For vectors with complex entries, using the given definition of the dot product would lead to quite different geometric properties. For instance, the dot product of a vector with itself can be an arbitrary complex number, and can be zero without the vector being the zero vector; this in turn would have severe consequences for notions like length and angle. Many geometric properties can be salvaged, at the cost of giving up the symmetric and bilinear properties of the scalar product, by alternatively defining
where "b" is the complex conjugate of "b". Then the scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, this scalar product is not linear in b (but rather conjugate linear), and the scalar product is not symmetric either, since
This type of scalar product is nevertheless quite useful, and leads to the notions of Hermitian form and of general inner product spaces.
The Frobenius inner product generalizes the dot product to matrices. It is defined as the sum of the products of the corresponding components of two matrices having the same size.
The dot product between a tensor of order n and a tensor of order m is a tensor of order n+m-2. The dot product is worked out by multiplying and summing across a single index in both tensors. If formula_32 and formula_33 are two tensors with element representation formula_34 and formula_35 the elements of the dot product formula_36 are given by
This definition naturally reduces to the standard vector dot product when applied to vectors, and matrix multiplication when applied to matrices.
Occasionally, a double dot product is used to represent multiplying and summing across two indices. The double dot product between two 2nd order tensors is a scalar.
Sarcocystis
Sarcocystis is a genus of protozoa. Species in this genus can live in reptiles, birds, mammals, and sometimes pigs. The name came from the Greek language, "sarx", which meant flesh, and "kystis", which meant bladder. The organism was first found by Miescher in 1843, in a mouse.
Zapatista Army of National Liberation
The Zapatista Army of National Liberation (EZLN; ) is a political and militant group based in –Chiapas, the southernmost state of Mexico. Its members are called the Zapatistas. They advocate for libertarian socialism, environmental sustainability and indigenous rights.
It is led by the indigenous people of the Chiapas Mountains. The name "Zapatista" refers to Emiliano Zapata who was a leader of the Mexican Revolution of 1910. The group was formed in the 1980s and led a large rebellion in 1994. Where it captured a small region of Mexico and was granted control over it.
Though the movement has many leaders, both women and men, Marcos has become the unofficial spokesperson for the movement. This is because he is very well educated, a good public speaker, and has written letters to and worked with writers and intellectuals from all over the world. He signs almost all of his letters with "Subcomandante Marcos, from the mountains of the southeast of Mexico." He is wearing a ski-mask in every picture and video of him and ski-masks have become a symbol of the EZLN in Mexico, as the majority of its supporters also wear them. Followers of the EZLN also paint murals of Marcos in his mask and Emiliano Zapata to show their support.
Since 1994, the Zapatistas have been in control of a region in Southern Chiapas. The Zapatistas created a 'Council of Good Government' which is a direct democracy representing between 120,000 to 300,000 people. The economy is made up of a coordinated network of communal stores, worker cooperatives and family farms. Their autonomous government provides low-interest loans, free education and universal health care. The Zapatistas mainly produce corn, beans, coffee, bananas, sugar, cattle, chicken, pigs and clothing. The Mexican military has attacked communities several times with drones, infantry and helicopters.
Lead glass
Lead glass, also known as lead crystal, is a type of glass that has lead in it. Normal glass has calcium oxide in it, while lead glass changes the calcium oxide with lead(II) oxide. It looks nice because the lead is very large and heavy and changes how the glass looks. Lead glass rings when tapped while ordinary glass does not. This glass can absorb some X-rays. It is very insulating.
It contains between 18 to 40% of lead (II) oxide by weight. The nomenclature of lead glass is regulated such that any glass that contains a minimum of 24% PbO by weight is referred to as “lead crystal”, and any glass that contains less than 24% PbO by weight is referred to as crystal glass. Lead glass has a chemical composition of SiO2 and PbO along with other impurities or additives added to achieve certain properties.
Lead glass is also known as X-ray glass or radiation shielding glass as one of its major applications is in the absorbance of high energy radiation while maintaining optical transparency. The presence and quantity of PbO are what determines some of lead glass’ most important optical and physical properties.
The properties of lead glass make it useful for many applications. Here are a few examples:
Summoner
The Summoner is one of the characters in the Canterbury Tales.
He is always drunk. Children are afraid of him. When he drinks too much wine, he starts speaking in Latin. He has pimples on his face but is rather intelligent. Chaucer used to write about the medieval age. Anachronism can be found in his writings easily. Everything that he has described about the Summoner is set in that time.
Lead(II) chloride
Lead(II) chloride, also known as plumbous chloride and lead dichloride, is a chemical compound. Its chemical formula is PbCl. It contains lead and chloride ions. The lead is in the +2 oxidation state. Its use may lead to lead poisoning.
Lead(II) chloride is a colorless crystalline solid. It does not dissolve well in water. It can dissolve in solutions that have chloride ions in them. It reacts with molten sodium nitrite to make lead(II) oxide.
Lead(II) chloride is made by reacting sodium chloride or hydrochloric acid with lead nitrate. It can also be made by reacting lead(IV) oxide with hydrochloric acid. It can also be made by reacting chlorine with lead.
It is used to make ceramics. A basic chloride is used in pigments.
Relic
A relic is an object, especially a piece of the body or a personal item of someone of religious importance, that was carefully preserved with an air of veneration as a memorial that you can touch. Relics are an important aspect of Buddhism, some denominations of Christianity, Hinduism, shamanism, and many other personal belief systems.
The word relic comes from the Latin "reliquiae" ('remains'). A reliquary is a shrine that houses one or more relics.
Some well-known religious relics include the Tooth of Buddha, the Holy Grail, the Crown of Thorns, the True Cross of Jesus, the Staff of Moses, and the Sword of Mohammed.
John F. Kennedy Jr.
John Fitzgerald Kennedy Jr. (November 25, 1960 – July 16, 1999) often called John-John or JFK Jr., was an American lawyer and magazine publisher. He was the son of American President John F. Kennedy.
Kennedy was born at Medstar Georgetown University Hospital in Washington, DC, to President elect John F. Kennedy and his wife Jacqueline. He spent his first three years at the White House until his father's assassination on November 22, 1963. The following year, he moved to New York City with his mother and older sister Caroline.
After high school, he attended Brown University. He graduated in 1983 with a Bachelor's Degree in history. He then attended the New York University of Law, earning his law degree in 1989. He founded the magazine "George" in 1995.