Syed Muhammad Fazal Agha (; (1946 – 20 May 2020) was a Pakistani politician. He was Governor of Balochistan from August 18 to October 12 1999. He also was Deputy Chairman of the Senate of Pakistan from 1988-1991. from Balochistan. He was a member of the Provincial Assembly of Balochistan from 13 August 2018 till his death.
Agha was born in Killi Huramzai, Pishin District. He died on 20 May 2020 from COVID-19 in Karachi, aged 78.
Euler's formula
In complex analysis, Euler's formula, also sometimes called Euler's relation, is an equation involving complex numbers and trigonometric functions. More specifically, it states that
formula_1
where "x" is a real number, "e" is Euler's number and "i" is the imaginary unit.
It makes a relation between trigonometric functions and exponential functions of complex numbers. It is named after Leonhard Euler, who published it in 1748. When he published it, Euler said that the angle must be a real number. Later, it turned out that the formula also works if the angle is not a real number, but a complex one.
When the angle is formula_2 and formula_3, Euler's formula becomes formula_4 and formula_5, respectively.
Atropatene
Atropatene was an ancient kingdom that existed in part of what is now Iran. It existed with Darius III of Persia and Alexander the Great. It started in the 4th century BCE, in what is now northern Iran. Its capital was Ganzak. It ended as a vassal kingdom of the Parthian empire. In 226, the Sasanian empire conquered it. At that time it was probably more of a province of the Parthian empire than a kingdom, even though its rulers were kings.
Today, many people see it as a predecessor of the modern state of Azerbaijan.
Virgin Komi Forests
The Virgin Komi Forests is a large woodland in the northern part of the Ural mountain range, in Russia. The forests are part of the Komi Republic of Russia. The forests cover an area of and are the largest virgin forests in Europe.
They were made a UNESCO World Heritage site in 1995. Most of the area is flat Taiga plain, about above sea level. They rise towards the Ural mountains.
They are threatened because parts of the area contain a lot of gold. Illegal logging is also a threat.
Leo the Last
Leo the Last is a 1970 British drama movie directed by John Boorman and based on the play "The Prince" by George Tabori. It stars Marcello Mastroianni, Billie Whitelaw, Calvin Lockhart, Brinsley Forde, Graham Crowden and was distributed by United Artists.
Knife in the Water
Knife in the Water () is a 1962 Polish drama movie directed by Roman Polanski in his directorial debut and stars Leon Niemczyk, Jolanta Umecka, Zygmunt Malanowi. It was nominated for an Academy Award in 1964.
Numerical methods for partial differential equations
Numerical methods for partial differential equations are computational schemes to obtain approximate solutions of partial differential equations (PDEs).
The scientific journal "Numerical Methods for Partial Differential Equations" is published to promote the studies of this area.
Chebfun is one of the most famous software in this field. They are also many libraries based on the finite element method such as:
Many PDEs appeared for the study of physics and other areas in science. Therefore, many mathematicians have challenged to make methods to solve them, but there is no method to mathematically solve PDEs except the Hirota direct method and the inverse scattering method. This is why numerical methods for PDEs are needed.
One of the most basic PDE solver is the finite difference method (FDM). This method approximates derivatives as differences:
This method works for easy problems. But it is powerless to some equations (such as the Navier–Stokes equations) because they are non-linear. Since this difficulty appeared, numerical analysts started to study other methods (just like the finite element method, FEM). On the other hand, some experts started to consider improvements for FDM.
Experts have discovered difference methods which preserves the property of the given PDE.
Ryogo Hirota, Mark Ablowitz and others have made methods that preserves the integrability (important mathematical property in the theory of dynamical systems) of PDEs. These methods are known to have better accuaracy than the original FDM.
Many PDEs have appeared from physics. So we can think about difference methods preserving physical properties. These difference methods are known as structure preserving numerical methods. The following list is the examples of them:
Some experts are studying their relation between numerical linear algebra.
The difference mthods in above have high accuracy, but their usage is limited because they depend on the behaviour of the given PDEs. This is why new types of FDM are still studied. For example, the following methods are studied:
Not only approximate solvers, but the study to "verify the existence of solution by computers" is also active. This study is needed because numerically obtained solutions could be phantom solutions (fake solutions). This kind of incident is already reported.
Literatures for specific solvers are described as follows.
Ivan's Childhood
Ivan's Childhood (), sometimes released as My Name Is Ivan is a 1962 Russian World War II drama movie directed by Andrei Tarkovsky and is based on the 1957 short story "Ivan" by Vladimir Bogomolov. It stars Nikolai Burlyaev, Valentin Zubkov, Evgeny Zharikov, Nikolai Grinko and was distributed by Mosfilm.
Strike (1925 movie)
Strike (, translit. "Stachka") is a 1925 Russian drama movie directed by Sergei Eisenstein and stars Maskim Shtraukh, Grigori Aleksandrov, Aleksandr Antonov. It was distributed by Goskino Proletkult.
Ariel
Ariel is a 1988 Finnish drama movie directed and written by Aki Kaurismäki and starring Turo Pajala, Susanna Haavisto, Matti Pellonpää. It was distributed by Finnkino.
Drifting Clouds (movie)
Drifting Clouds () is a 1996 Finnish comedy drama movie directed by Aki Kaurismäki and starring Kati Outinen, Kari Väänänen, Markku Peltola, Elina Salo. It was distributed by Sputnik.
Karen Asatrian
Karen Asatrian (: Կարեն Ասատրյան; born June 23, 1972) is Armenian jazz pianist and composer. His music is a creative synthesis of contemporary jazz and Armenian folk music. His arrangements are a jazzy exploration of the world of oriental melodies and western harmonies. The recent album “Noor” with Karen Asatrian & Armenian Spirit band is an expression of Armenian traditional songs rearranged with a contemporary jazz style.
Karen Asatrian received his first education in classical Violin and Piano at the Tchaikovsky Music School in Yerevan. He then continued his studies of both instruments as well as in composition at the Komitas State Conservatory of Yerevan. In 1999, he moved to Carinthia, Austria, where he finalized his Jazz-Piano studies under the direction of Dr. Prof. Harry Neuwirth, with distinction. In 2001, Karen successfully graduated earning a Bachelor of Fine Arts in Instrumental (Vocal) Pedagogy Studies receiving the state teaching certification for jazzpiano.