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Pilum. The pilum (Latin: [ˈpiːɫʊ̃]; pl.: pila) was a javelin commonly used by the Roman army in ancient times. It was generally about 2 m (6 ft 7 in) long overall, consisting of an iron shank about 7 mm (0.28 in) in diameter and 600 mm (24 in) long with a pyramidal head, attached to a wooden shaft by either a socket or a flat tang.
An alternative notation that is commonly used for the Pauli matrices is to write the vector index k in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the k-th Pauli matrix is σ k αβ. In this notation, the completeness relation for the Pauli matrices can be written
The pilum was a short-range javelin with an effective range of about 15 meters (50 ft), but could also be used as a spear in situations where an enemy had to be held back. [44] It was hurled at the enemy formations right before the charge and this hail of javelins was intended to break the force of the enemy charge as well as demoralize the ...
Modern reconstruction of the heavy pilum according to Polybius' specifications has shown that it would have weighed some 8.5 kg (19 lb), far too heavy to be of any practical use as a throwing weapon. The light pilum would have weighed a more serviceable 2.2 kg (4.9 lb). [49]
The Möbius–Kantor graph, the Cayley graph of the Pauli group with generators X, Y, and Z. In physics and mathematics, the Pauli group on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix and all of the Pauli matrices
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σ i, so that the Hermitian matrix is written as a Pauli vector. [2] In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as ...
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...
This javelin was used by the velites for skirmishing purposes, unlike the heavier pilum, which was used by the hastati and principes for weakening the enemy before advancing into close combat. The shafts were about 1.1 metres (3 ft 7 in) long, substantially shorter than the 2-metre (6 ft 7 in) pilum , and the point measured about 13 centimetres ...