Dirac delta function

In mathematics, the Dirac delta function is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one; as there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero; the Kronecker delta function, defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function. In engineering and signal processing, the delta function known as the unit impulse symbol, may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable.

The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit of a sequence of functions having a tall spike at the origin; the approximating functions of the sequence are thus "approximate" or "nascent" delta functions. The graph of the delta function is thought of as following the whole x-axis and the positive y-axis; the Dirac delta is used to model a tall narrow spike function, other similar abstractions such as a point charge, point mass or electron point. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a delta function. In doing so, one not only simplifies the equations, but one is able to calculate the motion of the ball by only considering the total impulse of the collision without a detailed model of all of the elastic energy transfer at subatomic levels. To be specific, suppose that a billiard ball is at rest.

At time t = 0 it is struck by another ball, in kg m / s. The exchange of momentum is not instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as instantaneous; the force therefore is P δ. To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval Δ t; that is, F Δ t = { P / Δ t 0 < t < Δ t, 0 otherwise. The momentum at any time t is found by integration: p = ∫ 0 t F Δ t d τ = { P t > Δ t P t / Δ t 0 < t < Δ t 0 otherwise. Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δ t → 0, giving p = { P t > 0 0 t ≤ 0. Here the functions F Δ t are thought of as useful approximations to the idea of instantaneous transfer of momentum; the delta function allows us to construct an idealized limit of these approximations. The actual limit of the functions lim Δ t → 0 F Δ t is zero everywhere but a single point, where it is infinite.

To make proper sense of the delta function, we should instead insist that the property ∫ − ∞ ∞ F Δ t d t = P, which holds for all Δ t > 0, should continue to hold in the limit. So, in the equation

List of Arka S─▒radakiler characters

Along with the Büyük Zafer crew, Arka Sıradakiler includes a large array of characters: students, families and fictional characters. The main characters, teacher Kemal and the crew, are listed first. Only main and recurring characters are listed. Bülent Emin Yarar as Kemal, an idealist teacher whose purpose is to recover the students who have problems, he tries to communicate with students. He cares about his students more than himself, he has dedicated his life to performing his duty up to his best and helping his students to be healthy, productive individuals. After the arc of Zehra's being kidnapped, he transferred to another school according to Arka Sıradakiler, he continued his career as a teacher in Öğretmen Kemal TV series. Bülent Çetinaslan as Oktay, a student, exposed to violence by his father, he reflects this violence to things around him. His friends regard him as a leader, he tries to create the justice. He is brave enough to take great risks and he does anything to help and protect his friends.

In the 160th episode, Pınar caused his death alongside with Gamze. Sinem Öztürk as Gamze, Oktay's girlfriend, she doesn't get herself involved in gang affairs. Her family pays any attention to her so she's unhappy at home and when her boyfriend Oktay puts himself into trouble at school she loses her temper. She's worried about his friends, her greatest aspiration is a happy family. In the 160th episode, Pınar caused her death alongside with Oktay. Barış Atay Mengüllü as one of the former students of Büyük Zafer High School, he was kicked out of school. He was left in an orphanage at a early age, he plays the vagabond with his friends around the school. While he had maintained his territory before appearance of Oktay and his gang, he waged war on them for the control of the area, he sneaky misses his school years. He is accepted to the school again in further episodes. In the fourth and last season, he gives up on Gamze and has an affair with Zehra, their relationship furthers and they fall in love with each other.

Their love ends under a train, after Zehra marries her rapist and kills him. Zehra and Saffet end their lives together unfortunately. Fırat Çöloğlu as İbo, one of Oktay's closest mates, a funny guy; as his father left them a long time ago, he is living alone with his mother. His father is an alcoholic and although he shares everything with his friends, he is ashamed of it, he has never known his father. His greatest aspiration is to save his mother by striking it rich the easy way. In further episodes, his father returns. In the fourth and last season, he loses the love of his life: Sibel. A woman, resembling her, named Şebnem becomes his lover. Deniz Sarıbaş as Sibel, the mysterious girl of Arka Sıradakiler. She's bothered with neither the things happening around the school, she has a great secret. She prostitutes herself, she has accepted this way of making life. She thinks, her greatest fear is. She starts going out with İbo in further episodes, but her stalker kills her, driving his boyfriend, İbo to lose his mind.

Deniz Sarıbaş as Şebnem, After Sibel dies in episode 150, Deniz Sarıbaş reappears in episode 155 as Şebnem. Temmuz Karikutal as Yadigar, a close friend of Oktay. A secretive, trustworthy guy. Although he's rather strict with his friends, he gets unexpectedly easygoing when the case is the girl he loves, he is so loyal to his friends. As keen as mustard on his friends, he's tough, cheerful, witty. Due to an unfortunate accident, he loses his ability to walk, he runs away, hiding this secret from his closest friends; when he returns, he wins back the advances of the girl: Eda. Meanwhile, he tries to walk again, with the help of Buket's mother. Sevda Dalgıç as Özge, a rebel girl determined not to live the rage of puberty in herself, she never holds back from reacting to situations. Pessimistically, she believes. Instead of working to change his bad luck to save herself, she chooses to rebelling to this situation and demonstrating her response by anger and violence. In the fourth and last season, she returned Pamuk's love.

On the 160th episode she and her boyfriend due to Şebnem. Pelin Akil as Zehra, Kemal's niece, she can't accommodate herself to the environment of Büyük Zafer High School after a childhood of freedom in the Netherlands. She sometimes steals to seek excitement, she doesn't hesitate to lie to reach her aim. Her only passion in life is music and she can get bored of anything else. Whimsey, unreliable, she is so fond of her freedom that anything can be expected from her at any moment, her attitude changes after Sedat's rape. After this incident she was forced for prostitution; as she refused, she was made addicted to heroine. She was saved by Ali and this made her fall in love with him, but this was a misunderstanding, as revealed with a kiss shared with Saffet. She chose Saffet, but her stalker Sedat and Ali and Saffet's another friend Memo prevented their happiness. So she decided to marry Sedat, confusing


Havuş is a village and municipality in the Sharur District of Nakhchivan Autonomous Republic, Azerbaijan. It is located 35 km in the west from the district center, in about 2,000 meters above sea level, on the slope of the Daralayaz ridge, the border with Armenia. There are club and a medical center in the village, it has a population of 114. In its territory are located the settlement of Tamaşalı and cemetery of the Middle Ages. According to some researchers, avuş is a distorted form of the "ağuş" word. In the Persian the word of ağuş means "embrace", "with open arms"; as if, the village is so called because it is situated at the foot of Mount Avuş. In fact, the village took its name from the same named spring. Avuş spring - According to information of local people, the name of the spring is from word of ovuc, it was so called as the shape it is similar to the handful. In the Turkic languages, avuş/avuj means "vaccine, alum", and this show the existence of the same material in the content of the water of spring.

According to official information dated on 1 January 1914 ethnic Tatars, consisting of a total of 346 people in both sexes lived in the Avuş village of Sharur-Daralayaz uyezd of Erivan Governorate. In May 1992, a part of its population have been forced to move to the other villages in associated with the armed aggression by Armenia. Havuş Maiden's Tower - the monument on the steepness rocks, in the east of the same named village in the Sharur region; the inside of the rock is consist from the ending of the shaped crown. It is called among the people the "Maiden House", "Maiden Tower", it is supposed. Havuş Necropolis - an ancient cemetery on the mountain slope near the same named village in the Sharur region; as a result of exploration researches in surface, were collected samples of material culture. According to the findings, it is supposed that the Havush necropolis is belongs to the II - IV centuries of AD. Havush Sanctuary - the cave among the steepness rocks in the north east of the same named village in the Sharur region.

Depth is more than 15 m. Width 3, in some places is 4 meters; the Entrance was carved in form of crown-shaped. There are image of several fighters with spears in hand which were engraved with a method of scratching and digging at the inside of the cave. There was carried out the cleaning work in size of 1х1 m, have been found human skeletons at a depth of 40 cm and sharp black arm fracture was found at a depth of 1 m in the cave; the cave is supposed to be an ancient human habitation, converted to sacred place is related to a belief in their ancestors and their spirits. Havuş at GEOnet Names Server