Sweden women's national football team
Sweden women's national football team () represents Sweden on the women's side in association football. Sweden won the 1984 UEFA European Championship. In the FIFA World Cup, the team ended up second in 2003, and won the bronze medals in 1991 and 2011.
Homeless shelter
A homeless shelter is a place for homeless people to take refuge. Shelters provide safety and protection from exposure to the weather. These shelters also provide food, warmth, and the basic needs for any person.
Determinant
The determinant of a square matrix is a scalar (a number) that indicates how that matrix behaves. It can be calculated from the numbers in the matrix.
The determinant of the matrix formula_1 is written as formula_2 or formula_3 in a formula. Sometimes, instead of formula_4 and formula_5, one simply writes formula_6 and formula_7.
There are a few ways to understand what the determinant says about a matrix.
An formula_8 matrix can be seen as describing a linear map in formula_9 dimensions. In which case, the determinant indicates the factor by which this matrix scales (grows or shrinks) a region of formula_9-dimensional space.For example, a formula_11 matrix formula_1, seen as a linear map, will turn a square in 2-dimensional space into a parallelogram. That parallellogram's area will be formula_2 times as big as the square's area.In the same way, a formula_14 matrix formula_15, seen as a linear map, will turn a cube in 3-dimensional space into a parallelepiped. That parallelepiped's volume will be formula_16 times as big as the cube's volume.
The determinant can be negative or zero. A linear map can stretch and scale a volume, but it can also reflect it over an axis. Whenever this happens, the sign of the determinant changes from positive to negative, or from negative to positive. A negative determinant means that the volume was mirrored over an odd number of axes.
One can think of a matrix as describing a system of linear equations. That system has a unique non-trivial solution "exactly" when the determinant is not 0 ("non-trivial" meaning that the solution is not just all zeros).
If the determinant is zero, then there is either no unique non-trivial solution, or there are infinitely many.
A matrix has an inverse matrix exactly when the determinant is not 0. For this reason, a matrix with a non-zero determinant is called invertible. If the determinant is 0, then the matrix is called non-invertible or singular.
Geometrically, one can think of a singular matrix as "flattening" the parallelepiped into a parallelogram, or a parallelogram into a line. Then the volume or area is 0, which means that there is no linear map that will bring the old shape back.
There are a few ways to calculate a determinant.
For larger matrices, the determinant is harder to calculate. One way to do it is called cofactor expansion.
Suppose that we have an formula_8 matrix formula_1. First, we choose any row or column of the matrix. For each number formula_24 in that row or column, we calculate something called its cofactor formula_25. Then formula_26.
To compute such a cofactor formula_25, we erase row formula_28 and column formula_29 from the matrix formula_1. This gives us a smaller formula_31 matrix. We call it formula_32. The cofactor formula_25 then equals formula_34.
Here is an example of a cofactor expansion of the left column of a formula_14 matrix:
formula_36
As illustrated above, one can simplify the computation of determinant by choosing a row or column that has many zeros; if formula_24 is 0, then one can skip calculating formula_25 altogether.
Philip the Arab
Philip the Arab ( 204 – September 249) was Roman emperor from 244 to 249 AD.
Philip () was born in around 204. He got the name "Philip the Arab" ( or ). (The "Chronicon Paschale", a Greek chronicle, and the Latin historian Aurelius Victor each have different dates for Philip's birth.) He was the son of Julius Marinus. Philip was probably born in the city he later named Philippopolis, in the Roman province of Arabia Petraea (now the city of Shahba, Syria).
Philip became emperor at the start of 244. (The "Justinianic Code"'s writing means that Philip became emperor after 13 January 244 and before 14 March the same year.) Philip's wife, Marcia Otacilia Severa, became "augusta" (empress) early in 244. After Philip became emperor, the Romans and the Sasanian Empire made peace. With this success, Philip got the name (or ) in the middle of 244. Philip came to Rome in 244, probably in early summer.
In 245, Philip was Roman consul for the first time. That year, the Romans went to war with the Carpi. On 12 November 245, Philip was at Aquae in Dacia (Vidonac, Serbia). Wars against the Carpi and Germanic peoples continued into 247. In 247, Philip was Roman consul for the second time. In late summer 247, Philip held a triumph in Rome. For his military success against the Carpi, Philip probably got the name at the end of 247.
In 248, Philip was Roman consul for the third time. In April 248, Philip held a festival for the one thousandth anniversary of Rome, a thousand years since Romulus became the first king of Rome. (The chronicle of the Latin historian Cassiodorus says that this started on 21 April and ended on the 23 April 248.)
John Chrysostom wrote that unlike other pagan rulers of the Romans, Philip was tolerant to the Christians and let them practise their faith openly. Eusebius wrote Philip and his wife received letters from Origen.
Some time before 11 September 249, soldiers in the Roman army killed Philip in a battle against the army of Decius. The battle was probably near Verona on the Italian Peninsula. Philip, Philip II, and Marcia Otacilia Severa may all have had "damnatio memoriae" on them, and the Romans removed their names from some inscriptions.
The Latin historian Eutropius says that Philip and Philip II were both deified (made into gods). This is probably not true. Where Philip's grave was is unknown.
Philip married Marcia Otacilia Severa before 238. At the start of 244, Marcia Otacilia Severa became "augusta" (empress) and got the Latin title: . Marcia Otacilia Severa probably died in 248.
They had a son, Philip II. The Latin history "Epitome de Caesaribus" says that Philip II was born in 236 or 237. Only the Greek historian John of Antioch says Philip had other sons. Philip made his son Philip II his "caesar" (junior emperor) in July or August 244. After July or August 244, Marcia Otacilia Severa also had the title . In July or August 247, Philip made his son his "augustus" (co-emperor).
Philip's father, Julius Marinus, died while his son was emperor. After that, people in Philippopolis in Arabia (Philip's home town) worshipped Julius Marinus as a god. However, the Roman Senate probably did not deify Julius Marinus, and he probably did not become one of the official gods.
Philip had a brother, Gaius Julius Priscus. He was a governor of Roman Mesopotamia () and then a praetorian prefect and governor of the East ().
Loneliness
Loneliness is a sad emotion that a person can feel if they are isolated or feel a sense of lack of friends and family members. Loneliness can make people feel anxious.
People can feel lonely even if they are surrounded by a lot of other people. Loneliness can occur even to people in marriages, relationships, families and successful jobs. One cause of loneliness is a lack of friends during childhood and teenage years. Loneliness can also be caused by chronic depression. Loneliness can also be caused when a relationship or marriage ends.
Loneliness is a theme in literature, going back to the Epic of Gilgamesh.
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