By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is possible to show that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i. e.,

By the end of the 18th century, three definitions of the gallon were in common use:

By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent.

By these definitions, mountains cover 64 % of Asia, 25 % of Europe, 22 % of South America, 17 % of Australia, and 3 % of Africa.

By most definitions, characters do not strictly require actual superhuman powers to be deemed superheroes, although terms such as costumed crime fighters are sometimes used to refer to those such as Batman and Green Arrow without such powers who share other common superhero traits.

By some definitions, the first supervillain was John Devil, a proto-Fantômas, created by Paul Féval, père in his eponymous 1860 novel, or Féval's nearly-immortal, machiavellian Colonel Bozzo-Corona, leader of Les Habits Noirs introduced in 1863.

By virtue of these interpretive connections, the network can function as a scientific theory: From certain observational data, we may ascend, via an interpretive string, to some point in the theoretical network, thence proceed, via definitions and hypotheses, to other points, from which another interpretive string permits a descent to the plane of observation.

By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is possible to show that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i. e.,

By some definitions of " society " and " homogenous ", virtually all contemporary national societies are multiethnic.

By a similar token, there are now definitions for " Combined Statistical Areas " ( CSA ).

By most definitions, it extends into Iowa, North and South Dakota, northern Illinois, and eastern Nebraska.

By present-day definitions some of these villagers would be defined as Syrians, some as Lebanese and some as Palestinians.

By the above definitions, there are no parallel geodesics to a given geodesic, all geodesics intersect.

By some definitions, the average size length of a sentence is given by " no.

By some definitions, a macro photograph is one in which the size of the subject on the negative or image sensor is life size or greater.

By the most strict definitions, no megastructures currently exist ( with the space elevator being the only such project under serious consideration ).

By more lenient definitions, the Great Wall of China ( 6. 7 Mm ) counts as a megastructure.

By most definitions, this is the first working gas turbine.

By some academic definitions, a traditional story about a historical human character like Muhammad would be a " legend ", not a " myth ".< ref > For some examples of this distinction between " legend " and " myth ", see < http :// falcon. jmu. edu /~ ramseyil / tradmays. htm > and ( for the classic distinction drawn by professional folklorists ) < http :// encarta. msn. com / text_761556135__1 / Folktales. html >.</ ref >

By most definitions, budget sets must be compact and convex.

By some definitions it contains the largest archipelago ( island group ) in the world by the number of islands, although many of the islands are very small and tightly clustered.

By some definitions WDM-PON is a dedicated wavelength for each ONU.

By incorporating the correct definitions for saturated thickness, specific storage, and specific yield, we can transform this into two unique governing equations for confined and unconfined conditions:

By some definitions, in Morningside was a former Edinburgh home of the author J. K. Rowling, writer of the Harry Potter series of books ; however by some definitions her house on Abbotsford Crescent was within the Merchiston area.

By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups.

By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions.

By the structure theorem for finite abelian groups, all such groups are products of cyclic groups.

By taking the dual category of which we call we get an exact and fully faithful embedding from our category to an abelian category which has enough projective objects and a cogenerator.

** By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable.

By design, these initials are very similar to the axioms for an abelian group, G1-G3 below.

By definition, a functor F from a triangulated category D into an abelian category A is a cohomological functor if for every distinguished triangle

By this point Apple had a wide variety of communications products under development, and many of these were announced along with AppleTalk Phase II.

By 2008, Dartmouth had graduated 238 classes of students and has over 60, 000 living alumni in a variety of fields.

By the time DDT was introduced in the U. S., the disease had already been brought under control by a variety of other means.

... By keeping all of the little people in focus, Coppola shows the variety of a young lawyer's life, where every client is necessary and most of them need a lot more than a lawyer.

By the time he composed the Gospel, he must have been a highly practiced and competent author-able to compose in a wide variety of literary forms according to the demands of the moment.

By combination of different α, β and γ subunits, a great variety (> 1000 ) G proteins can be produced.

By then, the Sumerian Peoples of south Mesopotamia were all organized into a variety of independent City-states, such as Ur and Uruk, which by around 26th century BC had begun to coalesce into larger political units.

By the turn of the decade, the punk rock movement had split deeply along cultural and musical lines, leaving a variety of derivative scenes and forms.

By 1904 he was clearly expressing his own understanding of these themes in his essays and books, though continuing to refer to a wide variety of historical sources.

By the late 1810s, it was not uncommon to dance a series of quadrilles during the evening, generally consisting of the same first three figures combined with a variety of different fourth and fifth figures.

By adopting and modifying local ideologies, Scouting has been able to find acceptance in a wide variety of cultures.

By the latter part of the century it had acquired a slightly arcane quality associated with the era of variety, but the term is still in active use .<

By the end of the 1960s, the idea of the Third World came to represent countries in Africa, Asia and Latin America that were considered underdeveloped by the West based on a variety of characteristics ( low economic development, low life expectancy, high rates of poverty and disease, etc .).

By 1985, a version of Emacs ( MicroEMACS ) was available for a variety of platforms, but it was not until June, 1987 that Stevie ( ST editor for VI enthusiasts ), a limited vi clone appeared.

By the 1860s, Syrah was established as an important variety in Australia.

By the late 19th century there was a general acceptance that emotionally traumatic experiences could cause long-term disorders which might display a variety of symptoms.

By the 12th century Italian flutes came in a variety of sizes, and fragments of 12th-century Norman bone whistles have been found in Ireland, and an intact 14 cm Tusculum clay whistle from the 14th century in Scotland.

By the 9th century most inhabitants north of Lille spoke a dialect of Middle Dutch, while the inhabitants to the south spoke a variety of Romance dialects.

By varying the extent and path of motion, a variety of effects can be obtained, with variable depth of field and different degrees of blurring of " out of plane " structures.

By the end of 2000, merchandising based on The Powerpuff Girls encompassed a whole variety of products, including T-shirts, toys, video games, lunchboxes, and dishware.

By drawing knowledge and experimental techniques from a wide variety of disciplines, biophysicists are often able to directly observe, model or even manipulate the structures and interactions of individual molecules or complexes of molecules.

By varying the amount of cream returned, producers can make a variety of low-fat milks to suit their local market.

By mobilizing these diverse resources, the UN Foundation works with many public and private partners and manages a variety of campaigns to help solve major problems facing the UN and the world community.

By itself, umami is not palatable, but it makes a great variety of foods pleasant especially in the presence of a matching aroma.

By night they hunt a variety of animals – mainly large invertebrates including scarab and huhu beetles, moths and caterpillars, wētā and spiders.