Rustic capitals

Rustic capitals is an ancient Roman calligraphic script. Because the term is negatively connotated supposing an opposition to the more'civilized' form of the Roman square capitals, Bernhard Bischoff prefers to call the script canonized capitals; the script was used between the 1st century and the 9th century, most between the 4th and 6th centuries. After the 5th century, rustic capitals began to fall out of use, but they continued to be used as a display script in titles and headings, along with uncial as the script of the main text. Rustic capitals are similar to Roman square capitals, but are less rigid, influenced more by pen and ink writing on papyrus or parchment than the writing used for inscriptions; the letters are thinner and more compressed, use many more curved lines than do square capitals, have descenders extending below the baseline. The scripts written in rustic capitals utilize punctus marks to denote word separation, contrary to the common practice of scriptura continua. About fifty manuscripts with rustic capitals survive, including four copies of works by Virgil, one copy of a work by Terence, one of a work by Prudentius.

The script was used for de luxe copies of pagan authors. Roman cursive'Manual of Latin Palaeography'

Elif (TV series)

Elif is a Turkish TV series that aired weekly on Kanal 7. It tells the story of a 6-year-old girl separated from her mother. Elif, a pretty and kind 6-year-old girl, is in danger of being sold to pay the gambling debts of her stepfather, Veysel Şimşek, her mother Melek escapes, but finds out that she is sick. She leaves Elif in the protection of her best friend, Ayşe Doğan, a maid in a wealthy family's household. Ayşe tries to get permission from the Emiroğlu family to raise Elif in the mansion. However, she is holding a well-hidden secret: Elif is the biological child of Kenan, the beloved eldest son of the wealthy family. Kenan is unaware, he is now married to Arzu Karapınar. They have a daughter, Duğçe Emiroğlu, a spoiled girl known for her bad temper. Elif is forced to live at the luxurious estate, away from her beloved mother, unaware of her proximity to her biological father. Kenan's wife, feels threatened, she and her father, Necdet Karapınar, conspire to hide a secret from Kenan about her daughter, Tuğçe.

Arzu would do anything to keep her position in the family safe. However, as crimes committed by Necdet and Arzu are exposed and Kenan reunite. Melek tries to ensure that Elif will not be separated again. Meanwhile, Kenan's brother Selim and Zeynep fall in love and support each other despite many obstacles, their happiness nearly comes to an end. The Emiroğlu family faces financial difficulties. Kenan and Tuğçe die when Arzu sets the house on fire, though Elif survive. With the death of Kenan, Selim becomes the head of the family-owned business. Gonca marries Serdar. Arzu is released from treatment at the mental hospital and seeks revenge against the Emiroğlu family. Selim is sent to jail. Arzu survives the nearly fatal incident, continues to threaten the safety and well-being of the family. Together with her new husband, Umit they take over their mansion. Melek and Zeynep work to contribute to the family's finances. Arzu changes her name. Selim gets diagnosed with cancer and together with Zeynep and Aliye they emigrate to America for his treatment.

Elif's life changes. During their move to another city, Yusuf is killed in a car accident. Melek and Elif are separated again and Elif finds herself in a hospital. Six months Elif goes to Macide's house and attempts to start a new life, but faces pressure from Tarik, Macide's son-in-law. In pursuit of Macide's legacy, Tarik wants to get rid of Elif. However, Elif has a supporter, who escaped from her family to study in Istanbul, she commits to protecting Elif against the family members. Meanwhile, Melek ends up homeless with lost memory, trying to find her daughter; when her temporary house gets destroyed, she goes to another house, where she's found by İnci, Tülay, Veysel. Tarik begins to atack Melek. Veysel dies killed by Tarik and Melek finds her daughter working in Macide and Kerem's house. Meanwhile and Inci are meeting with Elif and are moving into another city. Tarik is arrested. Someone from Tarik's guys tries to kill both Melek and Elif but they survive. Macide's sister, Kiymet comes after 30 years to revenge her because in the past her husband abandoned her pregnant and married Macide.

Together with her son Mahir they make evil plans against the Haktanir family, however Mahir meets with Melek by chance, falls in love with her and starts abandoning his evil goal while Kiymet puts Melek in jail because she learns that she has heir in the Haktanir family. Mahir sets her out of jail with the evidences pointing Tarik instead of Kiymet. Melek reveals everything about her starts coming out. Mahir confesses to Kerem that he is Kiymet's son and Kiymet being unable to accept the defeat shoots Mahir and Macide while before that she reveals to Mahir that he is not the son of Macide's husband and manipulated him with this lie in order to achieve her revenge plan. Macide dies, Kerem is transferred to hospital injured and Mahir is saved with a light wound. Kiymet is arrested. Mahir and Elif are together and happy. Isabella Damla Güvenilir as Elif Emiroğlu Selin Sezgin as Melek Özer Volkan Çolpan as Kenan Emiroğlu Cemre Melis Çınar as Arzu Karapınar Zeynep Öğren as Tuğçe Emiroğlu Hasan Ballıktaş as Veysel Şimşek Emre Kıvılcım as Selim Emiroğlu Gülçin Tunçok as Zeynep Emiroğlu Esin Benim as Ipek Emiroğlu Aysun Güven as Aliye Emiroğlu Batuhan Soncul as Murat Şimşek Aysegul Yalçiner as Kiraz Ozanay Alpkan as Ayşe Doğan İlker Gürsoy as Melih Özer Dilara Yüzer as Gonca Tunç Kıvılcım Kaya as Efruz Baba Kerem Akdeniz as Sadik Pelin Çalışkanoğlu as Pelin Sinem Akman as Feraye Gürhan Gülbahar as Necdet Karapınar Hak

Vector fields on spheres

In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, early work on the classification of division algebras. The question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in N-dimensional Euclidean space. A definitive answer was made in 1962 by Frank Adams, it was known, by direct construction using Clifford algebras, that there were at least ρ-1 such fields. Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found. In detail, the question applies to the'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers ρ determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere; the case of N odd is taken care of by the Poincaré–Hopf index theorem, so the case N is an extension of that.

Adams showed that the maximum number of continuous pointwise linearly-independent vector fields on the -sphere is ρ − 1. The construction of the fields is related to the real Clifford algebras, a theory with a periodicity modulo 8 that shows up here. By the Gram–Schmidt process, it is the same to ask for linear independence or fields that give an orthonormal basis at each point; the Radon–Hurwitz numbers ρ occur in earlier work of Johann Radon and Adolf Hurwitz on the Hurwitz problem on quadratic forms. For N written as the product of an odd number A and a power of two 2B, write B = c + 4d, 0 ≤ c < 4. Ρ = 2c + 8d. The first few values of ρ are: 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10... For odd n, the value of the function ρ is one; these numbers occur in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n×n matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix.

In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by Beno Eckmann, they are now applied in areas including theoretical physics. Porteous, I. R.. Topological Geometry. Van Nostrand Reinhold. Pp. 336–352. ISBN 0-442-06606-6. Zbl 0186.06304. Miller, H. R. "Vector fields on spheres, etc.". Retrieved 10 November 2018