From the commutativity of kets with ( complex ) scalars now follows that

From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of e < sup > x </ sup > to the complex plane.

From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space.

From these two basic types, a few modern designs have been created, such as the Hohner CBH-2016 chromatic and the Suzuki Overdrive diatonic, which have complex covers that allow for specific functions not usually available in the traditional design.

From this scarce notice, it has been concluded that a small church which encompassed a cave complex might have been located in Nazareth in the early 4th century ," although the town was Jewish until the 7th century AD.

From a systems perspective, a complex network structure can be decomposed into individual component firms ( Zhang and Dilts, 2004 ).

From 200, 000 BP onwards, Middle Paleolithic humans began to make complex stone blades with flaked edges which were used as spear heads.

From this building, which is still in use, the present hospital complex has grown.

From 2001 through 2008, the East Front of the Capitol ( site of most presidential inaugurations until Ronald Reagan began a new tradition in 1981 ) was the site of construction for this massive underground complex, designed to facilitate a more orderly entrance for visitors to the Capitol.

From this foundation arises a complex political and ideological superstructure, where economic development impacts upon societal progress.

From such a perspective, in different research contexts complex systems are defined on the basis of their different attributes.

) use an mboxrd variation with more complex From line quoting rules.

From the late 1960s until the mid-1990s, cadets also completed SERE training in the Jacks Valley complex between their fourth-and third-class years.

From south to north, the principal temple-cities, their principal temple complex, and the gods they served, were

From about 1854 to 1900, an industrial complex existed in and around Columbia, Marietta and Wrightsville that included 11 anthracite iron furnaces and related structures, as well as canal and railroad facilities servicing them.

From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry ; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.

From here, Route 120 heads northwest along the northern edge of the sports complex as a four-lane divided at-grade Paterson Plank Road, passing through industrial and commercial areas.

From the polemical contemporary works which describe his career have emerged the outlines of a complex figure, charting a dangerous course through the chaos of the late 5th century Athenian political scene ; although historians from ancient times to the present have offered far more specific portraits, of one form or another, it may be that nothing more than that outline will ever be known with certainty.

From that time, the pipe organ was the most complex man-made device, a distinction it retained until it was displaced by the telephone exchange in the late 19th century.

From its beginnings as the Wright brothers ' testing field, it has evolved into the headquarters for the Air Force's worldwide logistics system and all Air Force systems development and procurement, the aeronautical engineering center, a major research laboratory complex, the heart of Air Force graduate education, location of the second largest Air Force medical center, and home of the National Museum of the U. S. Air Force.

From these building blocks one can build more complex shapes like squares, triangles, circles and other composite figures.

From the ancient Greeks until the work of Carolus Linnaeus ( also known as Carl Linnaeus, or Carl von Linné ) and other 18th century naturalists, the main concept of natural history was the scala naturae or Great Chain of Being, a conceptual arrangement of minerals, vegetables, more primitive forms of animals, and more complex life forms on a linear scale of increasing " perfection ", culminating in our species.

From 1949 to 1991 the complex was used by the BBC.

From 1970 to 1984, Pratt lived mainly in France where Corto Maltese, a psychologically very complex character resulting from the travel experiences and the endless inventive capacity of his author, became the main character of a comics series.

From the primitive to the complex, those identifications indicate that the most primitive form of the music was associated with Marshall, Texas – and that the left-hand bass lines grew more complex as the distance from Marshall increased.

From the definition of matrix multiplication, there exists an × matrix, such that.

( From this Cox eventually infers that conjunction of plausibilities is associative, and then that it may as well be ordinary multiplication of real numbers.

From the associativity of multiplication, it follows that for any positive integers m and n,

From 1929 until the rapid multiplication in the number of swing bands from 1935 on, the Casa Loma Orchestra was one of the top North American dance bands.

From this principle the error rules of summation, multiplication etc.

From an estimated mass of 25 g for a zero-magnitude meteorite, the other masses are derived with the assumption of a mass decrease by a factor of 2.512 for each unit increase in magnitude.

From that point the first responder is quickly able to identify those in need of immediate attention, while not being distracted or overwhelmed by the magnitude of the situation.

From the perspective of an observer on a hypothetical planet around Vega, the Sun would appear as a faint 4. 3 magnitude star in the Columba constellation.

From Jupiter's surface — or rather, from just above its cloudtops — Amalthea would appear very bright, shining with a magnitude of − 4. 7, similar to that of Venus from Earth.

In 1954, in a letter to Yitzchak Ben Tzvi, Israel's second President, the Rebbe wrote: " From the time that I was a child attending cheder, and even before, the vision of the future Redemption began to take form in my imagination – the Redemption of the Jewish People from their final Exile, a redemption of such magnitude and grandeur through which the purpose of the suffering, the harsh decrees and annihilation of Exile will be understood ... "

From these tests, ratings of pleasantness can be obtained using either the direct scaling or magnitude matching method and may be of value in the diagnosis of dysgeusia.

From this definition and Coulomb's law, it follows that the magnitude of the electric field E created by a test charge Q is:

From a Balkan power of dominant magnitude Greece was thrown back into the unenviable position she occupied after the disastrous Greco-Turkish War of 1897.

From 40 Eridani, 16 light years away, the Sun would be an average looking star of about apparent magnitude 3. 3 in the constellation Serpens Caput.

From a planet orbiting Aldebaran, 65 light years away, the Sun would appear slightly above Antares in our constellation Scorpius, and at magnitude 6. 4 would barely be visible to the naked eye.

From equation ( 2 ) it follows, since T always has unit magnitude, that N is always perpendicular to T. From equation ( 3 ) it follows that B is always perpendicular to both T and N. Thus, the three unit vectors T, N, and B are all perpendicular to each other.

From the New York City boroughs outside Manhattan ( Brooklyn, Queens, Staten Island and the Bronx ), the limiting magnitude might be 3. 0, suggesting that at best, only about 50 stars might be seen at any one time.

From brightly lit Midtown Manhattan, the limiting magnitude is possibly 2. 0, meaning that from the heart of New York City only approximately 15 stars will be visible at any given time.

From relatively dark suburban areas, the limiting magnitude is frequently closer to 5 or somewhat fainter, but from very remote and clear sites, some amateur astronomers can see nearly as faint as 8th magnitude.

From the point of view of the interior of the car, an accelerating reference frame, there is a fictitious force pushing the passenger backwards, with magnitude equal to the mass of the passenger times the acceleration of the car.

From classical electrodynamics, a rotating electrically charged body creates a magnetic dipole with magnetic poles of equal magnitude but opposite polarity.

From a 17th magnitude star, it reached an apparent visual magnitude of – 0. 2 then began a rapid decline.