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Scalar (mathematics)

A scalar is an element of a field, used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector. In linear algebra, real numbers or other elements of a field are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. More a vector space may be defined by using any field instead of real numbers, such as complex numbers; the scalars of that vector space will be the elements of the associated field. A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an inner product space; the real component of a quaternion is called its scalar part. The term is sometimes used informally to mean a vector, tensor, or other "compound" value, reduced to a single component.

Thus, for example, the product of a 1×n matrix and an n×1 matrix, formally a 1×1 matrix, is said to be a scalar. The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix; the word scalar derives from the Latin word scalaris, an adjectival form of scala, from which the English word scale comes. The first recorded usage of the word "scalar" in mathematics occurs in François Viète's Analytic Art: Magnitudes that ascend or descend proportionally in keeping with their nature from one kind to another may be called scalar terms. According to a citation in the Oxford English Dictionary the first recorded usage of the term "scalar" in English came with W. R. Hamilton in 1846, referring to the real part of a quaternion: The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity. A vector space is defined as a set of vectors, a set of scalars, a scalar multiplication operation that takes a scalar k and a vector v to another vector kv.

For example, in a coordinate space, the scalar multiplication k yields. In a function space, kƒ is the function x ↦ k; the scalars can be taken from any field, including the rational, algebraic and complex numbers, as well as finite fields. According to a fundamental theorem of linear algebra, every vector space has a basis, it follows that every vector space over a scalar field K is isomorphic to a coordinate vector space where the coordinates are elements of K. For example, every real vector space of dimension n is isomorphic to n-dimensional real space Rn. Alternatively, a vector space V can be equipped with a norm function that assigns to every vector v in V a scalar ||v||. By definition, multiplying v by a scalar k multiplies its norm by |k|. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k. A vector space equipped with a norm is called a normed vector space; the norm is defined to be an element of V's scalar field K, which restricts the latter to fields that support the notion of sign.

Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four arithmetic operations. For this reason, not every scalar product space is a normed vector space; when the requirement that the set of scalars form a field is relaxed so that it need only form a ring, the resulting more general algebraic structure is called a module. In this case the "scalars" may be complicated objects. For instance, if R is a ring, the vectors of the product space Rn can be made into a module with the n×n matrices with entries from R as the scalars. Another example comes from manifold theory, where the space of sections of the tangent bundle forms a module over the algebra of real functions on the manifold; the scalar multiplication of vector spaces and modules is a special case of scaling, a kind of linear transformation. Operations that apply to a single value at a time. Scalar processor Scalar Hazewinkel, Michiel, ed. "Scalar", Encyclopedia of Mathematics, Springer Science+Business Media B.

V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Weisstein, Eric W. "Scalar". MathWorld. Mathwords.com – Scalar

Michael Lachanodrakon

Michael Lachanodrakon was a distinguished Byzantine general and fanatical supporter of Byzantine Iconoclasm under Emperor Constantine V. As a result of his iconoclast zeal, in 766 he rose to high office as governor of the Thracesian Theme, instigated a series of repressive measures against iconophile practices targeting the monasteries. A talented general, he led a series of campaigns against the Arabs of the Abbasid Caliphate before being dismissed from office in about 782. Restored to imperial favour in 790, he fell at the Battle of Marcellae against the Bulgars in 792. Nothing is known of Lachanodrakon's origins and early life, he receives a negative treatment in the historical sources, which were written after the final defeat of Byzantine Iconoclasm. Their profoundly iconophile perspective means that reports of his actions those relating to the suppression of icon worship, are untrustworthy. At the Council of Hieria in 754, Constantine V had declared the adoration of icons to be a heresy, had thereby elevated iconoclasm to official imperial policy.

No persecution of iconophiles was launched at first, but iconophile resistance grew, until from 765 on, Constantine began persecuting iconophiles, monks. The discovery of a wide-ranging iconophile plot against him involving some of the highest civil and military officials of the state in 766 provoked an extreme reaction. Patriarch Constantine II and other officials were deposed, publicly humiliated, executed, replaced by new, uncompromisingly iconoclast officials. In addition, the veneration of sacred relics and prayers to the saints and the Virgin Mary were condemned. By 763 or 764, according to the iconophile Life of St Stephen the Younger hagiography, Lachanodrakon had distinguished himself by his iconoclast fervour. On the emperor's orders, he led a group of soldiers on an invasion of the Pelekete monastery on the Propontis, where he arrested 38 monks and subjected the remainder to various tortures and mutilations. After burning down the monastery, he took the 38 captives to Ephesus. In 766/767, as part of the emperor's reshuffle of the senior echelons of the Byzantine Empire, Lachanodrakon was rewarded with the important post of strategos of the Thracesian Theme, given the rank of patrikios and imperial protospatharios according to his seal.

He soon began a harsh repression of the iconophiles. According to Theophanes the Confessor, in 769/770 he summoned the monks and nuns of his theme to Ephesus, gathered them in the city's tzykanisterion and forced them to marry, threatening them with blinding and exile to Cyprus if they refused. Although many resisted and "became martyrs" in Theophanes's words, many complied. Reports of exiled monks in Cyprus becoming Arab captives seem to corroborate this story. Theophanes reports further that in 771/772, Lachanodrakon dissolved all monasteries in the theme and expropriated their property, sent the proceeds to the emperor, who replied with a letter thanking him for his zeal. Lachanodrakon had relics, holy scriptures, monks' beards set on fire, killed or tortured those who venerated relics, prohibited the tonsure. Although embellished, these reports reflect actual events. At any rate, by 772, according to historian Warren Treadgold, Lachanodrakon seems to have succeeded in "eradicating monasticism within his theme".

Lachanodrakon was a capable general, winning fame for his campaigns against the Abbasids on the Byzantine Empire's eastern frontier. During the reign of Constantine V's son Leo IV he seems to have been the most prominent military commander leading expeditions comprising troops from several themes against the Arabs; the first such expedition occurred in 778 when, preempting an anticipated Arab raid, Lachanodrakon led a large army against Germanikeia. Although the city did not fall, the Byzantine army defeated a relief force, plundered the region, took many captives Jacobites, who were resettled in Thrace. In 780, Lachanodrakon ambushed and defeated an Arab invasion in the Armeniac Theme, killing the brother of the Arab commander Thumama ibn al-Walid; the Arab historian al-Tabari records that in 781 Lachanodrakon forced another Arab invasion, under'Abd al-Kabir, to withdraw without battle, although Theophanes ascribes the success to the sakellarios John. In 782, however, he was defeated by the Arab general al-Barmaqi during a large-scale invasion led by the future caliph Harun al-Rashid, losing some 15,000 men according to Theophanes.

In the aftermath of this defeat, because of his iconoclast past, he was removed from his command by the iconophile empress-regent Irene of Athens. Lachanodrakon reappears in 790, when the young emperor Constantine VI conspired to overturn the tutelage of Irene; the general was sent by Constantine to the Armeniac Theme to secure the allegiance of its soldiers. Constantine succeeded in toppling his mother in December 790. According to the account of Theophanes, he participated in the imperial campaign against the Bulgars in 792 that led to the disastrous defeat at the Battle of Marcellae on 20 July, where he was killed; the history of John Skylitzes records his death in the Battle of Versinikia, again against the Bulgars, in 813, but this is an error. Hollingsworth, Paul A.. "Lacha

Vagrant (software)

Vagrant is an open-source software product for building and maintaining portable virtual software development environments, e.g. for VirtualBox, KVM, Hyper-V, Docker containers, VMware, AWS. It tries to simplify the software configuration management of virtualizations in order to increase development productivity. Vagrant is written in the Ruby language. Vagrant was first started as a personal side-project by Mitchell Hashimoto in January 2010; the first version of Vagrant was released in March 2010. In October 2010, Engine Yard declared; the first stable version, Vagrant 1.0, was released in March 2012 two years after the original version was released. In November 2012, Mitchell formed an organization called HashiCorp to support the full-time development of Vagrant. HashiCorp now works on creating commercial editions and provides professional support and training for Vagrant. Vagrant was tied to VirtualBox, but version 1.1 added support for other virtualization software such as VMware and KVM, for server environments like Amazon EC2.

Vagrant is written in Ruby, but it can be used in projects written in other programming languages such as PHP, Java, C#, JavaScript. Since version 1.6, Vagrant natively supports Docker containers, which in some cases can serve as a substitute for a virtualized operating system. Vagrant uses "Provisioners" and "Providers" as building blocks to manage the development environments. Provisioners are tools. Puppet and Chef are the two most used provisioners in the Vagrant ecosystem. Providers are the services that Vagrant uses to create virtual environments. Support for VirtualBox, Hyper-V, Docker virtualization ships with Vagrant, while VMware and AWS are supported via plugins. Vagrant sits on top of virtualization software as a wrapper and helps the developer interact with the providers, it automates the configuration of virtual environments using Chef or Puppet, the user does not have to directly use any other virtualization software. Machine and software requirements are written in a file called "Vagrantfile" to execute necessary steps in order to create a development-ready box.

"Box" is a format and an extension for Vagrant environments, copied to another machine in order to replicate the same environment. Official website List of Vagrant boxes