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Philippe Pinel

Philippe Pinel (20 April 1745 – 25 October 1826) was a French physician, precursor of psychiatry and incidentally a zoologist. He was instrumental in the development of a more humane psychological approach to the custody and care of psychiatric patients, referred to today as moral therapy. He worked for the abolition of the shackling of mental patients by chains and, more generally, for the humanisation of their treatment. He also made notable contributions to the classification of mental disorders and has been described by some as "the father of modern psychiatry".

After the French Revolution, Dr. Pinel changed the way we look at the mentally ill (or "aliénés", "alienated" in English) by claiming that they can be understood and cured. An 1809 description of a case that Pinel recorded in the second edition of his textbook on insanity is regarded by some as the earliest evidence for the existence of the form of mental disorder later known as dementia praecox or schizophrenia, although Emil Kraepelin is generally accredited with its first conceptualisation.

"Father of modern psychiatry", he was credited with the first classification of mental illnesses. He had a great influence on psychiatry and the treatment of the alienated in Europe and the United States.

Early life

Pinel was born in Jonquières, the South of France, in the modern department of Tarn. He was the son and nephew of physicians. After receiving a degree from the faculty of medicine in Toulouse, he studied an additional four years at the Faculty of Medicine of Montpellier. He arrived in Paris in 1778.

He spent fifteen years earning his living as a writer, translator, and editor because the restrictive regulations of the old regime prevented him from practicing medicine in Paris. The faculty did not recognize a degree from a provincial university like Toulouse. He failed twice in a competition which would have awarded him funds to continue his studies. In the second competition, the jury stressed his ‘painful’ mediocrity in all areas of medical knowledge, an assessment seemingly so grossly incompatible with his later intellectual accomplishments that political motives have been suggested. Discouraged, Pinel considered emigrating to America. In 1784 he became editor of the medical journal the Gazette de santé, a four-page weekly. He was also known among natural scientists as a regular contributor to the Journal de physique. He studied mathematics, translated medical works into French, and undertook botanical expeditions.

At about this time he began to develop an intense interest in the study of mental illness. The incentive was a personal one. A friend had developed a ‘nervous melancholy’ that had ‘degenerated into mania’ and resulted in suicide. What Pinel regarded as an unnecessary tragedy due to gross mismanagement seems to have haunted him. It led him to seek employment at one of the best-known private sanatoria for the treatment of insanity in Paris. He remained there for five years prior to the Revolution, gathering observations on insanity and beginning to formulate his views on its nature and treatment.

Pinel was an Ideologue, a disciple of the abbé de Condillac. He was also a clinician who believed that medical truth was derived from clinical experience. Hippocrates was his model.

During the 1780s, Pinel was invited to join the salon of Madame Helvétius. He was in sympathy with the French Revolution. After the revolution, friends he had met at Madame Helvétius’ salon came to power. In August 1793 Pinel was appointed "physician of the infirmeries" at Bicêtre Hospital. At the time it housed about four thousand imprisoned men—criminals, petty offenders, syphilitics, pensioners and about two hundred mental patients. Pinel’s patrons hoped that his appointment would lead to therapeutic initiatives. His experience at the private sanatoria made him a good candidate for the job.

The Bicêtre and Salpêtrière
Soon after his appointment to Bicêtre Hospital, Pinel became interested in the seventh ward where 200 mentally ill men were housed. He asked for a report on these inmates. A few days later, he received a table with comments from the "governor" Jean-Baptiste Pussin. In the 1770s Pussin had been successfully treated for scrofula at Bicêtre; and, following a familiar pattern, he was eventually recruited, along with his wife, Marguerite Jubline, on to the staff of the hospice.

Appreciating Pussin's outstanding talent, Pinel virtually apprenticed himself to that unschooled but experienced custodian of the insane. His purpose in doing this was to "enrich the medical theory of mental illness with all the insights that the empirical approach affords". What he observed was a strict nonviolent, nonmedical management of mental patients that came to be called moral treatment or moral management, though psychological might be a more accurate term. Although Pinel always gave Pussin the credit he deserved, a legend grew up about Pinel single-handedly liberating the insane from their chains at Bicetre. This legend has been commemorated in paintings and prints, and has lived on for 200 years and is repeated in textbooks. In fact, it was Pussin who removed the iron shackles (but sometimes using straitjackets) at Bicêtre in 1797, after Pinel had left for the Salpêtrière. Pinel did remove the chains from patients at the Salpêtrière three years later, after Pussin joined him there. There is some suggestion that the Bicetre myth was actually deliberately fabricated by Pinel's son, Dr Scipion Pinel, along with Pinel's foremost pupil, Dr Esquirol. The argument is that they were 'solidists', which meant then something akin to biological psychiatry with a focus on brain disease, and were embarrassed by Pinel's focus on psychological processes. In addition, unlike Philippe, they were both royalists.

While at Bicêtre, Pinel did away with bleeding, purging, and blistering in favor of a therapy that involved close contact with and careful observation of patients. Pinel visited each patient, often several times a day, and took careful notes over two years. He engaged them in lengthy conversations. His objective was to assemble a detailed case history and a natural history of the patient's illness.

In 1795, Pinel became chief physician of the Hospice de la Salpêtrière, a post that he retained for the rest of his life. The Salpêtrière was, at the time, like a large village, with seven thousand elderly indigent and ailing women, an entrenched bureaucracy, a teeming market and huge infirmaries. Pinel missed Pussin and in 1802 secured his transfer to the Salpêtrière. It has also been noted that a Catholic nursing order actually undertook most of the day to day care and understanding of the patients at Salpêtrière, and there were sometimes power struggles between Pinel and the nurses.

Pinel created an inoculation clinic in his service at the Salpêtrière in 1799, and the first vaccination in Paris was given there in April 1800.

In 1795 Pinel had also been appointed as a professor of medical pathology, a chair that he held for twenty years. He was briefly dismissed from this position in 1822, with ten other professors, suspected of political liberalism, but reinstated as an honorary professor shortly thereafter.

A statue in honour of Pinel now stands outside the Salpêtrière.

Publications
In 1794 Pinel made public his essay 'Memoir on Madness', recently called a fundamental text of modern psychiatry. In it Pinel makes the case for the careful psychological study of individuals over time, points out that insanity isn't always continuous, and calls for more humanitarian asylum practices.

In 1798 Pinel published an authoritative classification of diseases in his Nosographie philosophique ou méthode de l'analyse appliquée à la médecine. Although he is properly considered one of the founders of psychiatry, this book also establishes him as the last great nosologist of the eighteenth century. While the Nosographie appears completely dated today, it was so popular in its time that it went through six editions between its initial publication and 1818. Pinel based his nosology on ideas of William Cullen, employing the same biologically-inspired terminology of 'genera' and 'species' of disorder. Pinel's classification of mental disorder simplified Cullen's 'neuroses' down to four basic types of mental disorder: melancholia, mania (insanity), dementia, and idiotism. Later editions added forms of 'partial insanity' where only that of feelings which seem to be affected rather than reasoning ability.

The first mental derangement is called melancholia. The symptoms are described as “taciturnity, a thoughtful pensive air, gloomy suspicions, and a love of solitude.” It is noted that Tiberius and Louis XI were subjected to this temperament. Louis was characterized by the imbalance between the state of bitterness and passion, gloom, love of solitude, and the embarrassment of artistic talents. However, Louis and Tiberius were similar in that they both were deceitful and planned a delusional trip to military sites. Eventually both were exiled, one to the Isle of Rhodes and the other to a province of Belgium. People with melancholia are often immersed with one idea that their whole attention is fixated on. On one hand they stay reserved for many years, withholding friendships and affection while on the other, there are some who make reasonable judgment and overcome the gloomy state. Melancholia can also express itself in polar opposite forms. The first is distinguished by an exalted sense of self-importance and unrealistic expectations such as attaining riches and power. The second form is marked by deep despair and great depression. Overall individuals with melancholia generally do not display acts of violence, though they may find it wildly fanciful. Depression and anxiety occurs habitually as well as frequent moroseness of character. Pinel remarks that melancholia can be explained by drunkenness, abnormalities in the structure of the skull, trauma in the skull, conditions of the skin, various psychological causes such as household disasters and religious extremism, and in women, menstruation and menopause.

The second mental derangement is called mania without delirium. It is described as madness independent of a disorder that impairs the intellectual faculties. The symptoms are described as perverse and disobedient. An instance where this type of species of mental derangement occurs where a mechanic, who was confined at the Asylum de Bicetre, experienced violent outbursts of maniacal fury. The paroxysms consisted of a burning sensation located in the abdominal area that was accompanied by constipation and thirst. The symptom spread to the chest, neck, and face area. When it reached the temples, the pulsation of the arteries increased in those areas. The brain was affected to some length but nonetheless, the patient was able to reason and cohere to his ideas. One time the mechanic experienced furious paroxysm at his own house where he warned his wife to flee to avoid death. He also experienced the same periodical fury at the asylum where he plotted against the governor. The specific character of mania without delirium is that it can either be perpetual or sporadic. However, there was no reasonable change in the cognitive functions of the brain; only perversive thoughts of fury and a blind tendency to acts of violence.

The third mental derangement is called mania with delirium. It is mainly characterized by indulgence and fury, and affects cognitive functions. Sometimes it may be distinguished by a carefree, gay humor that can venture off path in incoherent and absurd suggestions. Other times it can be distinguished by prideful and imaginary claims to grandeur. Prisoners of this species are highly delusional. For example, they would proclaim having fought an important battle, or witness the prophet Mohammad conjuring wrath in the name of the Almighty. Some declaim ceaselessly with no evidence of things seen or heard while others saw illusions of objects in various forms and colors. Delirium sometimes persists with some degree of frenzied uproar for a period of years, but it can also be constant and the paroxysm of fury repeat at different intervals. The specific character of mania with delirium is the same as mania without delirium in the sense that it can either be continued or cyclical with regular or irregular paroxysms. It is marked by strong nervous excitement, accompanied by a deficit of one or more of the functions of the cognitive abilities with feelings of liveliness, depression or fury.

The fourth mental derangement is called dementia, or otherwise known as the abolition of thinking. The characteristics include thoughtlessness, extreme incorrectness, and wild abnormalities. For instance, a man who had been educated on the ancient nobility was marching on about the beginning of the revolution. He moved restlessly about the house, talking endlessly and shouting passionately on insignificant reasons. Dementia is usually accompanied by raging and rebellious movement, by a quick succession of ideas formed in the mind, and by passionate feelings that are felt and forgotten without attributing it to objects. Those who are in captive of dementia have lost their memory, even those attributed to their loved ones. Their only memory consists of those in the past. They forget instantaneously things in the present – seen heard or done. Many are irrational because the ideas do not flow coherently. The characteristic properties of dementia are that there is no judgment value and the ideas are spontaneous with no connection. The specific character of dementia contains a rapid progression or continual succession of isolated ideas, forgetfulness of previous condition, repetitive acts of exaggeration, decreased responsiveness to external influence, and complete lack of judgment.

The fifth and last mental derangement is called idiotism, or otherwise known as “obliteration of the intellectual faculties and affections.” This disorder is derived from a variety of causes, such as extravagant and debilitating delight, alcohol abuse, deep sorrow, diligent study, aggressive blows to the head, tumors in the brain, and loss of consciousness due to blockage in vein or artery. Idiotism embodies a variety of forms. One such form is called Cretinism, which is a kind of idiotism that is relative to personal abnormalities. It is well known in the Valais and in parts of Switzerland. Most people who belong in this group are either deficient in speech or limited to the inarticulate utterances of sounds. Their expressions are emotionless, senses are dazed and motions are mechanical. Idiots also constitute the largest number of patients at hospitals. Individuals who have acute responsiveness can experience a violent shock to the extreme that all the activities of the brain can either be arrested in an action or eradicated completely. Unexpected happiness and exaggerated fear may likely occur as a result of a violent shock. As mentioned previously, idiotism is the most common among hospital patients and is incurable. At the Bicetre asylum, these patients constitute one fourth of the entire population. Many die after a few days of arrival, having been reduced to states of stupor and weakness. However, some who recover with the progressive regeneration of their strength also regain their intellectual capabilities. Many of the young people that have remained in the state of idiotism for several months or years are attacked by a spasm of active mania between twenty and thirty days. The specific character of idiotism includes partial or complete extermination of the intellect and affections, apathy, disconnected, inarticulate sounds or impairment of speech, and nonsensical outbursts of passion.

In his book Traité médico-philosophique sur l'aliénation mentale; ou la manie, published in 1801, Pinel discusses his psychologically oriented approach. This book was translated into English by D. D. Davis as a Treatise on Insanity in 1806, although Davis substituted Pinel's introduction for his own, leaving out among other things Pinel's strong praise for Alexander Crichton. Pinel's book had an enormous influence on both French and Anglo-American psychiatrists during the nineteenth century. He meant by alienation that the patient feels like a stranger (alienus) to the world of the 'sane'. A sympathetic therapist living in that world might be able to journey into the patient's experience, understand the 'alienated', their language, and possibly lead them back into society.

In 1802 Pinel published La Médecine Clinique which was based on his experiences at the Salpêtrière and in which he extended his previous book on classification and disease.

Pinel was elected to the Académie des Sciences in 1804 and was a member of the Académie de Médecine from its founding in 1820. He died in Paris on 25 October 1826.

Psychological understanding
The central and ubiquitous theme of Pinel's approach to etiology (causation) and treatment was "moral," meaning the emotional or the psychological not ethical. He observed and documented the subtleties and nuances of human experience and behavior, conceiving of people as social animals with imagination.

Pinel noted, for example, that: "being held in esteem, having honor, dignity, wealth, fame, which though they may be factitious, always distressing and rarely fully satisfied, often give way to the overturning of reason". He spoke of avarice, pride, friendship, bigotry, the desire for reputation, for conquest, and vanity. He noted that a state of love could turn to fury and desperation, and that sudden severe reversals in life, such as "from the pleasure of success to an overwhelming idea of failure, from a dignified state—or the belief that one occupies one—to a state of disgrace and being forgotten" can cause mania or 'mental alienation'. He identified other predisposing psychosocial factors such as an unhappy love affair, domestic grief, devotion to a cause carried to the point of fanaticism, religious fear, the events of the revolution, violent and unhappy passions, exalted ambitions of glory, financial reverses, religious ecstasy, and outbursts of patriotic fervor.

Treatments
Pinel developed specific practical techniques, rather than general concepts and assumptions. He engaged in therapeutic conversations to dissuade patients from delusions. He offered benevolent support and encouragement, although patients who persistently resisted or caused trouble might be threatened with incarceration or punishment if they were not able to control themselves.

Pinel argued that psychological intervention must be tailored to each individual rather than be based solely on the diagnostic category, and that it must be grounded in an understanding of the person's own perspective and history. He noted that "the treatment of insanity (l’aliénation mentale) without considering the differentiating characteristics of the patients [la distinction des espèces] has been at times superfluous, rarely useful, and often harmful", describing the partial or complete failures of some psychological approaches, as well as the harm that the usual cruel and harsh treatments caused to patients before they came to his hospital. He saw improvement as often resulting from natural forces within the patient, an improvement that treatment could at best facilitate and at worst interfere with.

Pinel’s approach to medical treatments has been described as ambiguous, complex, and ambivalent. He insisted that psychological techniques should always be tried first, for example "even where a violent and destructive maniac could be calmed by a single dose of an antispasmodic [he referred to opium], observation teaches that in a great number of cases, one can obtain a sure and permanent cure by the sole method of expectation, leaving the insane man to his tumultuous excitement... ...and [furthermore] seeing, again and again, the unexpected resources of nature left to itself or wisely guided, has rendered me more and more cautious with regard to the use of medications, which I no longer employ—except when the insufficiencies of psychological means have been proven."

For those cases regarded as psychologically incurable, Pinel would employ baths, showers, opium, camphor and other antispasmodics, as well as vesicants, cauterization, and bloodletting in certain limited cases only. He also recommended the use of laxatives for the prevention of nervous excitement and relapse.

Pinel often traced mental states to physiological states of the body, and in fact could be said to have practiced psychosomatic medicine. In general, Pinel traced organic causes to the gastrointestinal system and peripheral nervous system more often than to brain dysfunction. This was consistent with his rarely finding gross brain pathology in his post-mortem examinations of psychiatric patients, and his view that such findings that were reported could be correlational rather than causative.

Management
Pinel was concerned with a balance between control by authority and individual liberty. He believed in "the art of subjugating and taming the insane" and the effectiveness of "a type of apparatus of fear, of firm and consistent opposition to their dominating and stubbornly held ideas", but that it must be proportional and motivated only by a desire to keep order and to bring people back to themselves. The straitjacket and a period of seclusion were the only sanctioned punishments. Based on his observations, he believed that those who were considered most dangerous and carried away by their ideas had often been made so by the blows and bad treatment they had received, and that it could be ameliorated by providing space, kindness, consolation, hope, and humor.

Because of the dangers and frustrations that attendants experienced in their work, Pinel put great emphasis on the selection and supervision of attendants in order to establish a custodial setting dedicated to norms of constraint and liberty that would facilitate psychological work. He recommended that recovered patients be employed, arguing that "They are the ones who are most likely to refrain from all inhumane treatment, who will not strike even in retaliation, who can stand up to pleading, menaces, repetitive complaining, etc. and retain their inflexible firmness." Pinel also emphasized the necessity for leadership that was "thoughtful, philanthropic, courageous, physically imposing, and inventive in the development of maneuvers or tactics to distract, mollify, and impress" and "devoted to the concept of order without violence", so that patients are "led most often with kindness, but always with an inflexible firmness." He noted that his ex-patient and superintendent Pussin had showed him the way in this regard, and had also often been better placed to work with patients and develop techniques due to his greater experience and detailed knowledge of the patients as individuals.

Moral judgments
Pinel generally expressed warm feelings and respect for his patients, as exemplified by: "I cannot but give enthusiastic witness to their moral qualities. Never, except in romances, have I seen spouses more worthy to be cherished, more tender fathers, passionate lovers, purer or more magnanimous patriots, than I have seen in hospitals for the insane, in their intervals of reasonableness and calm; a man of sensibility may go there any day and take pleasure in scenes of compassion and tenderness". He argued that otherwise positive character traits could cause a person to be vulnerable to the distressing vicissitudes of life, for example "those persons endowed with a warmth of imagination and a depth of sensitivity, who are capable of experiencing powerful and intense emotions, [since it is they] who are most predisposed to mania".

Pinel distanced himself from religious views, and in fact considered that excessive religiosity could be harmful.

However, he sometimes took a moral stance himself as to what he considered to be mentally healthy and socially appropriate. Moreover, he sometimes showed a condemnatory tone toward what he considered personal failings or vice, for example noting in 1809: "On one side one sees families which thrive over a course of many years, in the bosom of order and concord, on the other one sees many others, especially in the lower social classes, who offend the eye with the repulsive picture of debauchery, arguments, and shameful distress!". He goes on to describe this as the most prolific source of alienation needing treatment, adding that while some such examples were a credit to the human race many others are "a disgrace to humanity!"

Influence
Pinel is generally seen as the physician who more than any other transformed the concept of 'the mad' into that of patients needing care and understanding, establishing a field that would eventually be called psychiatry. His legacy included improvement of asylum conditions; broadly psychosocial (incl. milieu) therapeutic approaches; history-taking; nosography (the science of the description of syndromes); broadly-numerical assessments of courses of illness and treatment responses; and a record of clinical teaching.

Pinel's actions took place in the context of the Enlightenment, and a number of others were also reforming asylums along humanitarian lines. For example, Vincenzo Chiarugi, in the 1780s in Italy, removed metal chains from patients but did not enjoy the same renown bestowed on the more explicitly humanitarian Pinel who was so visible from late 18th century revolutionary France. In France, Joseph D'Aquin in Chambéry permitted patients to move about freely and published a book in 1791 urging humanitarian reforms, dedicating the second edition in 1804 to Pinel. The movement as a whole become known as moral treatment or moral management, and influenced asylum development and psychological approaches throughout the Western world.

Pinel's most important contribution may have been the observation and conviction that there could be sanity and rationality even in cases that seemed on the surface impossible to understand, and that this could appear for periods in response to surrounding events (and not just because of such things as the phase of the moon, a still common assumption and the origin of the term lunatic). The influential philosopher Hegel praised Pinel for this approach.

"The right psychical treatment therefore keeps in view the truth that insanity is not an abstract loss of reason (neither in the point of intelligence nor of will and its responsibility), but only derangement, only a contradiction in a still subsisting reason; − just as physical disease is not an abstract, i.e. mere and total, loss of health (if it were that, it would be death), but a contradiction in it. This humane treatment, no less benevolent than reasonable (the services of Pinel towards which deserve the highest acknowledgement), presupposes the patient's rationality, and in that assumption has the sound basis for dealing with him on this side − just as in the case of bodily disease the physician bases his treatment on the vitality which as such still contains health."

Pinel also started a trend for diagnosing forms of insanity that seemed to occur 'without delerium' (confusion, delusions or hallucinations). Pinel called this Manie sans délire, folie raisonnante or folie lucide raisonnante. He described cases who seemed to be overwhelmed by instinctive furious passions but still seemed sane. This was influential in leading to the concept of moral insanity, which became an accepted diagnosis through the second half of the 19th century. Pinel's main psychiatric heir, Esquirol, built on Pinel's work and popularised various concepts of monomania.

However, Pinel was also criticised and rejected in some quarters. A new generation favoured pathological anatomy, seeking to locate mental disorders in brain lesions. Pinel undertook comparisons of skull sizes, and considered possible physiological substrates, but he was criticized for his emphasis on psychology and the social environment. Opponents were bolstered by the discovery of tertiary syphilis as the cause of some mental disorder. Pinel's humanitarian achievements were emphasized and mythologised instead.

With increasing industrialization, asylums generally became overcrowded, misused, isolated and run-down. The moral treatment principles were often neglected along with the patients. There was recurrent debate over the use of psychological-social oppression even if some physical forces were removed. By the mid-19th century in England, the Alleged Lunatics' Friend Society was proclaiming the moral treatment approach was achieved "by mildness and coaxing, and by solitary confinement", treating people like children without rights to make their own decisions.

Similarly in the mid-20th century, Foucault's influential book, Madness and Civilization: A History of Insanity in the Age of Reason, also known as History of Madness, focused on Pinel, along with Tuke, as the driving force behind a shift from physical to mental oppression. Foucault argued that the approach simply meant that patients were ignored and verbally isolated, and were worse off than before. They were made to see madness in others and then in themselves until they felt guilt and remorse. The doctor, despite his lack of medical knowledge about the underlying processes, had all powers of authority, and defined insanity. Foucault also suggested that a focus on the rights of patients at Bicetre was partly due to revolutionary concerns that it housed and chained victims of arbitrary or political power, or alternatively that it might be enabling refuge for anti-revolutionary suspects, as well as just 'the mad'.

Scull argues that the "...manipulations and ambiguous 'kindness' of Tuke and Pinel..." may nevertheless have been preferable to the harsh coercion and physical "treatments" of previous generations, though he does recognise its "...less benevolent aspects and its latent potential ... for deterioration into a repressive form...." Some have criticised the process of deinstitutionalization that took place in the 20th century and called for a return to Pinel's approach, so as not to underestimate the needs that mentally ill people might have for protection and care.

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title: Philippe Pinel


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Space syntax

Space syntax is a set of theories and techniques for the analysis of spatial configurations. It was conceived by Bill Hillier, Julienne Hanson, and colleagues at The Bartlett, University College London in the late 1970s to early 1980s to develop insights into the mutually constructive relation between society and space. As space syntax has evolved, certain measures have been found to correlate with human spatial behavior, and space syntax has thus come to be used to forecast likely effects of architectural and urban space on users.

Thesis
The general idea is that spaces can be broken down into components, analyzed as networks of choices, then represented as maps and graphs that describe the relative connectivity and integration of those spaces. It rests on three basic conceptions of space:
 * an isovist (popularised by Michael Benedikt at University of Texas), or viewshed or visibility polygon, the field of view from any particular point
 * axial space (idea popularized by Bill Hillier at UCL), a straight sight-line and possible path
 * convex space (popularized by John Peponis, and his collaborators at Georgia Tech), an occupiable void where, if imagined as a wireframe diagram, no line between two of its points goes outside its perimeter: all points within the polygon are visible to all other points within the polygon.

The three most popular ways of analysing a street network are integration, choice and depth distance.

Integration
Integration measures the amount of street-to-street transitions needed from a street segment, to reach all other street segments in the network, using shortest paths. The graph analysis could also limit measure integration at radius 'n', for segments further than this radius not to be taken into account. The first intersecting segment requires only one transition, the second two transitions and so on. The result of the analysis finds street segments that require fewest turns to reach all other streets, which are called 'most integrated' and are usually represented with hotter colors, such as red or yellow. Integration can also be analyzed in local scale instead of the scale of the whole network. In the case of radius 4, for instance, only four turns are counted departing from each street segment. Measure also is highly related to network analysis Centrality.

Theoretically, the integration measure shows the cognitive complexity of reaching a street, and is often argued to 'predict' the pedestrian use of a street: the easier it is to reach a street, the more popular it should be.

While there is some evidence of this being true, the method is biased towards long, straight streets that intersect with many other streets. Such streets, as Oxford Street in London, come out as especially strongly integrated. However, a slightly curvy street of the same length would typically be segmented into individual straight segments, not counted as a single line, which makes curvy streets appear less integrated in the analysis.

Choice
The choice measure is easiest to understand as a 'water-flow' in the street network. Imagine that each street segment is given an initial load of one unit of water, which then starts pouring from the starting street segment to all segments that successively connect to it. Each time an intersection appears, the remaining value of flow is divided equally amongst the splitting streets, until all the other street segments in the graph are reached. For instance, at the first intersection with a single other street, the initial value of one is split into two remaining values of one half, and allocated to the two intersecting street segments. Moving further down, the remaining one half value is again split among the intersecting streets and so on. When the same procedure has been conducted using each segment as a starting point for the initial value of one, a graph of final values appears. The streets with the highest total values of accumulated flow are said to have the highest choice values.

Like integration, choice analysis can be restricted to limited local radii, for instance 400m, 800m, 1600m. Interpreting Choice analysis is trickier than integration. Space syntax argues that these values often predict the car traffic flow of streets, but, strictly speaking, choice analysis can also be thought to represent the number of intersections that need to be crossed to reach a street. However, since flow values are divided (not subtracted) at each intersection, the output shows an exponential distribution. It is considered best to take a log of base two of the final values in order to get a more accurate picture.

Depth distance
Depth distance is the most intuitive of the analysis methods. It explains the linear distance from the center point of each street segment to the center points of all the other segments. If every segment is successively chosen as a starting point, a graph of cumulative final values is achieved. The streets with lowest Depth Distance values are said to be nearest to all the other streets. Again, the search radius can be limited to any distance.

Applications
From these components it is thought to be possible to quantify and describe how easily navigable any space is, useful for the design of museums, airports, hospitals, and other settings where wayfinding is a significant issue. Space syntax has also been applied to predict the correlation between spatial layouts and social effects such as crime, traffic flow, and sales per unit area.

In general, the analysis uses one of many software programs that allow researchers to analyse graphs of one (or more) of the primary spatial components.

History
Space syntax originated as a programme research in the early 1970s when Bill Hillier, Adrian Leaman and Alan Beattie came together at the School of Environmental Studies at University College London (now part of the Bartlett School of Architecture). Bill Hillier had been appointed Director of the Unit for Architectural Studies (UAS) as successor to John Musgrove. They established a new MSc programme in Advanced Architectural Studies and embarked on a programme of research aimed at developing a theoretical basis for architecture. Previously Bill Hillier had written papers with others as secretary to the RIBA, notably 'Knowledge and Design' and 'How is Design Possible'. These laid the theoretical foundation for a series of studies that sought to clarify how the built environment relates to society. One of the first cohorts of students on the MScAAS was Julienne Hanson who went on to co-author The Social Logic of Space (SLS) with Bill Hillier (CUP, 1984). This brought together in one place a comprehensive review of the programme of research up to that point, but also developed a full theoretical account for how the buildings and settlements we construct an not merely the product of social processes, but also play a role in producing social forms. SLS also developed an analytic approach to representation and quantification of spatial configuration at the building and the settlement scale, making possible both comparative studies as well as analysis of the relationship between spatial configuration and aspect of social function in the built environment. These methods coupled to the social theories have turned out to have a good deal of explanatory power. Space syntax has grown to become a tool used around the world in a variety of research areas and design applications in architecture, urban design, urban planning, transport and interior design. Many prominent design applications have been made by the architectural and urban planning practice Space Syntax Limited, which was founded at The Bartlett, University College London in 1989. These include the redesign of Trafalgar Square with Foster and Partners and the Pedestrian Movement Model for the City of London.

Over the past decade, Space syntax techniques have been used for research in archaeology, information technology, urban and human geography, and anthropology. Since 1997, the Space syntax community has held biennial conferences, and many journal papers have been published on the subject, chiefly in Environment and Planning B.

Criticism
Space syntax's mathematical reliability has come under scrutiny because of a seeming paradox that arises under certain geometric configurations with 'axial maps', one of the method's primary representations of spatial configuration. This paradox was proposed by Carlo Ratti at the Massachusetts Institute of Technology, but comprehensively refuted in a passionate academic exchange with Bill Hillier and Alan Penn. There have been moves to combine space syntax with more traditional transport engineering models, using intersections as nodes and constructing visibility graphs to link them, by researchers including Bin Jiang, Valerio Cutini and Michael Batty. Recently there has also been research development that combines space syntax with geographic accessibility analysis in GIS, such as the place syntax-models developed by the research group Spatial Analysis and Design at the Royal Institute of Technology in Stockholm, Sweden. A series of interdisciplinary works published in 2006 by Vito Latora, Sergio Porta and colleagues, proposing a network approach to street centrality analysis and design, have highlighted space syntax' contribution to decades of previous studies in the physics of spatial complex networks.

title: Space syntax


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Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

Diophantus
The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:

 * $$A = x + y\ $$
 * $$B = x^2 + y^2.\ $$

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).

Diophantus' major work was the Arithmetica, of which only a portion has survived.

Fermat
Fermat's Last Theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.

Gauss
One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae (Latin: Arithmetical Investigations) is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.

The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular.

Dirichlet
In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Leopold Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.

He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.

Dedekind
Richard Dedekind's study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has been written that: "'Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death.' (Edwards 1983)"

1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by Hilbert, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem.

Hilbert
David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.

Artin
Emil Artin established the Artin reciprocity law in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

Modern theory
Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

It was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence supporting it, yet no proof; as a result the "astounding" conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands program, a list of important conjectures needing proof or disproof.

From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.

Failure of unique factorization
An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers $O$ of an algebraic number field $K$.

A prime element is an element $p$ of $O$ such that if $p$ divides a product $ab$, then it divides one of the factors $a$ or $b$. This property is closely related to primality in the integers, because any positive integer satisfying this property is either $1$ or a prime number. However, it is strictly weaker. For example, $−2$ is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as
 * $$6 = 2 \cdot 3 = (-2) \cdot (-3).$$

In general, if $u$ is a unit, meaning a number with a multiplicative inverse in $O$, and if $p$ is a prime element, then $up$ is also a prime element. Numbers such as $p$ and $up$ are said to be associate. In the integers, the primes $p$ and $−p$ are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When K is not the rational numbers, however, there is no analog of positivity. For example, in the Gaussian integers $Z[i]$, the numbers $1 + 2i$ and $−2 + i$ are associate because the latter is the product of the former by $i$, but there is no way to single out one as being more canonical than the other. This leads to equations such as
 * $$5 = (1 + 2i)(1 - 2i) = (2 + i)(2 - i),$$

which prove that in $Z[i]$, it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used in unique factorization domains (UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering.

However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an irreducible element. An irreducible element $x$ is an element such that if $x = yz$, then either $y$ or $z$ is a unit. These are the elements that cannot be factored any further. Every element in O admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring $Z[√\overline{-5}]$. In this ring, the numbers $√\overline{-5}$, $3$ and $2 + √\overline{-5}$ are irreducible. This means that the number $2 - √\overline{-5}$ has two factorizations into irreducible elements,
 * $$9 = 3^2 = (2 + \sqrt{-5})(2 - \sqrt{-5}).$$

This equation shows that $9$ divides the product $3$. If $(2 + √\overline{-5})(2 - √\overline{-5}) = 9$ were a prime element, then it would divide $3$ or $2 + √\overline{-5}$, but it does not, because all elements divisible by $2 - √\overline{-5}$ are of the form $3$. Similarly, $3a + 3b√\overline{-5}$ and $2 + √\overline{-5}$ divide the product $2 - √\overline{-5}$, but neither of these elements divides $3^{2}$ itself, so neither of them are prime. As there is no sense in which the elements $3$, $3$ and $2 + √\overline{-5}$ can be made equivalent, unique factorization fails in $2 - √\overline{-5}$. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.

Factorization into prime ideals
If $Z[√\overline{-5}]$ is an ideal in $I$, then there is always a factorization
 * $$I = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_t^{e_t},$$

where each $$\mathfrak{p}_i$$ is a prime ideal, and where this expression is unique up to the order of the factors. In particular, this is true if $O$ is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers are Dedekind domains.

When $I$ is a UFD, every prime ideal is generated by a prime element. Otherwise, there are prime ideals which are not generated by prime elements. In $O$, for instance, the ideal $Z[√\overline{-5}]$ is a prime ideal which cannot be generated by a single element.

Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension field $(2, 1 + √\overline{-5})$ of $E$. This extension field is now known as the Hilbert class field. By the principal ideal theorem, every prime ideal of $K$ generates a principal ideal of the ring of integers of $O$. A generator of this principal ideal is called an ideal number. Kummer used these as a substitute for the failure of unique factorization in cyclotomic fields. These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals.

An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding ideals $E$ are prime ideals of the ring $pZ$. However, when this ideal is extended to the Gaussian integers to obtain $Z$, it may or may not be prime. For example, the factorization $pZ[i]$ implies that
 * $$2\mathbf{Z}[i] = (1 + i)\mathbf{Z}[i] \cdot (1 - i)\mathbf{Z}[i] = ((1 + i)\mathbf{Z}[i])^2;$$

note that because $2 = (1 + i)(1 − i)$, the ideals generated by $1 + i = (1 − i) ⋅ i$ and $1 + i$ are the same. A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by Fermat's theorem on sums of two squares. It implies that for an odd prime number $1 − i$, $p$ is a prime ideal if $pZ[i]$ and is not a prime ideal if $p ≡ 3 (mod 4)$. This, together with the observation that the ideal $p ≡ 1 (mod 4)$ is prime, provides a complete description of the prime ideals in the Gaussian integers. Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q (that is, a Galois extension with abelian Galois group).

Ideal class group
Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group. Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so that they admit a group structure. This is done by generalizing ideals to fractional ideals. A fractional ideal is an additive subgroup $(1 + i)Z[i]$ of $J$ which is closed under multiplication by elements of $K$, meaning that $O$ if $xJ ⊆ J$. All ideals of $x ∈ O$ are also fractional ideals. If $O$ and $I$ are fractional ideals, then the set $J$ of all products of an element in $IJ$ and an element in $I$ is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The group identity is the ideal $J$, and the inverse of $(1) = O$ is a (generalized) ideal quotient:

 * $$J^{-1} = (O:J) = \{x \in K: xJ \subseteq O\}.$$

The principal fractional ideals, meaning the ones of the form $J$ where $Ox$, form a subgroup of the group of all non-zero fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional ideals $x ∈ K^{×}$ and $I$ represent the same element of the ideal class group if and only if there exists an element $J$ such that $x ∈ K$. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted $xI = J$, $Cl K$, or $Cl O$ (with the last notation identifying it with the Picard group in algebraic geometry).

The number of elements in the class group is called the class number of K. The class number of $Pic O$ is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as $Q(√\overline{-5})$.

The ideal class group has another description in terms of divisors. These are formal objects which represent possible factorizations of numbers. The divisor group $(2, 1 + √\overline{-5})$ is defined to be the free abelian group generated by the prime ideals of $Div K$. There is a group homomorphism from $O$, the non-zero elements of $K^{×}$ up to multiplication, to $K$. Suppose that $Div K$ satisfies
 * $$(x) = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_t^{e_t}.$$

Then $x ∈ K$ is defined to be the divisor
 * $$\operatorname{div} x = \sum_{i=1}^t e_i[\mathfrak{p}_i].$$

The kernel of $div x$ is the group of units in $div$, while the cokernel is the ideal class group. In the language of homological algebra, this says that there is an exact sequence of abelian groups (written multiplicatively),
 * $$1 \to O^\times \to K^\times \xrightarrow{\text{div}} \operatorname{Div} K \to \operatorname{Cl} K \to 1.$$

Real and complex embeddings
Some number fields, such as $O$, can be specified as subfields of the real numbers. Others, such as $Q(√\overline{2})$, cannot. Abstractly, such a specification corresponds to a field homomorphism $Q(√\overline{−1})$ or $K → R$. These are called real embeddings and complex embeddings, respectively.

A real quadratic field $K → C$, with $Q(√\overline{a})$, and $a ∈ Q, a > 0$ not a perfect square, is so-called because it admits two real embeddings but no complex embeddings. These are the field homomorphisms which send $a$ to $√\overline{a}$ and to $√\overline{a}$, respectively. Dually, an imaginary quadratic field $−√\overline{a}$ admits no real embeddings but admits a conjugate pair of complex embeddings. One of these embeddings sends $Q(√\overline{−a})$ to $√\overline{−a}$, while the other sends it to its complex conjugate, $√\overline{−a}$.

Conventionally, the number of real embeddings of $−√\overline{−a}$ is denoted $K$, while the number of conjugate pairs of complex embeddings is denoted $r_{1}$. The signature of K is the pair $r_{2}$. It is a theorem that $(r_{1}, r_{2})$, where $r_{1} + 2r_{2} = d$ is the degree of $d$.

Considering all embeddings at once determines a function $$M \colon K \to \mathbf{R}^{r_1} \oplus \mathbf{C}^{r_2}$$, or equivalently $$M \colon K \to \mathbf{R}^{r_1} \oplus \mathbf{R}^{2r_2}.$$ This is called the Minkowski embedding.

The subspace of the codomain fixed by complex conjugation is a real vector space of dimension $K$ called Minkowski space. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements of $d$ by an element $K$ corresponds to multiplication by a diagonal matrix in the Minkowski embedding. The dot product on Minkowski space corresponds to the trace form $$\langle x, y \rangle = \operatorname{Tr}(xy)$$.

The image of $x ∈ K$ under the Minkowski embedding is a $O$-dimensional lattice. If $d$ is a basis for this lattice, then $B$ is the discriminant of $det B^{T}B$. The discriminant is denoted $O$ or $Δ$. The covolume of the image of $D$ is $$\sqrt{|\Delta|}$$.

Places
Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based on valuations. Consider, for example, the integers. In addition to the usual absolute value function |·| : Q → R, there are p-adic absolute value functions |·|p : Q → R, defined for each prime number p, which measure divisibility by p. Ostrowski's theorem states that these are all possible absolute value functions on Q (up to equivalence). Therefore, absolute values are a common language to describe both the real embedding of Q and the prime numbers.

A place of an algebraic number field is an equivalence class of absolute value functions on K. There are two types of places. There is a $$\mathfrak{p}$$-adic absolute value for each prime ideal $$\mathfrak{p}$$ of O, and, like the p-adic absolute values, it measures divisibility. These are called finite places. The other type of place is specified using a real or complex embedding of K and the standard absolute value function on R or C. These are infinite places. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there are $O$ real places and $r_{1}$ complex places. Because places encompass the primes, places are sometimes referred to as primes. When this is done, finite places are called finite primes and infinite places are called infinite primes. If $r_{2}$ is a valuation corresponding to an absolute value, then one frequently writes $$v \mid \infty$$ to mean that $v$ is an infinite place and $$v \nmid \infty$$ to mean that it is a finite place.

Considering all the places of the field together produces the adele ring of the number field. The adele ring allows one to simultaneously track all the data available using absolute values. This produces significant advantages in situations where the behavior at one place can affect the behavior at other places, as in the Artin reciprocity law.

Places at infinity geometrically
There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let $$k = \mathbb{F}_q$$ and $$X/k$$ be a smooth, projective, algebraic curve. The function field $$F = k(X)$$ has many absolute values, or places, and each corresponds to a point on the curve. If $$X$$ is the projective completion of an affine curve $$\hat{X} \subset \mathbb{A}^n$$ then the points in $$X - \hat{X}$$ correspond to the places at infinity. Then, the completion of $$F$$ at one of these points gives an analogue of the $$p$$-adics.

For example, if $$X = \mathbb{P}^1$$ then its function field is isomorphic to $$k(t)$$ where $$t$$ is an indeterminant and the field $$F$$ is the field of fractions of polynomials in $$t$$. Then, a place $$v_p$$ at a point $$p \in X$$ measures the order of vanishing or the order of a pole of a fraction of polynomials $$p(x)/q(x)$$ at the point $$p$$. For example, if $$p = [2:1]$$, so on the affine chart $$x_1 \neq 0$$ this corresponds to the point $$2 \in \mathbb{A}^1$$, the valuation $$v_2$$ measures the order of vanishing of $$p(x)$$ minus the order of vanishing of $$q(x)$$ at $$2$$. The function field of the completion at the place $$v_2$$ is then $$k((t-2))$$ which is the field of power series in the variable $$t-2$$, so an element is of the form"$\begin{align} &a_{-k}(t-2)^{-k} + \cdots + a_{-1}(t-2)^{-1} + a_0 + a_1(t-2) + a_2(t-2)^2 + \cdots \\ &=\sum_{n = -k}^{\infty} a_n(t-2)^n \end{align}$"for some $$k \in \mathbb{N}$$. For the place at infinity, this corresponds to the function field $$k((1/t))$$ which are power series of the form"$\sum_{n=-k}^\infty a_n(1/t)^n$"

Units
The integers have only two units, $v$ and $1$. Other rings of integers may admit more units. The Gaussian integers have four units, the previous two as well as $−1$. The Eisenstein integers $±i$ have six units. The integers in real quadratic number fields have infinitely many units. For example, in $Z[exp(2πi / 3)]$, every power of $Z[√\overline{3}]$ is a unit, and all these powers are distinct.

In general, the group of units of $2 + √\overline{3}$, denoted $O$, is a finitely generated abelian group. The fundamental theorem of finitely generated abelian groups therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of the roots of unity that lie in $O^{×}$. This group is cyclic. The free part is described by Dirichlet's unit theorem. This theorem says that rank of the free part is $O$. Thus, for example, the only fields for which the rank of the free part is zero are $r_{1} + r_{2} − 1$ and the imaginary quadratic fields. A more precise statement giving the structure of O× ⊗Z Q as a Galois module for the Galois group of K/Q is also possible.

The free part of the unit group can be studied using the infinite places of $Q$. Consider the function

 * $$\begin{cases} L: K^\times \to \mathbf{R}^{r_1 + r_2} \\ L(x) = (\log |x|_v)_v \end{cases}$$

where $K$ varies over the infinite places of $v$ and |·|v is the absolute value associated with $K$. The function $v$ is a homomorphism from $L$ to a real vector space. It can be shown that the image of $K^{×}$ is a lattice that spans the hyperplane defined by $$x_1 + \cdots + x_{r_1 + r_2} = 0.$$ The covolume of this lattice is the regulator of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, the idele class group, that describes both the quotient by this lattice and the ideal class group.

Zeta function
The Dedekind zeta function of a number field, analogous to the Riemann zeta function, is an analytic object which describes the behavior of prime ideals in $O^{×}$. When $K$ is an abelian extension of $K$, Dedekind zeta functions are products of Dirichlet L-functions, with there being one factor for each Dirichlet character. The trivial character corresponds to the Riemann zeta function. When $Q$ is a Galois extension, the Dedekind zeta function is the Artin L-function of the regular representation of the Galois group of $K$, and it has a factorization in terms of irreducible Artin representations of the Galois group.

The zeta function is related to the other invariants described above by the class number formula.

Local fields
Completing a number field K at a place w gives a complete field. If the valuation is Archimedean, one obtains R or C, if it is non-Archimedean and lies over a prime p of the rationals, one obtains a finite extension $$K_w/\mathbf{Q}_p:$$ a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example, the Kronecker–Weber theorem can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.

Finiteness of the class group
One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. This is a consequence of Minkowski's theorem since there are only finitely many Integral ideals with norm less than a fixed positive integer page 78. The order of the class group is called the class number, and is often denoted by the letter h.

Dirichlet's unit theorem
Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units O× of the ring of integers O. Specifically, it states that O× is isomorphic to G × Zr, where G is the finite cyclic group consisting of all the roots of unity in O, and r = r1 + r2 − 1 (where r1 (respectively, r2) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of K). In other words, O× is a finitely generated abelian group of rank r1 + r2 − 1 whose torsion consists of the roots of unity in O.

Reciprocity laws
In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states
 * $$ \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.$$

A reciprocity law is a generalization of the law of quadratic reciprocity.

There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin's reformulated reciprocity law states that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.

Class number formula
The class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.

Related areas
Algebraic number theory interacts with many other mathematical disciplines. It uses tools from homological algebra. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry. Moreover, the study of higher-dimensional schemes over Z instead of number rings is referred to as arithmetic geometry. Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds.

title: Algebraic number theory


text:
Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols $$\nabla\cdot\nabla$$, $$\nabla^2$$ (where $$\nabla$$ is the nabla operator), or $$\Delta$$. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian $Δf (p)$ of a function $f$ at a point $p$ measures by how much the average value of $f$ over small spheres or balls centered at $p$ deviates from $f (p)$.

The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation $Δf = 0$ are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.

The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow; the wave equation describes wave propagation; and the Schrödinger equation describes the wave function in quantum mechanics. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology.

Definition
The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence ($$\nabla \cdot$$) of the gradient ($$\nabla f$$). Thus if $$f$$ is a twice-differentiable real-valued function, then the Laplacian of $$f$$ is the real-valued function defined by:

where the latter notations derive from formally writing: $$\nabla = \left ( \frac{\partial }{\partial x_1}, \ldots , \frac{\partial }{\partial x_n} \right ).$$ Explicitly, the Laplacian of $f$ is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates $x_{i}$:

As a second-order differential operator, the Laplace operator maps $C$ functions to $C$ functions for $k ≥ 2$. It is a linear operator $Δ : C(R^{n}) → C(R^{n})$, or more generally, an operator $Δ : C(Ω) → C(Ω)$ for any open set $Ω ⊆ R^{n}$.

Diffusion
In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium. Specifically, if $u$ is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of $u$ through the boundary $∂V$ of any smooth region $V$ is zero, provided there is no source or sink within $V$: $$\int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0,$$ where $n$ is the outward unit normal to the boundary of $V$. By the divergence theorem, $$\int_V \operatorname{div} \nabla u\, dV = \int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0.$$

Since this holds for all smooth regions $V$, one can show that it implies: $$\operatorname{div} \nabla u = \Delta u = 0.$$ The left-hand side of this equation is the Laplace operator, and the entire equation $Δu = 0$ is known as Laplace's equation. Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.

The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation. This interpretation of the Laplacian is also explained by the following fact about averages.

Averages
Given a twice continuously differentiable function $$f : \R^n \to \R $$ and a point $$p\in\R^n$$, the average value of $$f $$ over the ball with radius $$h$$ centered at $$p$$ is: $$\overline{f}_B(p,h)=f(p)+\frac{\Delta f(p)}{2(n+2)} h^2 +o(h^2) \quad\text{for}\;\; h\to 0$$

Similarly, the average value of $$f $$ over the sphere (the boundary of a ball) with radius $$h$$ centered at $$p$$ is: $$\overline{f}_S(p,h)=f(p)+\frac{\Delta f(p)}{2n} h^2 +o(h^2) \quad\text{for}\;\; h\to 0.$$

Density associated with a potential
If $φ$ denotes the electrostatic potential associated to a charge distribution $q$, then the charge distribution itself is given by the negative of the Laplacian of $φ$: $$q = -\varepsilon_0 \Delta\varphi,$$ where $ε_{0}$ is the electric constant.

This is a consequence of Gauss's law. Indeed, if $V$ is any smooth region with boundary $∂V$, then by Gauss's law the flux of the electrostatic field $E$ across the boundary is proportional to the charge enclosed: $$\int_{\partial V} \mathbf{E}\cdot \mathbf{n}\, dS = \int_V \operatorname{div}\mathbf{E}\,dV=\frac1{\varepsilon_0}\int_V q\,dV.$$ where the first equality is due to the divergence theorem. Since the electrostatic field is the (negative) gradient of the potential, this gives: $$-\int_V \operatorname{div}(\operatorname{grad}\varphi)\,dV = \frac1{\varepsilon_0} \int_V q\,dV.$$

Since this holds for all regions $V$, we must have $$\operatorname{div}(\operatorname{grad}\varphi) = -\frac 1 {\varepsilon_0}q$$

The same approach implies that the negative of the Laplacian of the gravitational potential is the mass distribution. Often the charge (or mass) distribution are given, and the associated potential is unknown. Finding the potential function subject to suitable boundary conditions is equivalent to solving Poisson's equation.

Energy minimization
Another motivation for the Laplacian appearing in physics is that solutions to $Δf = 0$ in a region $U$ are functions that make the Dirichlet energy functional stationary: $$ E(f) = \frac{1}{2} \int_U \lVert \nabla f \rVert^2 \,dx.$$

To see this, suppose $f : U → R$ is a function, and $u : U → R$ is a function that vanishes on the boundary of $U$. Then: $$\left. \frac{d}{d\varepsilon}\right|_{\varepsilon = 0} E(f+\varepsilon u) = \int_U \nabla f \cdot \nabla u \, dx = -\int_U u \, \Delta f\, dx $$

where the last equality follows using Green's first identity. This calculation shows that if $Δf = 0$, then $E$ is stationary around $f$. Conversely, if $E$ is stationary around $f$, then $Δf = 0$ by the fundamental lemma of calculus of variations.

Two dimensions
The Laplace operator in two dimensions is given by:

In Cartesian coordinates, $$\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$$ where $x$ and $y$ are the standard Cartesian coordinates of the $xy$-plane.

In polar coordinates, $$\begin{align} \Delta f &= \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} \\ &= \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2}, \end{align}$$ where $r$ represents the radial distance and $θ$ the angle.

Three dimensions
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.

In Cartesian coordinates, $$\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.$$

In cylindrical coordinates, $$\Delta f = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2 },$$ where $$\rho$$ represents the radial distance, $φ$ the azimuth angle and $z$ the height.

In spherical coordinates: $$\Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},$$ or $$\Delta f = \frac{1}{r} \frac{\partial^2}{\partial r^2} (r f) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},$$ by expanding the first term, these expressions read $$\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},$$

where $φ$ represents the azimuthal angle and $θ$ the zenith angle or co-latitude.

In general curvilinear coordinates ($ξ^{1}, ξ^{2}, ξ^{3}$): $$\Delta = \nabla \xi^m \cdot \nabla \xi^n \frac{\partial^2}{\partial \xi^m \, \partial \xi^n} + \nabla^2 \xi^m \frac{\partial}{\partial \xi^m } = g^{mn} \left(\frac{\partial^2}{\partial\xi^m \, \partial\xi^n} - \Gamma^{l}_{mn}\frac{\partial}{\partial\xi^l} \right),$$

where summation over the repeated indices is implied, $g^{mn}$ is the inverse metric tensor and $Γ^{l} _{mn}$ are the Christoffel symbols for the selected coordinates.

$N$ dimensions
In arbitrary curvilinear coordinates in $N$ dimensions ($ξ^{1}, ..., ξ^{N}$), we can write the Laplacian in terms of the inverse metric tensor, $$ g^{ij} $$: $$\Delta = \frac 1{\sqrt{\det g}}\frac{\partial}{\partial\xi^i} \left( \sqrt{\det g} g^{ij} \frac{\partial}{\partial \xi^j}\right) ,$$ from the Voss-Weyl formula for the divergence.

In spherical coordinates in $N$ dimensions, with the parametrization $x = rθ ∈ R^{N}$ with $r$ representing a positive real radius and $θ$ an element of the unit sphere $S^{N−1}$, $$ \Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{N-1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \Delta_{S^{N-1}} f$$ where $Δ_{S^{N−1}}|undefined$ is the Laplace–Beltrami operator on the $(N − 1)$-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as: $$\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \left(r^{N-1} \frac{\partial f}{\partial r} \right).$$

As a consequence, the spherical Laplacian of a function defined on $S^{N−1} ⊂ R^{N}$ can be computed as the ordinary Laplacian of the function extended to $R^{N}∖{0}$ so that it is constant along rays, i.e., homogeneous of degree zero.

Euclidean invariance
The Laplacian is invariant under all Euclidean transformations: rotations and translations. In two dimensions, for example, this means that: $$\Delta ( f(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)) = (\Delta f)(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)$$ for all θ, a, and b. In arbitrary dimensions, $$\Delta (f\circ\rho) =(\Delta f)\circ \rho$$ whenever ρ is a rotation, and likewise: $$\Delta (f\circ\tau) =(\Delta f)\circ \tau$$ whenever τ is a translation. (More generally, this remains true when ρ is an orthogonal transformation such as a reflection.)

In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.

Spectral theory
The spectrum of the Laplace operator consists of all eigenvalues $λ$ for which there is a corresponding eigenfunction $f$ with: $$-\Delta f = \lambda f.$$

This is known as the Helmholtz equation.

If $Ω$ is a bounded domain in $R^{n}$, then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space $L^{2}(Ω)$. This result essentially follows from the spectral theorem on compact self-adjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and the Rellich–Kondrachov theorem). It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When $Ω$ is the $n$-sphere, the eigenfunctions of the Laplacian are the spherical harmonics.

Vector Laplacian
The vector Laplace operator, also denoted by $$\nabla^2$$, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.

The vector Laplacian of a vector field $$ \mathbf{A} $$ is defined as $$ \nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A}). $$ This definition can be seen as the Helmholtz decomposition of the vector Laplacian.

In Cartesian coordinates, this reduces to the much simpler form as $$ \nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z), $$ where $$A_x$$, $$A_y$$, and $$A_z$$ are the components of the vector field $$\mathbf{A}$$, and $$ \nabla^2 $$ just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product.

For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates.

Generalization
The Laplacian of any tensor field $$\mathbf{T}$$ ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor: $$\nabla ^2\mathbf{T} = (\nabla \cdot \nabla) \mathbf{T}.$$

For the special case where $$\mathbf{T}$$ is a scalar (a tensor of degree zero), the Laplacian takes on the familiar form.

If $$\mathbf{T}$$ is a vector (a tensor of first degree), the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the Jacobian matrix shown below for the gradient of a vector: $$\nabla \mathbf{T}= (\nabla T_x, \nabla T_y, \nabla T_z) = \begin{bmatrix} T_{xx} & T_{xy} & T_{xz} \\ T_{yx} & T_{yy} & T_{yz} \\ T_{zx} & T_{zy} & T_{zz} \end{bmatrix} , \text{ where } T_{uv} \equiv \frac{\partial T_u}{\partial v}.$$

And, in the same manner, a dot product, which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices: $$ \mathbf{A} \cdot \nabla \mathbf{B} = \begin{bmatrix} A_x & A_y & A_z \end{bmatrix} \nabla \mathbf{B} = \begin{bmatrix} \mathbf{A} \cdot \nabla B_x & \mathbf{A} \cdot \nabla B_y & \mathbf{A} \cdot \nabla B_z \end{bmatrix}.$$ This identity is a coordinate dependent result, and is not general.

Use in physics
An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow: $$\rho \left(\frac{\partial \mathbf{v}}{\partial t}+ ( \mathbf{v} \cdot \nabla ) \mathbf{v}\right)=\rho \mathbf{f}-\nabla p +\mu\left(\nabla ^2 \mathbf{v}\right),$$ where the term with the vector Laplacian of the velocity field $$\mu\left(\nabla ^2 \mathbf{v}\right)$$ represents the viscous stresses in the fluid.

Another example is the wave equation for the electric field that can be derived from Maxwell's equations in the absence of charges and currents: $$\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0.$$

This equation can also be written as: $$\Box\, \mathbf{E} = 0,$$ where $$\Box\equiv\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2,$$ is the D'Alembertian, used in the Klein–Gordon equation.

Generalizations
A version of the Laplacian can be defined wherever the Dirichlet energy functional makes sense, which is the theory of Dirichlet forms. For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.

Laplace–Beltrami operator
The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The Laplace–Beltrami operator, when applied to a function, is the trace ($tr$) of the function's Hessian: $$\Delta f = \operatorname{tr}\big(H(f)\big)$$ where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative, in terms of which the "geometer's Laplacian" is expressed as $$ \Delta f = \delta d f .$$

Here $δ$ is the codifferential, which can also be expressed in terms of the Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on differential forms $α$ by $$\Delta \alpha = \delta d \alpha + d \delta \alpha .$$

This is known as the Laplace–de Rham operator, which is related to the Laplace–Beltrami operator by the Weitzenböck identity.

D'Alembertian
The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic.

In Minkowski space the Laplace–Beltrami operator becomes the D'Alembert operator $$\Box$$ or D'Alembertian: $$\square = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}.$$

It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy particle physics. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the wave equations, and it is also part of the Klein–Gordon equation, which reduces to the wave equation in the massless case.

The additional factor of $c$ in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the $x$ direction were measured in meters while the $y$ direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that $c = 1$ in order to simplify the equation.

The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds.

title: Laplace operator


text:
Div

Div or DIV may refer to:

Science and technology

 * Division (mathematics), the mathematical operation that is the inverse of multiplication
 * and, HTML tags that implement generic elements
 * , a C mathematical function
 * Divergence, a mathematical operation in vector calculus
 * Days in vitro, for example see Cultured neuronal network
 * Desquamative inflammatory vaginitis, an uncommon acute inflammation of the vagina; see Vulva disease

Other uses

 * Diversity Immigrant Visa, a United States congressionally mandated lottery program for receiving a United States Permanent Resident Card
 * 504 (number) (DIV), in Roman numerals
 * Div (mythology), a demon in Middle Eastern mythology
 * Divisi or div., a music term used in orchestral scores
 * Div, a character in the Penny Arcade
 * Divorce, a process in which a married couple breaks up and their marriage license is nullified.

title: Div


text:
Grad

Grad or grads may refer to:

Places

 * Grad (toponymy) (Cyrillic: Град) is a Slavic word meaning "town", "city", "castle" or "fortified settlement" that appears in numerous Slavic toponyms

 * Specific places named Grad:
 * Grad (Dubrovnik) - colloquial name for Grad (Ragusa) and present-day Dubrovnik by its inhabitants, neighboring and hinterlands population.
 * Grad, Visoko, Bosnia and Herzegovina
 * Grad, Delčevo, North Macedonia
 * Grąd, Podlaskie Voivodeship, Poland
 * Grąd, West Pomeranian Voivodeship, Poland
 * Grad, Cerklje na Gorenjskem, Slovenia
 * Municipality of Grad, Slovenia
 * Grad, Grad, a village, the seat of the municipality
 * Grad, Split, an administrative division of Split, Croatia

People

 * Grad (surname)

Arts, entertainment, and media

 * Grad (EP), a Cactus Jack EP
 * Grad, the dragon in Ral Grad

Education

 * Grad, short for a graduate, one who has graduated from an education program

Geometry and measurement

 * Gradian, a unit of angular measurement
 * Gradient of a scalar field, a differential operator in mathematics
 * Grad, a small unit in tuning very close to the schisma, which it is also called
 * abbreviation of gigaradian (Grad), a unit of angle
 * abbreviation of gigarad (Grad), a unit of radiation dose

Other uses

 * BM-21 Grad, a Soviet multiple-launch rocket system and an associated series of 122 mm artillery rockets
 * Grad Associates, an architectural firm based in Newark, New Jersey
 * GrADS, a free software for geophysical data visualization

title: Grad


text:
Frederick I of Denmark

Frederick I (Danish and Frederik; Friedrich; Fredrik; 7 October 1471 – 10 April 1533) was King of Denmark and Norway. He was the last Catholic monarch to reign over Denmark and Norway, when subsequent monarchs embraced Lutheranism after the Protestant Reformation. As king of Norway, Frederick is most remarkable in never having visited the country and was never crowned as such. Therefore, he was styled King of Denmark, the Vends and the Goths, elected King of Norway. Frederick's reign began the enduring tradition of calling kings of Denmark alternatively by the names Christian and Frederick.

Background
Frederick was the younger son of the first Oldenburg King Christian I of Denmark, Norway and Sweden (1426–81) and of Dorothea of Brandenburg (1430–95). Soon after the death of his father, the underage Frederick was elected co-Duke of Schleswig and Holstein in 1482, the other co-duke being his elder brother, King John of Denmark. In 1490 at Frederick's majority, both duchies were divided between the brothers.

In 1500, he had convinced his brother King John to conquer Dithmarschen. A great army was called from not only the duchies, but with additions from all of the Kalmar Union for which his brother briefly was king. In addition, numerous German mercenaries took part. The expedition failed miserably, however, in the Battle of Hemmingstedt, where one-third of all knights of Schleswig and Holstein lost their lives.

Reign
When his brother, King John died, a group of Jutish nobles had offered Frederick the throne as early as 1513, but he had declined, rightly believing that the majority of the Danish nobility would be loyal to his nephew Christian II. In 1523, Christian was forced by disloyal nobles to abdicate as king of Denmark and Norway, and Frederick took the throne of Denmark in 1523 and was elected king of Norway in 1524. It is not certain that Frederick ever learned to speak Danish. After becoming king, he continued spending most of his time at Gottorp, a castle and estate in the city of Schleswig.

In 1524 and 1525, Frederick had to suppress revolts among the peasants in Agder, Jutland and Scania who demanded the restoration of Christian II. The high point of the rebellion came in 1525 when Søren Norby, the governor (statholder) of Gotland, invaded Blekinge in an attempt to restore Christian II to power. He raised 8000 men who besieged Kärnan (Helsingborgs slott), a castle in Helsingborg. Frederick's general, Johann Rantzau, moved his army to Scania and defeated the peasants soundly in April and May 1525.

Frederick played a central role in the spread of Lutheran teachings throughout Denmark. In his coronation charter, he was made the solemn protector (værner) of the Catholic Church in Denmark. In that role, he asserted his right to select bishops for the Catholic dioceses in the country. Christian II had been intolerant of Protestant teaching, but Frederick took a more opportunist approach. For example, he ordered that Catholics and Lutherans share the same churches and encouraged the first publication of the Bible in the Danish language. In 1526, when Lutheran Reformer Hans Tausen was threatened with arrest and trial for heresy, Frederick appointed him his personal chaplain to give him immunity.

Starting in 1527, Frederick authorized the closure of Franciscan houses and monasteries in 28 Danish cities. He used the popular anti-establishment feelings that ran against some persons of the Catholic hierarchy and nobility of Denmark as well as keen propaganda to decrease the power of bishops and Catholic nobles. During his reign, Frederick was skillful enough to prevent all-out warfare between Catholics and Protestants. In 1532, he succeeded in capturing Christian II who had tried to invade Norway, and to make himself king of the country. Frederick died on 10 April 1533 in Gottorp, at the age of 61, and was buried in Schleswig Cathedral. Upon Frederick's death, tensions between Catholics and Protestants rose to a fever pitch which would result in the Count's Feud (Grevens Fejde).

Family and children
On 10 April 1502, Frederick married Anna of Brandenburg (1487–1514), the daughter of John Cicero, Elector of Brandenburg and Margaret of Thuringia. The couple had two children:

 * 1) Christian III, King of Denmark and Norway (12 August 1503 – 1 January 1559)
 * 2) Dorothea of Denmark (1 August 1504 – 11 April 1547), married 1 July 1526 to Albert, Duke of Prussia.

Frederick's wife Anna died on 5 May 1514, 26 years old. Four years later on 9 October 1518 at Kiel, Frederick married Sophie of Pomerania (20 years old; 1498–1568), a daughter of Bogislaw "the Great", Duke of Pomerania. Sophie and Frederick had six children:

 * 1) John II of Denmark, Duke of Schleswig-Holstein-Haderslev (28 June 1521 – 2 October 1580)
 * 2) Elizabeth of Denmark (14 October 1524 – 15 October 1586), married:
 * 3) on 26 August 1543 to Magnus III of Mecklenburg-Schwerin.
 * 4) on 14 February 1556 to Ulrich III, Duke of Mecklenburg-Güstrow.
 * 5) Adolf of Denmark, Duke of Holstein-Gottorp (25 January 1526 – 1 October 1586)
 * 6) Anna of Denmark (1527 – 4 June 1535)
 * 7) Dorothea of Denmark (1528 – 11 November 1575), married on 27 October 1573 to Christopher, Duke of Mecklenburg-Gadebusch.
 * 8) Frederick of Denmark (13 April 1532 – 27 October 1556), Prince-Bishop of Hildesheim and Bishop of Schleswig.

title: Frederick I of Denmark