The first step is usually to take the derivative of the initial function. From here maxima or minima can be found within a domain by simply finding the critical points. These include domain (x) values where the differentiated function equals 0 or does not exist. Simply finding critical points is not enough to determine maxima or minima. The final step to determining a maximum, minimum, or neither can be one of two methods.
(1) X values lesser and lesser than the critical point must be substituted into the differentiated function to determine if the values change sign. If they go from negative to positive, the critical point is a minimum. If they go from positive to negative, the critical point is a maximum. If neither occurs, the point is neither. This is known as the first derivative test.
(2) The derivative of the differentiated function can be taken, creating a second-order derivative. The critical point must be substituted into the second derivative. If the output is positive, the critical point is a minimum. If the output is negative, the critical point is a maximum. This is known as the second derivative test.
The application of optimization span many fields. A simple example is attempting to find the smallest possible difference in the distance of two objects in two-dimensional space (x and y). In this context, the derivative of the function that gives the difference is taken in order to find the minimum. A more complicated example is in Machine Learning, in which the optimization function attempts to find the global minimum of the loss function in order to minimize the difference or loss between the algorithm’s predictions and the actual values. This example is more difficult as Machine learning algorithms often utilize multidimensional data usually in the form of tensors yielding more complicated functions.
Today, there are many tools to support optimization studies:
Arab slave trade
Arab slave trade refers to the practice of slavery in the Arab world. This is mainly in Western Asia, North Africa, Southeast Africa, the Horn of Africa. It also includes parts of Europe (such as Iberia and Sicily) beginning during the era of the Muslim conquests and continuing into the 20th century. The trade was conducted through slave markets in the Middle East, North Africa and the Horn of Africa. Most of the slaves were from Africa's interior.
Historians estimate that between 650 AD and the 1960s, 10 to 18 million people were enslaved by Arab slave traders. They were taken from Europe, Asia and Africa across the Red Sea, Indian Ocean, and Sahara desert.
The Arab trade of Zanj (Bantu) slaves in Southeast Africa is one of the oldest slave trades. It was going on 700 years before the European Atlantic slave trade. Male slaves were often employed as servants, soldiers, or workers by their owners. Most male slaves were castrated. It is estimated that as many as 6 out of every 10 boys bled to death during the process. But the high price of Eunuchs made it worthwhile. Woman and children taken as slaves were mainly used as servants and concubines. While the later Atlantic slave trade concentrated on men for labor, the Arab slave trade shifted over time to concentrate more on woman and young girls for sexual purposes.
In Mauritania slavery was abolished legally in 1980. But Muslim Berbers still own an estimated 90,000 African slaves. This is despite the fact that African Mauritanians converted to Islam over a hundred years ago and the Qur’an does not allow Muslims to enslave other Muslims. Slaves are used for farm labor, concubines and domestic servants. The children remain the property of their masters. They can be bought, sold, or exchanged for trucks, camels or guns.
Slavery in Sudan is active again with the Muslim north waging war against Animists and Christians in the south. In these raids almost all slaves are taken from the tribes in the Nuba Mountains. Government sponsored Arab militias often kill the men and enslave women and children. Those taken slaves are forced to convert to Islam. Those that refuse are put to death. The trafficking in woman and children in Western and Southern Africa violates the 1948 UN Universal Declaration of Human Rights.
John King Jr.
John B. King Jr. (born 1975 in Brooklyn, New York City) is an American politician. He was the Secretary of Education at the U.S. Department of Education from 2016 to 2017. Before, he served as its Acting Deputy Secretary, and from 2011 to 2014 he was the New York State Education Commissioner.
The former Secretary of Education, Arne Duncan, was given the job of putting into effect the No Child Left Behind Act. However, King is required to put into effect the law that replaces that law, the Every Student Succeeds Act.
Sally Jewell
Sarah Margaret Roffey Jewell (born February 21, 1956) is a British-born American businesswoman. She is the 51st United States Secretary of the Interior. She served in the government of President Barack Obama from April 2013 to 2017. She is the second woman to hold the position, after Gale Norton.
Antennae galaxies
The Antennae galaxies (NGC 4038/NGC 4039) are two colliding galaxies. They are in the constellation Corvus. The process takes hundreds of millions of years, maybe even a billion years.
The collision, with its clouds of gas, dust and magnetic fields, causes extremely rapid star formation. This is their starburst phase, and might last a few hundred million years. They were discovered by William Herschel in 1785.
They are in the NGC 4038 group with five other galaxies. Before The Collision, NGC 4038 was a barred spiral galaxy and NGC 4039 was a spiral galaxy. These two galaxies are known as the Antennae galaxies because they have two long tails of stars, gas and dust ejected from the galaxies as a result of tidal force in the collision that look like an insect’s antennae. The nuclei of the two galaxies are joining to become one giant galaxy, sometime in the future to form an Elliptical galaxy.
These interacting galaxies are nearer and less remote to the Milky Way than previously thought—at 45 million light-years instead of 65 million light-years.
Yisroel Belsky
Chaim Yisroel Belsky (August 22, 1938 – January 28, 2016) was an American rabbi and posek of Orthodox and Haredi Judaism. He was one of the roshei yeshivas at Yeshiva Torah Vodaas, and rabbi of the summer camp network run by Agudath Israel of America. He was born in Brooklyn, New York.
Belsky was criticized by other rabbis for his repeated defense of sexual abusers. Some have claimed that he intimidated victims of abuse. Belsky has publicly stated that victims should not report their rape to the police.
Belsky was taken to hospital on January 22, 2016 in Brooklyn. He had a "serious illness". He died eight days later on January 30, at the age of 77.
Jean-Marie Doré
Jean-Marie Doré (12 June 1938 – 29 January 2016) was a Guinean politician. He was the Prime Minister of Guinea from January 2010 until December 2010.
Doré was the President of the Union for the Progress of Guinea (UPG). He was opposition leader for years before being chosen to lead a transitional government that was in place during the preparation and conduct of the 2010 presidential election.
Doré died in Conakry on 29 January 2016, at the age of 77.
T. N. Gopakumar
T. N. Gopakumar (1957 – 30 January 2016) was an Indian writer, movie director and journalist. He was the news editor in chief of Asianet News. Gopakumar was also the producer and anchorperson of the popular weekly programme "Kannadi" on Asianet.
Gopakumar died of cancer in a private hospital in Thiruvananthapuram on 30 January 2016. He was aged 58.
Georgia Davis Powers