elevation data from the Mars Orbiter Laser Altimeter (MOLA) on board the NASA Mars Global Surveyor mission.
ESA's Mars Express provides a new mosaic of Mars.
A new mosaic from ESA's Mars Express shows off the Red Planet's north polar ice cap and its distinctive dark spiralling troughs. The mosaic was generated from 32 individual orbit 'strips' captured between 2004 and 2010, and covers an area of around a million square kilometers.
The ice cap is a permanent fixture, but in the winter season – as it is now in early 2017 – temperatures are cold enough for around 30 percent of the carbon dioxide in the planet's atmosphere to precipitate onto the cap, adding a seasonal layer up to a meter thick.
Strong winds are thought to have played an important role in shaping the ice cap over time, blowing from the elevated center towards its lower edges and twisted by the same Coriolis force that causes hurricanes to spiral on Earth.
1 Comment on "New Mosaic Shows Off the Red Planet's North Polar Ice Cap"


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Farmington Hills-based H.W. Kaufman Group has added Kori M. Johanson to its executive team as corporate vice president and chief compliance officer.
Johanson will oversee the company's compliance team and be responsible for corporate and regulatory compliance, government affairs, and state insurance taxation for Kaufman and all of its subsidiaries.
With more than 20 years of insurance experience in corporate compliance, regulatory, and government affairs, Johanson was responsible for legal, compliance, and regulatory relations at PURE, a personal insurance carrier. There, she served as vice president, associate general counsel, and led the compliance and regulatory function. Johanson was also chief compliance officer for ACORD.
Johanson holds a juris doctorate degree from the Western New England University School of Law and a bachelor's degree from Trinity College. She is the 2018-2019 vice chair of the annual conference for the Association of Insurance Compliance Professionals, is co-editor of the Federation of Regulatory Counsel journal publication, and is active in many industry organizations.
H.W. Kaufman Group is a global network of companies that work with insurance.


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Q: Manipulation HTML DOM of Vue3 List Item i use vue 3 and i want to manipulate a single list item by button click
this is my html:
<socialDiv v-for="(follower, i) in followerList" :key="follower.id" :ref="el => { followerDiv[i] = el }">

{{ follower.name }}

{{ follower.username }}

<button @click="handleBtnClick" id="fbtn">{{ follower.btn }}</button>

this is my script:

this is my output:

i'm able to access followerDiv.value[0] but i cannot manipulate the item itself as a DOM element.
how can i access the child items as DOM elements?
Update:
To access the list items i have to add "$el" after value.
I can access the values via:
followerDiv.value[i].$el.style.background = "red"

A: If I understood you correctly, you can pass index to your function :

const { ref } = Vue
const app = Vue.createApp({
  setup() {
    const followerDiv = ref({})
    const followerList = ref([{id: 1, name: 'aa', username: 'aa', btn: 'aa'}, {id: 2, name: 'bb', username: 'bb', btn: 'bb'}])

    function handleBtnClick(i) {
      followerDiv.value[i].style.color = "red"
    }
    return {
      followerList, followerDiv, handleBtnClick
    }
  },
})
app.mount('#demo')

 { followerDiv[i] = el }">

{{ follower.name }}

{{ follower.username }}

    <button @click="handleBtnClick(i)" id="fbtn">{{ follower.btn }}</button>

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Wall-mount Hair Steamer ION II utilizes computerized micro mist technology to diffuse Anions when operating. The anions can decrease the drying phenomenon of curly and regular hair by neutralizing the positive charge that's found in and around our hair. Unit also features a extra large hood for comfortable fittings for all clients. Various temperature, modes and angle adjustments on unit for easy & friendly usage.
Power requirements: 110V-120V ~ 50/60 Hz 1300W.


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High Profile Vessel Casualties Ignite 'Place of Refuge' Call to Action
The MSC Flaminia on fire in the North Atlantic in 2012.
Shipowners, Salvors and Insurers represented by at least three trade associations are jointly calling for the implementation of international measures to provide a Place of Refuge for stricken vessels following a series of high-profile incidents where casualty vessels have been delayed or denied in accessing a safe harbor.
The International Chamber of Shipping (ICS) says that it has noted "with dismay" the refusal by some coastal States to make places of refuge available, risking lives and the environment in the process. In particular, the ICS referenced recent incidents such as the Stolt Valor and MSC Flaminia fires in 2012 and the more recent plight of the Maritime Maisie chemical tanker which is currently being held at sea by six tugs after a collision and fire on December 29, 2013. The vessel has been seeking a place of refuge in either the Republic of Korea or Japan for more than a month.
The three trade associations -including the International Union of Marine Insurance (IUMI), the International Salvage Union, (ISU) and ICS- have all recognized that the issue of Places of Refuge for casualty vessels is sensitive and that the risk of pollution from casualties cannot be completely removed. In addition, it is also recognized that handling casualty vessels can carry political implications and risk to coastal communities.
At the same time, however, the associations say that the failure to offer a suitable Place of Refuge may prevent successful salvage intervention and therefore allow a casualty's condition to worsen and ultimately lead to pollution that might otherwise have been prevented.
"Guidance on the handling of requests for places of refuge was agreed at IMO but often when a case arises the coastal states concerned take a 'not in my backyard' attitude," said Peter Hinchliffe, ICS Secretary General. "This is in marked contrast to attitudes to aircraft in need of assistance. This current case shows that recent lessons have simply not been learned."
Commenting on the matter, President of the ISU, Leendert Muller said: "Our members are right on the front line of this issue. Too often they are unable to follow the best course of action which is to take the casualty into shelter, which does not necessarily have to be a port. We have seen infamous cases like Castor and Prestige and more recently the MSC Flaminia and Stolt Valor and now the Maritime Maisie where our members, attending damaged vessels, experienced great difficulty in finding an authority willing to accept the casualty."
Ole Wikborg, President of the IUMI, points out: "The potential impact of environmental damage has to be reduced as much as possible and the safety of crews is paramount and we have to minimize material damage to ships and equipment. Coastal states must be able to make the best possible decision to prevent further damage following a maritime accident. Some countries have a system that seems to be functioning. IUMI is of the strong opinion that the prevailing regulations as set out, for example, by IMO and the EU are sufficient but that the necessary steps have to be taken to make the rules work."
ISU, ICS and IUMI all note the international legal context for the issue and the significant relevant legislation that is in place internationally and regionally, in particular, IMO Resolution A.949, "Guidelines on Places of Refuge for ships in need of assistance"; Resolution A.950 (23) (recommending all coastal States to establish a Maritime Assistance Service) and the 1989 Salvage Convention as well as the European Union vessel traffic monitoring and information system (Directive 2002/59/EC as amended by Directive 2009/17/EC) which prevents member States from issuing an outright refusal to provide a place of refuge and states that safety of human life and the environment are of over-riding concern.
The industry bodies say they do not see merit in pursuing additional international legislation as it will be a lengthy process and will consume resources. Instead they will campaign for better application of, compliance with and enforcement of existing rules and guidance. ISU has already formally presented views on Places of Refuge to EU member states through the EU Commission; the issue will be raised in IMO fora this year and there will be direct engagement with the governments of individual coastal states.
In short, coastal states should be encouraged to recognize that granting a Place of Refuge to a casualty vessel may be the most appropriate course. States should establish an authority to assess each case on its merits without political interference. Such an assessment must include a visual inspection and conclude with recommendations for managing and mitigating the risk of any impact on local coastlines and communities. The assumption should be that a Place of Refuge will be granted if needed and that there should be "no rejection without inspection".
IUMI, ICS and ISU would like to see wider adoption by coastal states of simple, robust, "single point" command and control models.


I'm not a religious guy, but I've been preaching about the virtues of moving to the cloud for a long time. Well before the idea became popular, I saw the potential cost savings offered by the cloud, as well as the opportunities to leverage it to generate new innovations. Now, nearly every industry is recognizing these potential benefits, including church pastors and other religious leaders. Over the last few years, I've been pleased to see corporate executives and end-users increasingly seek cloud-based alternatives to traditional, on-premises software and systems to achieve their business objectives. However, I was surprised to see the level of adoption that has already occurred in the religious community, according to a recent Intacct survey. Click here to read my views in E-Commerce Times about the industry implications of the survey findings.
Can Trade Associations Be Good Channels To Market?
Ever since my early days as an industry analyst at IDC in the 1980s, I've been convinced that one of the most potent, yet under-utilized resources in the IT market are the major trade associations.
When I first looked at these associations, I thought of them as targets for research purposes. Now, I view them as a terrific channel to market.
Intacct is putting my ideas to the test by entering into a new alliance with the American Institute of Certified Public Accountants (AICPA) and its CPA2Biz subsidiary to encourage the association's 45,000 member CPA firms and their 350,000 small and mid-sized businesses (SMBs) to adopt Software-as-a-Service (SaaS) and "cloud computing" alternatives to traditional financial applications.


Can Child Support, Alimony, Time-Sharing be Temporarily Established During Florida Divorce?
Child support, alimony, financial accounts, visitation, and the marital home are factors in a Florida divorce, but are also factors when first separating from you a spouse. When separating from a spouse there are many questions that often arise, including who will stay in the house, how bills will get paid, etc. The other question that faces many couples is when child support should be paid and how much child support will be. As a Jacksonville divorce lawyer, I have many clients that need information about how to proceed once the choice is made that separation is necessary. If you decide to file for divorce, then you can also file an action with the court to address these issues on a temporary basis until the court can finalize the divorce. The idea is to get from point A, the filing of the divorce, to point B, the end of the divorce.
How do you have temporary needs established in a divorce? When you file for divorce in Florida, the other party has 20 days to respond to the petition. While you can file a Motion for Temporary Needs to request the court provide you with child support, spousal support, the home, etc., it cannot be scheduled for a hearing until either the 20 days expires or the other party responds to the petition, whichever occurs first. Once your spouse responds to the petition, the motion can be scheduled for a hearing. The hearing will either be held in front a court magistrate, which is allowed to hear cases and provide a ruling to a judge, or the judge him/herself will hear the motion.
Can we reach an agreement before the hearing? Yes. If your attorney and your spouse's attorney can reach an agreement between you both prior to the hearing, then the agreement can be formalized into a written document and both parties will need to sign it. Once the agreement is signed, it is sent to the judge so that it may be signed an entered by the court as order regarding temporary needs. The order can address anything from support issues to a temporary time-sharing plan, so that the parties have a workable schedule for visitation with the children, if there are children involved.
If we cannot reach an agreement, how do we move forward? Once the hearing is scheduled with the court there will be a date assigned for both parties to be heard on the issues. At the hearing, much like at trial, both parties may have witnesses and present evidence regarding their financial needs, issues involving the children, the marital home, etc. The judge or magistrate will listen to the evidence and then decide what the best scenario is for the parties while the divorce is pending. Often, due to the temporary nature of the order, alimony may be awarded at an increased amount then what will happen in the final hearing, and child support may factor in certain things to help keep the parties in a sustainable place from the date of the hearing through the finalization of the divorce. What this means is that the court may be more creative with the outcome on a temporary basis so that the parties can make a transition without harming either party financially.
Before moving forward with your divorce, you should speak with an experienced family law attorney to better understand your rights and options.


text:
\section{Introduction\label{sec:1}}\enlargethispage{3pt}
The standard approach to quantum mechanical eigenvalue problems is
based on matrix representation of the operator defined in a complex Hilbert space and its subsequent diagonalization in a chosen basis. A modern approach to
quantum mechanics \cite{Hestenes66,Snygg97,Doran03,Perwass09} is
based on Clifford's geometric algebra (GA), where the Clifford
space filled with geometric objects such as oriented lines, planes
etc is used instead of abstract Hilbert space. Selection of a
particular GA to solve the physical problem, in contrast to
Hilbert space approach, automatically then takes into account Clifford space
properties such as dimension, signature and symmetry relations of
the physical space or spacetime.

Despite the advantages that GA offers to quantum mechanics, in practice the methods needed to solve concrete
problems  are not elaborated enough. So, for the
physicists and engineers the GA may appear a nice but exotic
theory only. In this report a practical method to calculate eigenvalues and eigenmultivectors
is proposed that is based on the reduction of the  multivector
eigenvalue equation to well known rotor equation. In particular the primary goal of this paper is to shown by fully elaborated examples how GA can be used to find eigenvalues and eigenmultivectors in 3D~Euclidean space with \textit{Cl}$_{3,0}$ and in relativistic Minkowski space with \textit{Cl}$_{3,1}$ algebra without explicitly involving Hamiltonian matrix diagonalization procedure. Both mentioned algebras are of paramount importance in physics.
An overview on eigenvalue problems in the framework of Clifford analysis is
given in Ref.~\cite{Sprossig01}. Some examples of GA approach to
eigenblades can be found in books~\cite{Snygg97,Doran03,Perwass09,Hestenes99}.

In the Sec.~\ref{sec:2} we shall remind the pertinent properties
of rotors and methods used to find the eigens. In Sec.~\ref{sec:3} we demonstrate how the
proposed method works in practice where we solve four problems of
increasing complexity. Relations with matrix representation
and symbols are summarized in Appendix. The following fonts are
used to denote GA elements:  $a$ or $\ba$ -- vector, $\mathcal{A}$ --
bivector, $A$ -- multivector, $\mathsf{A}$ -- transformation.

\section{Rotors and eigens in GA\label{sec:2}}

\textit{Rotation of vector}. In GA the rotation of vector $a$ to
vector $c$ is described by a linear transformation
\begin{equation}\label{rotvec}
\sR(a)=Ra R^{-1}=c,
\end{equation}
where $R$ is the rotor that satisfies $R R^{-1}=R^{-1}R=1$,
$R^{-1}=\tilde{R}$, with tilde denoting the reversion operation.
Since   $c^2=a^2$ the inverse transformation is $a=\tilde{R}c R$.
Usually the rotor $R$ is constructed applying two reflections in planes
perpendicular to unit vectors $\hat{m}$ and $\hat{n}$,
\begin{equation}
R=\hat{m}\hat{n},\quad \tilde{R}=\hat{n}\hat{m},\quad
\hat{m}^2=\hat{n}^2=1.
\end{equation}
In this way defined rotor can be put into more convenient exponential form
\begin{equation}\label{rotmn}
R=\hat{m}\hat{n}=\hat{m}\d\hat{n}+\frac{\hat{m}\w\hat{n}}{|\hat{m}\w\hat{n}|}|\hat{m}\w\hat{n}|=
\cos\frac{\theta}{2}+\hat{\cB}\sin\frac{\theta}{2}=\ee^{\hat{\mathcal{B}}\,\theta/2},
\end{equation}
where $\theta$ is the angle of rotation in a plane made up of
$\hat{m}$ and $\hat{n}$, $\cos(\theta/2)=\hat{m}\,\d\,\hat{n}$ and
$\sin(\theta/2)=|\hat{m}\w\hat{n}|$.
$\hat{\cB}=\frac{\hat{m}\w\hat{n}}{|\hat{m}\w\hat{n}|}$ denotes the
unit bivector, $\hat{\cB}^2=-1$.

In practice, however, it is required to have a rotor which brings
the end of a unit radius-vector $\hat{a}$ to a new position
$\hat{b}$. If $\theta$ is the angle between $\hat{a}$ an $\hat{b}$
then the rotation plane can be written as the outer product
$\hat{a}\w\hat{b}=(\hat{a}\hat{b}-\hat{b}\hat{a})/2$ and angle
$\theta$ can be expressed through the inner product
$\hat{a}\cdot\hat{b}=(\hat{a}\hat{b}+\hat{b}\hat{a})/2=\cos\theta$.
The rotor in this case takes the following form
\begin{equation}\label{rotab1}
R=\ee^{\pm\frac{\hat{a}\w\hat{b}}{|\hat{a}\w\hat{b}|}\frac{\theta}{2}}=
\cos\frac{\theta}{2}\pm\frac{\hat{a}\w\hat{b}}{|\hat{a}\w\hat{b}|}\sin\frac{\theta}{2},
\end{equation}
The trigonometric functions of half angle,
$\cos(\theta/2)=2^{-1/2}(1+\cos\theta)^{1/2}$ and
$\sin(\theta/2)=2^{-1/2}(1-\cos\theta)^{1/2}$, appear in the
formulas \eqref{rotmn} and \eqref{rotab1}. If $\hat{a}$ and
$\hat{b}$ are mutually orthogonal, then the
equation~\eqref{rotab1} simplifies to
\begin{equation}\label{rotab2}
R = \ee^{\pm\hat{a}\hat{b}\,\pi/4}=
\big(1\pm\hat{a}\hat{b}\big)\big/\sqrt{2},\quad \hat{a}\bot\hat{b}.
\end{equation}
The rotor can also be put~\cite{Doran03} in the form
$R=(1+\hat{a}\hat{b})/(|\hat{a}+\hat{b}|)$, which reduces
to~\eqref{rotab1} after half angle formulas are applied.

It is important to note, that two successive rotations can be
always described by the composite rotation
$\sR_3(a)=\sR_2\big(\sR_1(a)\big)\equiv\sR_2\sR_1a$. However, the
result of product of two rotors is not necessarily a rotor. The
exception happens, in particular, when $\sR_1$ and $\sR_2 $ belong
to nonintersecting subspaces.

\textit{Rotation of blade and multivector}. The rotation of a
blade relies on the extension of the outermorphism from vectors to
multivectors~\cite{Hestenes84}.  Since the outermorphism preserves
the outer product and as a result the shape of geometric objects,
this transformation induces a more general  linear transformation
in the full Clifford space. For example, the rotation of the
bivector $a\w b$ gives new bivector $\mathsf{R}(a\w
b)=\mathsf{R}(a)\w\mathsf{R}(b)$ having the same properties. To
rotate general blade $A_r=a_1\w a_2\w\dots\w a_r$ one rotates all
the vectors $a_i$ in it
$\underline{\sR}(A_r)=\sR(a_1)\w\sR(a_2)\w\dots\w \sR(a_r)$ (the
underbar denotes the outermorphism), so the rotation formula for
blade $A_r$ has the same structure as for vector~\eqref{rotvec},
$\mathsf{R}(A_r)=R A_r\tilde{R}=B_r$.  In particular, the
 pseudoscalar is rotationally invariant $RIR^{-1}=I$.
The formula remains valid if the blade is replaced by any
multivector. General properties of rotation transformation $\sR$
of multivectors $M$ and $N$ are listed below.
\[\begin{array}{lll}
1.& \sR(\alpha M)=\alpha\sR(M)&\textrm{Preserves scalar multiplication} \\
2.& \sR(M+N)=\sR(M)+\sR(N)&\textrm{Preserves multivector addition} \\
3.& \sR(M\d N)=\sR(M)\d\sR(N)&\textrm{Preserves inner product} \\
4.& \sR(M\w N)=\sR(M)\w\sR(N)&\textrm{Preserves outer product} \\
5.& \sR(MN)=\sR(M)\sR(N)&\textrm{Preserves geometric product} \\
6.&\textrm{grade}\big(\sR(A_r)\big)=\textrm{grade}(A_r)&\textrm{Preserves grade $r$}\\
\end{array}\]
An important property of the outermorphism is that the
outermorphism of the product of two functions $\sF$ and $\sG$ is
the product of the outermorphisms,
\begin{equation}
\underline{\sF\sG}(a\w b\dots\w c)=\sF\sG(a)\w \sF\sG(b)\dots\w
\sF\sG(c).
\end{equation}

\textit{Blade eigens}. The outermorphism allows to
characterize the invariant subspaces under linear transformations.
In particular, it allows to generalize the concept of the
eigenblade~\cite{Hestenes84}. Indeed, if $\sF$ is a linear function then
the equation
\begin{equation}\label{eigen}
\underline{\sF}(A_r)=\lambda A_r
\end{equation}
is called the eigenblade equation or collineation transformation,
where blade $A_r$ of grade $r$ is called eigenblade and $\lambda$ is
a real eigenvalue. Frequently the outermorphism symbol (underbar)
is omitted.  For example, if $a$ and $b$ are vectors and function
$\sF$ satisfies $\sF(a)=b$ and $\sF(b)=a$, then by outermorphism
extension we have $\underline{\sF}(a\w b)=\sF(a)\w \sF(b)=b\w
a=-a\w b$, or $\sF(a\w b)=- a\w b$. Thus $a\w b$ is the
eigenbivector with eigenvalue $-1$.

If applied to pseudoscalar $I$ the outermorphism can be used to define
the determinant $\text{det}(\sF)$ of linear transformation $\sF$,
\begin{equation}\label{det}
\underline{\sF}(I)=\text{det}(\sF)I,
\end{equation}
without any need of matrices. For example, in
\textit{Cl}$_{3,0}$ the determinant of $\sF$ as follows from
\eqref{det}  is
 $\text{det}(\sF)=\big(\sF(\e{1})\w\sF(\e{2})\w
\sF(\e{3})\big)\big(\e{3}\w\e{2}\w\e{1}\big)=\text{scalar}$.

The eigenvalue equation~\eqref{eigen} can be rewritten in the form
\begin{equation}
(\sF-\lambda)(A_r)=0
\end{equation}
showing that the operator $(\sF-\lambda)$ is singular. Every
singular operator has a vanishing determinant:
\begin{equation}\label{deteq}
\det(\sF-\lambda)(A_r)=\big(\sF(I_r)-\lambda I_r\big)I_r^{-1}=0,
\end{equation}
where the last expression follows from~\eqref{det}, the $I_r$ being
$r$-grade pseudoscalar. This equation is called the secular
equation for~$\sF$, the solution of which with respect to
$\lambda$ gives the eigenvalues. For example, let $\sF$ be a
vector-valued function $\bbf(\mathbf{x})$ of vector $\mathbf{x}$ in
\textit{Cl}$_{3,0}$. If $\bbf_i=\bbf(\e{i})$, then
\[\text{det}(\bbf-\lambda)(\mathbf{x})=(\bbf_1-\lambda\e{1})\w
(\bbf_2-\lambda\e{2})\w(\bbf_3-\lambda\e{3})I_3^{-1}=0,\] where
$I_3=\e{1}\w\e{2}\w\e{3}=\e{1}\e{2}\e{3}$ and $I_3^{-1}=-I_3$.
After expansion this equation reduces to scalar cubic equation
\[\lambda^3+\alpha_2\lambda^2+\alpha_1\lambda+\alpha_0=0 \]
where the scalar coefficients are
$\alpha_0=(\bbf_1\w\bbf_2\w\bbf_3)\d I_3$,
$\alpha_1=-(\bbf_1\w\bbf_2\w\e{3}+\bbf_1\w\e{2}\w\bbf_3+
\e{1}\w\bbf_2\w\bbf_3)\d I_3$, and
$\alpha_2=(\bbf_1\w\e{2}\w\e{3}+
\e{1}\w\bbf_2\w\e{3}+\e{1}\w\e{2}\w\bbf_3)\d I_3$. After the
eigenvalues have been determined, the eigenblade
equation~\eqref{eigen} becomes algebraic multivector equation with
respect to  $A_r$ which can be solved by GA methods. Example of
applications of this approach can be found, for example, in the
textbook~\cite{Hestenes99}. The described method, however, is not
a coordinate free, and is based on expansion of the
transformation in a particular frame.

\textit{Spinor eigens}. The eigens, however, in many cases could
be calculated in a compact way without any reference to
determinant equation~\eqref{deteq}. In particular, as we shall
see, this is easy to do if the eigenequation
\begin{equation}\label{eigspin}
\sH(\psi)=E\psi
\end{equation}
for spinor $\psi$ that consists of a sum of even-grade blades can
be transformed to an eigenequation for a corresponding rotor. In quantum
mechanics the transformation $\sH$ is called the Hamiltonian and
$E$ is the energy (scalar). The reformulation of eigen
problem~\eqref{eigspin} may be realized if one observes that in
many quantum mechanical problems there is a specific direction
called the quantization axis ($z$-axis). As we shall see this
allows to reduce quantum mechanical eigenmultivector equation to a
single rotor equation, where the quantization axis is rotated to
some final direction determined by the Hamiltonian of the problem.
It is important to note that, in contrast to Hilbert space
formulation where the quantization axis is hidden in the structure
of the Hamiltonian, in GA this axis appears
explicitly. In other words the quantization axis, represented below by $\e{3}$
basis vector, appears directly in the Hamiltonian function. The
proposed method is based on this quantum mechanical
Hamiltonian property and allows us to construct a single optimal
rotor expressed in coordinate-free way, which is very similar to
rotor~\eqref{rotab1} or~\eqref{rotab2}, where the vectors
$a=\e{3}$ and $b$ are given by
eigenequation~\eqref{eigspin}.

The realization of the described procedure is straightforward in case
of \textit{Cl}$_{3,0}$ algebra (Schr{\"o}dinger-Pauli equation). In
relativistic \textit{Cl}$_{3,1}$ and \textit{Cl}$_{1,3}$ algebras
the spinor of Dirac equation is the sum of eight terms: scalar,
six bivectors and pseudoscalar. The reduction of relativistic
spinors to rotors of 3D~Euclidean space can be accomplished
after splitting of $\psi$ into even $\psi_{+}$ and odd $\psi_{-}$
parts with the help of spatial inversion operator described in
Appendix~\ref{append},
\begin{equation}
\psi=\psi_{+}+\psi_{-},\quad\inv{\psi}=\psi_{+}-\psi_{-}.
\end{equation}
As we shall see the eigenvalue equations
$\sH(\psi_{+}+\psi_{-})=(\psi_{+}+\psi_{-})E$ and
$\inv{\sH}(\psi_{+}+\psi_{-})=(\psi_{+}-\psi_{-})E$ in this case
 reduce to two multivector equations for even $\psi_{+}$
and odd $\psi_{-}$ (or $I\psi_{-}$) parts. Reduced
equations then can be interpreted as rotors which rotate quatization axis $\e{3}$ to the final direction determined by the Hamiltonian.

\section{Examples from quantum mechanics\label{sec:3}}
\begin{figure}
\centering
\qquad a)\includegraphics[width=3.cm]{Fig1a_Acus.eps}
b)\includegraphics[width=3.7cm]{Fig1b_Acus.eps}\newline
c)\includegraphics[width=4cm]{Fig1c_Acus.eps}
d)\includegraphics[width=4cm]{Fig1d_Acus.eps}
\caption{a) Energy vs wave vector (spectrum) for a monolayer graphene. CB
 and VB are conduction and valence bands. The arrows show
 direction of pseudospin.
 b) Spectrum of the quantum well with spin-orbit interaction present. c) Energy of two coupled  two-level atoms vs coupling
constant $\Gamma$. d) Spectrum  of bilayer graphene with
interlayer voltage $2U$ applied.}\label{f:DispEx}
\end{figure}

In this section four characteristic examples from quantum
mechanics  are considered to illustrate how the proposed method
works in practice. The spectra of all considered Hamiltonians are
shown in Fig.~\ref{f:DispEx}.

\subsection{Monolayer graphene\label{sec:4a}}
Monolayer graphene is a 2D crystal consisting of a single layer of
carbon atoms. The intense interest in graphene was  stimulated by
2010 Nobel prize in physics (A.~Geim and K.~Novoselov) and
potential application of graphene in construction of novel 2D
nanodevices~\cite{Katsnelson12}. From theoretical point of view the most
interesting problem is the ultrarelativistic electron behavior in 2D
lattice at very small energies ($<1$~eV), Fig.~\ref{f:DispEx}a. The spectrum represents an ultrarelativistic limit of eigenenergy $E=\bigl(\mathbf{p}^2 +m^{\star 2}\bigr)^{1/2}$ of the solution of the Dirac equation in the case when fraction of electron energy due to kinetic momentum $\mathbf{p}$ is much larger than effective electron mass, $\mathbf{p}^2 \gg m^{\star 2}$, where $m^{\star}\approx 0.012 m_0$ ($m_0$ being free electron mass).

In the Hilbert space formalism the massless (ultrarelativistic)
Hamiltonian in the vicinity of critical $K$ point of graphene (the momentum space point where electron transport takes place) reduces to simple form
\begin{equation}\label{Hgmat}
\hat{H}_{MG}= \hbar v_F\begin{bmatrix}
 0& k_x-\ii k_y\\
 k_x+\ii k_y&0
\end{bmatrix},
\end{equation}
where $v_F\approx 10^{6}$~m/s is the Fermi velocity. The wave
vector $\mathbf{k}=(k_x,k_y)$ lies in the graphene plane and is measured
with respect to $K$ valley minimum. In the following the natural
units $\hbar=1$ along with $v_F=1$ are used. The eigenenergies of
the Hamiltonian~\eqref{Hgmat} are
\begin{equation}\label{dispK2}
E_{\pm}=\pm\sqrt{k_x^2+k_y^2}=\pm|\mathbf{k}|,
\end{equation}
where plus/minus sign  corresponds to conduction/valence bands
shown in Fig.~\ref{f:DispEx}a.  As discussed in~\cite{Dargys13a} a
minimal algebra needed  to describe the monolayer Hamiltonian is
\textit{Cl}$_{3,0}$, where the basis vectors $\e{i}$  play the
role of space coordinates and satisfy $\e{1}^2=\e{2}^2=\e{3}^2=1$,
Fig.~\ref{f:grapheneAxes}. The orthogonal oriented planes are
described by bivectors $\Ie{1}=\e{2}\e{3}$, $\Ie{2}=\e{3}\e{1}$,
and $\Ie{3}=\e{1}\e{2}$. With scalar and pseudoscalar
$I=\e{1}\e{2}\e{3}$ included, there are  8 basis elements.

Now we shall transform the Hamiltonian~\eqref{Hgmat} to
\textit{Cl}$_{3,0}$ algebra using the following mapping rules from
Hilbert to Clifford  space~\cite{Doran03},
\begin{equation}\label{repRules}
\begin{split}
|\psi\rangle &\longleftrightarrow\psi=a_0+a_1\Ie{1}+a_2\Ie{2}+a_3\Ie{3},\\
\lambda|\psi\rangle &\longleftrightarrow\lambda\psi,\quad
\hat{\sigma}_i|\psi\rangle \longleftrightarrow\e{i}\psi\e{3},
\end{split}
\end{equation}
where $\lambda$ is the scalar and $\hat{\sigma}_i$ is one of $2\times 2$ complex Pauli
matrices $\hat{\sigma}_1=\hat{\sigma}_x$, $\hat{\sigma}_2=\hat{\sigma}_y$ or $\hat{\sigma}_3=\hat{\sigma}_z$. Then, $\hat{H}_{MG}|\psi\rangle$ maps to the following
simple GA function of spinor $\psi$
\begin{equation}\label{HkinGA}
\sH_{MG}(\psi)=\mathbf{k}\psi\e{3},
\end{equation}
where the wave vector $\mathbf{k}=k_x\e{1}+k_y\e{2}$ lies in the graphene
plane, Fig.~\ref{f:grapheneAxes}. It should be noted that the
Hamiltonian function~\eqref{HkinGA} has a coordinate-free form.
The appearance of vector $\e{3}$ in Hamiltonian~\eqref{HkinGA} follows from the existence the pseudospin quantization axis, which is
perpendicular to graphene plane, Fig.~\ref{f:grapheneAxes}. This
is not so evident from the Hilbert space Hamiltonian~\eqref{Hgmat}.

The eigenvalue equation for monolayer graphene then reads
\begin{equation}
\mathbf{k}\psi\e{3}=\psi E ,
\end{equation}
which can be easily rewritten in a form of the rotor equation
\begin{equation}\label{rotMG}
\frac{\psi\e{3}\tilde{\psi}}{\psi\tilde{\psi}}=\frac{\mathbf{k}}{|\mathbf{k}|^2}E,
\end{equation}
where $|\mathbf{k}|\equiv k=\sqrt{\mathstrut{\vphantom{x^2}}\smash{k_x^2}+\smash{k_y^2}}$ is the magnitude of the
wave vector. Since~$\e{3}$ is a unit vector, it follows that after
rotation by $\psi$ the right hand side of \eqref{rotMG} must be
unit vector too. This means that the square of it satisfies
algebraic equation $(E/k)^2=1$, which in fact represents the quantization condition, and
from which the dispersion law, $E_{\pm}=\pm k$, immediately follows.

The corresponding eigenmultivectors can be deduced from the same
equation~\eqref{rotMG} after insertion of $E_{\pm}$ values and
requiring the rotor to satisfy $\psi\tilde{\psi}=1$. Then, the
equation~\eqref{rotMG} reduces to
\begin{equation}\label{rot2}
\psi_{\pm}\e{3}\tilde{\psi}_{\pm}=\pm\hat{\mathbf{k}},
\end{equation}
where $\hat{\mathbf{k}}=\mathbf{k}/|\mathbf{k}|$ is the unit vector. Thus, according to
GA the eigenspinors $\psi_{\pm}$ of a monolayer graphene are
simply rotors that bring  the unit vector $\e{3}$ which is
parallel to the quantization axis to new direction $\pm\hat{\mathbf{k}}$,
which is either parallel, as shown in Fig.~\ref{f:grapheneAxes},
or antiparallel to the electron wave vector $\mathbf{k}$. No such
clear-cut geometrical interpretation for eigenspinors exists in
the Hilbert space. Using \eqref{dispK2}, solution of the rotor equation~\eqref{rot2} can
be written automatically as a rotation of $\e{3}$ in the unit
bivector plane $\hat{\mathbf{k}}\w\e{3}$ by angle $\theta=\pi/2$ [see
Eq.~\eqref{rotab2}],%
\begin{figure}
 \centering
 \begin{minipage}[c]{0.35\textwidth}
 \raggedright
\includegraphics[width=5cm]{Fig2_Acus.eps}
 \end{minipage}%
 \begin{minipage}[c]{0.75\textwidth}
 \raggedleft
\caption{\label{f:grapheneAxes}2D electron (dot)  moving with
velocity $\mathbf{v}$ and coordinate system
$\{\e{1},\e{2},\e{3}\}$. The planes represent either two graphene
layers, or in the case of quantum well, two potential walls so that
the electron is squeezed between the planes. $\mathbf{k}$ is the  wave
vector, $\theta=\pi/2$, $\cos\varphi=k_1/|\mathbf{k}|$,
$\sin\varphi=k_2/|\mathbf{k}|$.}
\end{minipage}%
\end{figure}%
\begin{equation}\label{psiMG}
\psi_{\pm}=\ee^{\pm\hat{\mathbf{k}}\w\e{3}\,\pi/4}
=\frac{1}{\sqrt{2}}\Big(1\pm\hat{\mathbf{k}}\w\e{3}\Big)=\frac{1}{\sqrt{2}}\Big(1\pm\hat{\mathbf{k}}\e{3}\Big).
\end{equation}

It should be noted that the eigenspinors~\eqref{psiMG} are
coordinate-independent. The appearance of $\e{3}$ in~\eqref{psiMG}
is indispensable since the quantum mechanics requires to indicate
the direction of the pseudospin (or spin) quantization axis which
in our case was selected to be $\e{3}$. If, as shown in
Fig.~\ref{f:grapheneAxes},  angle $\varphi$ between $\e{1}$ and
$\mathbf{k}$ is introduced the equation~\eqref{psiMG} can be written in
coordinate form as
$\psi_{\pm}=\big(1\pm\sin\varphi\Ie{1}\mp\cos\varphi\Ie{2}\big)/\sqrt{2}$.

From quantum mechanical point of view the pseudospin in graphene
has the same properties as spin. However, the spin is an intrinsic
property of electron, while the pseudospin is the property of
interacting $p$-orbitals of carbon atoms that make up the lattice.
Thus, the average of pseudospin\pagebreak vector may be found in exactly the
same way as for electron spin~\cite{Doran03},
\begin{equation}
\langle\mathbf{P}\rangle=\psi_{\pm}\e{3}\tilde{\psi}_{\pm}=\pm\mathbf{k}/|\mathbf{k}|=
\pm(\cos\varphi\,\e{1}+\sin\varphi\,\e{2}).
\end{equation}
We see, that the pseudospin is either parallel or antiparallel to
electron propagation direction, Fig.~\ref{f:DispEx}a. This allows to introduce the helicity
operator~\cite{Dargys13a}.

\subsection{Electron in a quantum well (QW)\label{sec:4b}}
Let us now consider a potential well where the electron is
squeezed between two stepped-potential planes. The Hamiltonian for
electron in the  QW is
\begin{equation}\label{ham2D}
\hat{H}_{QW}=\big(\hbar^2/(2m^*)\big)(k_x^2+k_y^2)\hat{1}+\alpha_{\rm{R}}(k_y\hat{\sigma}_x-k_x\hat{\sigma}_y)
\end{equation}
where $m^*$ is the effective mass of electron, $k_x$ and $k_y$ are
the in-plane components and $\hat{1}$ denotes $2\times 2$ identity matrix. The first term is the kinetic energy and
the second term describes interaction of electron spin with its
orbital motion via spin-orbit interaction constant
$\alpha_{\rm{R}}$. For simplicity we shall assume that
$m^*=\hbar=1$. Using the replacement rules~\eqref{repRules} the
Hamiltonian~\eqref{ham2D} can be mapped into \textit{Cl}$_{3,0}$,
\begin{equation}\label{HQW}
\sH_{QW}(\psi)=
\mathbf{k}^2/2+\alpha_{\rm{R}}\e{12}\mathbf{k}\psi\e{3},\quad\mathbf{k}=k_x\e{1}+k_y\e{2}.
\end{equation}
This expression is a coordinate-free. The blades $\e{12}$ and
$\e{3}$ represent the QW plane and spin quantization axis which
may be arbitrary.

To find the eigens one has to solve the multivector equation
$\sH_{QW}(\psi)=E\psi$, where $E$ is the eigenenergy. Assuming
that the spinor is normalized $\psi\tilde{\psi}=1$, the equation
\eqref{HQW} can be rearranged into the rotor form
\begin{equation}\label{roteq12A}
\psi\e{3}\tilde{\psi}=\frac{k^2/2-E}{\alpha_R k^2}\mathbf{k}\e{12},
\end{equation}
where $\mathbf{k}\e{12}=\mathbf{k}\d\e{12}$ is the vector and $k=|\mathbf{k}|$ is $\mathbf{k}$ norm. Since the right hand
side of \eqref{roteq12A} must be unit vector, its square must be
equal to $1$. This again results in a scalar equation (the quantization condition) the solution of which gives the spectrum (solid and dashed lines in
Fig.~\ref{f:DispEx}b),
\begin{equation}\label{eig12}
E_{\pm}=k^2/2\pm k\alpha_R.
\end{equation}
Insertion of eigenenergies $E_{\pm}$  back into \eqref{roteq12A} then
gives
\begin{equation}\label{roteq12B}
\psi_{\pm}\e{3}\tilde{\psi}_{\pm}=\pm(k_y\e{1}-k_x\e{2})/k\equiv\hat{\ba}_{\pm}.
\end{equation}
Since $\e{3}$ is perpendicular to, while $\hat{\ba}_{\pm}$ lies in
the quantum well plane the rotation angle is $\theta=\pm\pi/2$.
Then the solution of equation~\eqref{roteq12B} is the rotor
\begin{equation}
\psi_{\pm}=\exp\Big(\pm\frac{\e{3}\w\hat{\ba}_{\pm}}{|\e{3}\w\hat{\ba}_{\pm}|}\frac{\pi}{4}\Big).
\end{equation}
In  coordinate form the eigenspinors are
$\psi_{\pm}=(1\mp\cos\varphi\,\e{23}\mp\sin\varphi\,\e{31})\big/\sqrt{2}$.
The eigenspinors $\psi_{\pm}$ can be used to calculate the average spin
$\langle\mathbf{s}_{\pm}\rangle=\psi_{\pm}\e{3}\tilde{\psi}_{\pm}$,
\begin{equation}
\langle\mathbf{s}_{\pm}\rangle=\pm(\sin\varphi\,\e{1}-\cos\varphi\,\e{2}).
\end{equation}
Thus the spins of  SO split energy bands, as shown in Fig.~\ref{f:DispEx}b,
are parallel to confining planes and have opposite directions, and
at the same time (since $\mathbf{k}\d\langle\mathbf{s}\rangle=0$) they
are perpendicular to the wave vector.

\subsection{A pair of coupled two-level atoms}
In \textit{Cl}$_{3,0}$ algebra, a single two-level atom
Hamiltonian is $\hat{H}_1=\hbar\omega\hat{\sigma}_z/2$ which
 can be coupled to dipole moment
$\hat{\mathbf{d}}_1=\boldsymbol{\mu}\hat{\sigma}_x$, where
$\hbar\omega$ is the energy difference between the energy levels
and $\boldsymbol{\mu}$ is the vectorial coupling constant. Let us
take a pair of two-level atoms coupled to each other through
dipole-dipole interaction. This kind of interaction may produce
entanglement. Let the dipole coupling interaction energy be of the
form $\Gamma\hat{\sigma}^{(1)}\hat{\sigma}^{(2)}$, where $\Gamma$
is the coupling constant and the superscript indicates the first
and second two-level atom. Since the coupling energy is
proportional to $\boldsymbol{\mu}^{(1)}\d\boldsymbol{\mu}^{(2)}$,
the coupling constant  $\Gamma$ depends on angle $\theta$ between
the vectors $\boldsymbol{\mu}^{(1)}$ and $\boldsymbol{\mu}^{(2)}$.
The composite Hamiltonian describing two two-level atom then is
\begin{equation}\label{twoatoms}
\hat{H}_2=\frac{\hbar\omega}{2}\Big(\hat{\sigma}_z^{(1)}\otimes\hat{1}+\hat{1}\otimes\hat{\sigma}_z^{(2)}\Big)+
\Gamma\hat{\sigma}_x^{(1)}\otimes\hat{\sigma}_x^{(2)}.
\end{equation}
The eigenvalues of $\hat{H}_2$ form four energy levels,
$\pm\Gamma$ and $\pm\sqrt{\Gamma^2+\omega^2}$,  as shown in
Fig.~\ref{f:DispEx}c.

Since \textit{Cl}$_{3,0}$ algebra is too small for the four level
system, the \textit{Cl}$_{3,1}$ algebra will be used in the
following. Its matrix representation by $4\times 4$ complex
matrices is given in the Appendix~\ref{append}.  After mapping of
matrix equation~\eqref{twoatoms} onto \textit{Cl}$_{3,1}$ algebra
one finds the respective GA Hamiltonian ($\hbar=1$)
\begin{equation}\label{H2}
\sH_2(\psi)=\frac{\omega}{2}\e{34}\psi\e{34}-\Gamma\e{2}\psi\e{3}+\frac{\omega}{2}\e{3}\psi\e{3}.
\end{equation}
To find the eigens of the equation $\sH_2(\psi)=E\psi$ we will
utilize spatial inversion operation $\inv{\psi}=-\e{4}\psi\e{4}$
(see Appendix~\ref{append}). With the help of it the spinor can be
split into a sum of even $\inv{\psi}_{+}=\psi_{+}$ and odd
$\inv{\psi}_{-}=-\psi_{-}$ parts. Since $\e{34}\psi\e{34} =
\e{3}\bar{\psi} \e{3}$ and the eigenequation
$\sH_2(\psi)=E\psi$ and its inversion
$\inv{\sH}_2(\psi)=E\inv{\psi}$ give
$\big(\sH_2(\psi)+\inv{\sH}_2(\psi)\big)/2=E\psi_{+}$ and
$\big(\sH_2(\psi)-\inv{\sH}_2(\psi)\big)/2=E\psi_{-}$, we find
that the equations for even and odd states decouple into two independent equations,
\begin{align}
-\Gamma\e{2}\psi_{+}\e{3}&=E\psi_{+},\label{eq1}\\
\omega\e{3}\psi_{-}\e{3}-\Gamma\e{2}\psi_{-}\e{3}&=E\psi_{-}.\label{eq2}
\end{align}
Assuming that $\psi_{+}\tilde{\psi}_{+}=1$, the first
equation~\eqref{eq1} can be rearranged into rotor form
\begin{equation}
\psi_{+}\e{3}\tilde{\psi}_{+}=-\frac{E}{\Gamma}\e{2},
\end{equation}
from which we find the eigenvalues and respective rotor equation
\begin{equation}
E^{(1,2)}=\pm\Gamma,\quad \psi_{+}\e{3}\tilde{\psi}_{+}=\mp\e{2}.
\end{equation}
The solution of the latter gives the first pair of spinor-rotors
\begin{equation}
\psi_{+}^{(1,2)}=\ee^{\pm\e{23}\pi/4}=\big(1\pm\e{23}\big)/\sqrt{2}\,.
\end{equation}
The rotor $\psi_{+}^{(1)}$  gives the eigenenergy
$E_{+}^{(1)}=\langle\tilde{\psi}_{+}^{(1)}\sH(\psi_{+}^{(1)})\rangle=-\Gamma$
and rotates $\e{3}$ to $\e{2}$, while $\psi_{+}^{(2)}$ gives
$E_{+}^{(2)}=\langle\tilde{\psi}_{+}^{(2)}\sH(\psi_{+}^{(2)})\rangle=\Gamma$
and rotates  $\e{3}$ to $-\e{2}$.

In a similar fashion from the second eigenequation~\eqref{eq2} one
finds
\begin{equation}
E^{(3,4)}=\pm\sqrt{\omega^2+\Gamma^2},\quad
\psi_{-}\e{3}\tilde{\psi}_{-}=\pm\frac{\omega\e{3}-\Gamma\e{2}}{\sqrt{\omega^2+\Gamma^2}}\equiv\hat{\ba}.
\end{equation}
The solution of the second equation gives the second pair of
spinor-rotors
\begin{equation}\
\psi_{-}^{(3,4)}=\exp\Bigl({\pm\frac{\e{3}\w\hat{\ba}}{|\e{3}\w\hat{\ba}|}
\frac{\theta}{2}}\Bigr)=\cos\frac{\theta}{2}\pm\frac{\e{3}\w\hat{\ba}}{|\e{3}\w\hat{\ba}|}\sin\frac{\theta}{2},
\end{equation}
where $\cos\theta=\e{3}\d\hat{\ba}$,
$\cos(\theta/2)=2^{-1/2}\sqrt{1+\cos\theta}$,
$\sin(\theta/2)=2^{-1/2}\sqrt{1-\cos\theta}$.  In our case
$\cos\theta=\omega/\sqrt{\omega^2+\Gamma^2}$ and
$\frac{\e{3}\w\hat{\ba}}{|\e{3}\w\hat{\ba}|}=\e{23}$, therefore,
the rotation acts in $\e{23}$ plane.

\subsection{Bilayer graphene\label{sec:4d}}
\begin{figure}
 \centering
 \begin{minipage}[c]{0.4\textwidth}
 \raggedright
\includegraphics[width=5.5cm]{Fig3_Acus.eps}
 \end{minipage}%
 \begin{minipage}[c]{0.7\textwidth}
 \raggedleft
\caption{Basis vectors $\{\e{1},\e{2},\e{3}\}$, the wave vector
$\mathbf{k}$, and the rotation plane
$\hat{\mathcal{B}}=\e{3}\w\hat{\ba}$. The vector $\e{3}$ is
perpendicular to bilayer graphene plane (square). The line that
connects $K$ and $K^{\prime}$ valleys in the Brillouin zone is
perpendicular to $\e{2}$.}\label{f:grapheneAxesab}
 \end{minipage}%
\end{figure}
In previous example the rotors $\psi_{+}$ and $\psi_{-}$ were
decoupled. Here we shall consider more complex case of bilayer
graphene (BG)  where the rotors are coupled. A unique feature of
the BG is its tunable energy band structure~\cite{Katsnelson12}.
In \textit{Cl}$_{3,1}$ the BG Hamiltonian function
reads~\cite{Dargys14c}
\begin{equation}\label{HBG}
\sH_{BG}(\psi)=\eta\mathbf{k}\psi\Ie{3}-\frac{\gamma_1}{2}
\e{2}\big(\psi-\inv{\psi}\big)\e{3}+\eta U\e{3}\psi\e{3},
\end{equation}
where $\gamma_1$ is the interlayer coupling energy (constant),
$2U$ is the potential difference between the layers, and $\eta=+1$ is for
$K$-valley and $\eta=-1$ for $K^{\prime}$-valley, i.~e. the points in the momentum space where electron transport takes place. The presence of
two basis vectors in~\eqref{HBG} comes from the geometry. One of them represents pseudospin quantization axis which is aligned with $\e{3}$, the other being
$K-K^{\prime}$ axis which is perpendicular to $\e{2}$,
Fig.~\ref{f:grapheneAxesab}. Thus, the geometry again comes explicitly in GA formulation of
quantum problems.

To transform  the  eigenequation $\sH_{BG}(\psi)=E\psi$ into rotors
the spinor is again split into even and odd parts,
$\psi=\psi_{+}+\psi_{-}$, and then the  eigenequation is
decomposed into two coupled multivector equations for $\psi_{+}$
and $\psi_{-}$,
\begin{equation}\label{coupled}
\begin{split}
E\psi_{+}&=\eta\mathbf{k}\psi_{-}\Ie{3}+\eta U\e{3}\psi_{+}\e{3},\\
E\psi_{-}&=\eta\mathbf{k}\psi_{+}\Ie{3}+\eta
U\e{3}\psi_{-}\e{3}-\gamma_1\e{2}\psi_{-}\e{3}.
\end{split}
\end{equation}
The method of solution of the coupled system~\eqref{coupled} uses
the fact that $\psi_{+}$ and $I\psi_{-}$, where $I$ is the
pseudoscalar of \textit{Cl}$_{3,1}$, are rotors in 3D
Euclidean space. If $\psi_{-}$ is solved from the first
equation and inserted into the second equation of~\eqref{coupled}
then after some algebraic manipulations one can construct the following
rotor equation $\psi_{+}\e{3}\tilde{\psi}_{+}=\hat{\ba}$ for
$\psi_{+}$ (and analogous equation for $I\psi_{-}$). The
requirement that the final vector $\hat{\ba}$ in
Fig.~\ref{f:grapheneAxesab} after rotation will not change its
length is expressed by condition $\hat{\ba}^2= 1$ which
gives~\cite{Dargys14c}
\begin{equation}\label{conda}
\hat{\ba}^2=\frac{(E^2-k^2+U^2)^2+U^2\gamma_1^2}{E^2(4U^2+\gamma_1^2)}=1.
\end{equation}
This fourth order  polynomial equation yields BG spectrum shown in
Fig.~\ref{f:DispEx}d,
\begin{equation}\label{spectrum}
E^{(i)}=\pm\sqrt{k^2+U^2+\frac{\gamma_1^2}{2}\pm\frac{1}{2}\sqrt{\gamma_1^4+4k^2(4U^2+\gamma_1^2)}}\,.
\end{equation}
Knowing  $\hat{\ba}^{(i)}=\ba^{(i)}/|\ba^{(i)}|$ and $E^{(i)}$,
one can  construct the rotor for the $i$-th energy band. For
details and properties of Berry's phase calculated in terms of GA
the reader should refer to~\cite{Dargys14c}.

In conclusion, we have shown that presence of a specific direction
called the quantization axis in the Hamiltonian can be employed to
solve quantum problems in GA.  Since GA approach is
coordinate-free the formulas found in this way appear to be
compact and may be interpreted geometrically.

\section{Appendix\label{append}}
\enlargethispage{8pt}
\textit{Matrix representation}. The basis vectors $\e{i}$ of
\textit{Cl}$_{3,1}$ algebra satisfy $\e{1}^2=\e{2}^2=\e{3}^2=1$
and $\e{4}^2=-1$. The following $4\times 4$ matrix representation
of $\e{i}$ was used in transforming Hilbert space Hamiltonians to
GA
\begin{equation}\label{e1e2}
\hat{e}_{1}=
\begin{bmatrix}
0&\hat{1}\\
\hat{1}&0\\
\end{bmatrix}
, \quad
\hat{e}_{2}=\ii
\begin{bmatrix}
0&-\hat{1}\\
\hat{1}&0\\
\end{bmatrix}
,\quad
\hat{e}_{3}=
\begin{bmatrix}
\hat{\sigma}_y&0\\
0&-\hat{\sigma}_y\\
\end{bmatrix}, \quad
\hat{e}_{4}=\ii
\begin{bmatrix}
\hat{\sigma}_z&0\\
0&-\hat{\sigma}_z\\
\end{bmatrix},
\end{equation}
where $\ii=\sqrt{-1}\,$. $\hat{1}$ is $2\times 2$ unit matrix and
$\hat{\sigma}_x$, $\hat{\sigma}_y$, $\hat{\sigma}_z$  are Pauli
matrices. Representations of higher blades can be found from
matrix products, for example,
$\e{12}\rightarrow\hat{e}_{12}=\hat{e}_{1}\hat{e}_{2}$.

\textit{Replacement rules} The following mapping is assumed
between the complex spinor $|\psi\rangle$ in a form of column and
GA spinor $\psi$,
\begin{equation}\label{psipsi}
 |\psi\rangle=
\begin{bmatrix}
\ \,a_0+\ii a_3\\
- b_3+\ii b_0\\
-b_2-\ii b_1\\
-a_1+\ii a_2
\end{bmatrix}
\longleftrightarrow\begin{cases}\psi=&a_0+a_1\e{23}-a_2\e{31}+a_3\e{12}\\
&\ -b_0I-b_1\e{14}+b_2\e{24}+b_3\e{34},
\end{cases}
\end{equation}
where $a_i$'s and $b_i$'s are scalars. Matrix representation of
basis elements \eqref{e1e2} and rule~\eqref{psipsi}  allow to construct the following replacement
rules between the action of matrices on column vectors and
\textit{Cl}$_{3,1}$ multivector functions,
\begin{equation}\label{repRulesAA}
\hat{e}_i|\psi\rangle\leftrightarrow\e{i}\psi\Ie{3},\quad
\hat{e}_{ij}|\psi\rangle\leftrightarrow\e{ij}\psi,\quad
\hat{I}\hat{e}_i|\psi\rangle\leftrightarrow\Ie{i}\psi\Ie{3},
\end{equation}
where $i,j=1,2,3,4$. Also, additional replacement rules may be
useful
\begin{equation}\label{repRulesA}
\begin{split}
&\ii|\psi\rangle\leftrightarrow I\psi\e{34},\quad
\langle\psi|\psi\rangle\leftrightarrow\langle\psi^{\dagger}\psi\rangle,\quad
\langle\varphi|\psi\rangle\leftrightarrow\langle\varphi^{\dagger}\psi\rangle-\langle\varphi^{\dagger}\psi\,\e{12}\rangle\e{12}.
\end{split}
\end{equation}
As always, in GA the angled brackets, for example $\langle
M\rangle$, indicate that only the scalar part of the
multivector~$M$ should be taken.

\textit{Spatial inversion, reversion and dagger operations}. The
spatial inversion  (denoted by overbar) changes signs of all
spatial vectors to opposite but  leaves ``time'' vector $\e{4}$
invariant: $\inv{\e{}}_i=-\e{i}$ for $i$=1,2,3 and
$\inv{\e{}}_4=\e{4}$. In \textit{Cl}$_{3,1}$  the inversion of
general multivector $M$ is defined by
\begin{equation}
\inv{M}=-\e{4}M\e{4}.
\end{equation}
Properties of inversion:  $\inv{M_1+M_2}=\inv{M}_1+\inv{M}_2$, and
$\inv{M_1M_2}=\inv{M}_1\,\,\inv{M}_2$. The spatial inversion
allows to split the general spinor~\eqref{psipsi} into even and
odd parts, $\psi=\psi_{+}+\psi_{-}$, which satisfy
$\inv{\psi}_{+}=\psi_{+}$ and $\inv{\psi}_{-}=-\psi_{-}$.

The reversion (denoted by tilde) changes the order of basis
vectors in the multivector. The dagger operation is a
combination of the reversion and spatial inversion
\begin{equation}\label{dagger}
\psi^{\dagger}=-\e{4}\tilde{\psi}\e{4}.
\end{equation}
If applied to a general bispinor it allows to find the square of
the module
\begin{equation}
\langle\psi^{\dagger}\psi\rangle=
a_0^2+a_1^2+a_2^2+a_3^2+b_0^2+b_1^2+b_2^2+b_3^2.
\end{equation}

redpajama_set_name: RedPajamaArXiv


text:
California police officer killed, suspect in US illegally
California police officer killed, suspect in US illegally /node/1426951/world
Police Cpl. Ronil Singh, who was gunned down the day after Christmas (Twitter/@StanSheriff)
Updated 28 December 2018
The suspect is believed the be in the US illegally
The officer was an immigrant and a devoted patriot
SAN FRANCISCO: Authorities are searching for a man believed to be living in the country illegally who they say gunned down a California police officer the day after Christmas, inflaming the nationwide debate on immigration.
President Donald Trump referred to the death of police Cpl. Ronil Singh in a tweet Thursday that ended with the demand: "Build the wall!"
Singh died Wednesday after stopping a suspected drunken driver in the tiny town of Newman in California's agricultural San Joaquin Valley.
A suspect identified from convenience store video before the killing, remains at large and is considered armed and dangerous.
Newman's police chief, Randy Richardson, held back tears Thursday as he described Singh — a native of Fiji — as a patriot who worked hard to serve his adopted country.
Topics: US California Gun Violence
Police: 3 dead, including shooter, at Florida yoga studio
One killed, one wounded in Vienna shooting
No-deal Brexit looms as race for new British PM wraps up /node/1527186/world
Many lawmakers, business community fear dire economic outcome
A majority of lawmakers in the House of Commons are opposed to a no-deal Brexit
LONDON: The battle to become Britain's next prime minister enters the home straight on Wednesday with both candidates hardening their positions on Brexit, putting the future government on a collision course with Brussels.
Ex-London Mayor Boris Johnson, the favourite to replace Theresa May, and Foreign Secretary Jeremy Hunt, are now both referring to Britain's departure with no overall deal in place as a realistic prospect.
The business community and many lawmakers fear dire economic consequences from a no-deal Brexit, which would lead to immediate trade tariffs for some sectors including the automotive industry.
Johnson and Hunt are taking part in a final question-and-answer session later on Wednesday before the result of the vote by Conservative Party members is announced next Tuesday.
The new party leader will be confirmed as prime minister by Queen Elizabeth II on the following day.
Britain has twice delayed its scheduled departure from the European Union after 46 years of membership as May tried and failed to get her deal with Brussels through parliament.
The two candidates vying to replace her have vowed to scrap a "backstop" provision in the agreement that Brussels insisted upon to keep the Irish border open.
Their latest attacks on the measure during a debate on Monday prompted a plunge in the value of the British pound.
The currency fell again Wednesday to its lowest level against the US dollar in over two years.
"The tougher stance from both Boris Johnson and Jeremy Hunt in terms of their rhetoric on Brexit is clearly weighing on the pound," said market analyst Neil Wilson.
"Make no mistake, this decline in the pound is down to traders pricing in a higher chance of a no-deal exit."
The backstop has proved a key stumbling block in the Brexit process.
The measure would keep open the post-Brexit border between British-ruled Northern Ireland and EU member the Republic of Ireland whatever the outcome of negotiations over the future relationship between London and Brussels.
Johnson announced early in his campaign that he would not sign up to it and would pursue a no-deal Brexit if required, leading his opponent to follow suit.
However, European leaders have been adamant that the backstop must remain a part of any divorce deal, raising the prospect of a no-deal Brexit.
German Defence Minister Ursula von der Leyen, who will become European Commission president in November, said the draft withdrawal agreement provided "certainty".
She also broached a possible further delay to Britain's departure, saying: "I stand ready for a further extension of the withdrawal date, should more time be required for a good reason."
Johnson has pledged that under his leadership, Britain will leave "do or die" on the current deadline of October 31.
A majority of lawmakers in the House of Commons are opposed to a no-deal Brexit, but attempts to pass legislation blocking the scenario have failed.
Reports this week suggested Johnson is considering plans to end the current session of parliament in early October, leaving MPs powerless.
Finance Minister Philip Hammond said Wednesday it was "terrifying" that some Brexit supporters thought that no deal would leave Britain better off.
And in a speech in London, May said the "best route" for Britain was to leave with a deal.
Delivering her last major address, she railed against the trend towards "absolutism" in Britain and abroad, and urged her successor to compromise.
"Whatever path we take must be sustainable for the long term, so that delivering Brexit brings our country back together. That has to mean some kind of compromise," she said.
Topics: Brexit British PM Theresa may Boris Johnson Jeremy Hunt Britain
UK economy feels the strain of global slowdown as well as Brexit
UK PM candidate Hunt: Boris Johnson is a 'coward' for avoiding debates on Brexit


Being listed among the `50 Most Impactful Leaders in Water & Water Management' in the world is no mean feat. And for scientist Suhas P Wani, director of the development center at the International Crops Research Institute for the Semi -Arid Tropics (ICRISAT), the honour is just another notch in his long list of accomplishments. The Hyderabadi scientist was honoured for his work on rainwater harvesting and soil conservation methods in Mumbai recently .
"The feeling hasn't quite sunk in yet," says Suhas who has been working towards improving rural livelihoods and sustainable agriculture for 40 years now.
Born in a farming family in a small town in Maharashtra, Wani witnessed the hardships faced by farmers since childhood.A bright student in school, he set his mind on studying agricultural science. "My parents wanted me to become a doctor or engineer and were initially disappointed with my choice as they felt it would bring me back to the fields but I persuaded them that it won't happen," says Wani. After his post-graduation, he moved to Bangalore to pursue his PhD.
After a brief stint with the Rajasthan government and working in partnership with a few NGOs in water management, Suhas joined ICRISAT in Hyderabad in 1980. "In between I had left for a short time to work in corporate sector but I wasn't able to do much for the farmers there. So I returned to Hyderabad," explains Wani.
Throughout his career, the 63-year-old has devoted himself to finding sustainable solutions to the water crisis in regions of Andhra Pradesh, Madhya Pradesh, Karnataka and Rajasthan to name a few. His work has not just been limited to India, he even received a medal from the Vietnamese government in 2004 for his work towards improving the agriculture and irrigation methods in upland Vietnam.
With water shortage plaguing different parts of India, the road to water conservation is a long one. "The biggest challenge we face today is the attitude of people.Woh kehte hain na paani ki tarah paisa beh raha hai (money is flowing like water), the reverse may come true soon," says Wani."At the rate, we are going, there will be a scarcity of water by 2025 in atleast two-thirds of the world. The climate change has led to 3 million hectare area turning into semi-arid and dryland region," says Wani.
Wani's next mission is to work with the government towards improving farming conditions. "It's difficult to get the government to put in systems benefiting the farmers. But it's not impossible.Once we achieve that, the possibilities are going to be huge," he signs off.


text:
north-rhine-westphalia
Summary: The Jewish community of Dueren was decimated during the Black Death pogroms of 1348/49, but a Jewish presence was re-established there in 1370. In the mid-1880s, after two centuries during which large numbers of Jews settled in the area, more than 600 Jews lived in the Dueren region, approximately 10% of whom lived in the city of Dueren. Although the Jews of Dueren were not recognized as an official Jewish community, they maintained a prayer room and a school. In 1847, the authorities permitted the community to build a synagogue which would also serve the surrounding towns; and in 1869, construction commenced on the large house of worship (inaugurated in 1872). On Pogrom Night, SA troops tried to set the synagogue building on fire; they failed, and instead busied themselves by breaking windows, tearing the Torah scrolls and holy books and breaking furniture. Unappeased, the men returned the following morning, doused the building with kerosene and set it on fire, after which they did the same to the community center, the school building and the sexton�s house. According to records, the blaze was so intense that several neighboring homes caught fire, the inhabitants just managing to escape unharmed. The Dueren municipality billed the Jewish community for the cost of demolishing the synagogue ruins. At the former synagogue site, now a parking lot, a memorial stone with an engraving of the synagogue commemorates the destruction.
Photo: The synagogue of Dueren. Courtesy of: City Archive of Dueren.
Sources: LJG, SG-NRW, SIA
Located in: north-rhine-westphalia


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Founded in 1882 by civil engineer William H. Hardy, Hattiesburg was named in honor of Hardy's wife Hattie. The town was incorporated two years later with a population of 400. Hattiesburg's population first expanded as a center of the lumber and railroad industries, from which was derived the nickname "The Hub City". It now attracts newcomers to the area because of the diversity of the economy, strong neighborhoods and the central location in South Mississippi.
Hattiesburg is home to The University of Southern Mississippi. South of Hattiesburg is Camp Shelby, the largest National Guard training base east of the Mississippi River.
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As of the 2010 United States Census, there were 45,989 people residing in the city. 52.8% were African American, 40.5% White, 0.2% Native American, 0.9% Asian, 0.0% Pacific Islander, 0.1% from some other race and 1.1% from two or more races. 4.3% were Hispanic or Latino of any race.
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