Thus, the electron and positron are denoted simply as and respectively.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

Thus the necessary and sufficient conditions for the truth of a proposition of law simply involved internal logic and consistency, and that the state's agents used state power with responsibility.

Thus the dragoon regiments of the Imperial German Army wore the pickelhaube ( spiked helmet ) of the same design as those of the infantry and the British dragoons wore scarlet tunics, In other respects however dragoons had adopted the same tactics, roles and equipment as other branches of the cavalry and the distinction had become simply one of traditional titles.

Thus, though knowledge requires justification, truth, and belief, the word " knowledge " can't be, according to Williamson's theory, accurately regarded as simply shorthand for " justified true belief.

Thus, the challenge is to create agile or discovery-driven implementations of the EVM principle, and not simply to reject the notion of measuring technical performance objectively.

Thus the 32-bit integer and 32-bit floating-point objects can simply use the value directly ; whereas the 64-bit integer and 64-bit floating-point cannot use this method.

Thus, seemingly irreducibly complex structures are merely " very complex ", or they are simply misunderstood or misrepresented.

Thus having been " Mirandized " a suspect may avow explicitly the invocation of these rights, or, alternatively, simply remain silent.

Thus the past, present, future, and anything in and of, would simply exists without possible creation of, or deviation from.

Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others.

Thus, in the very large cases, the HTSC cost can not be offset by simply reducing the coil size at a higher magnetic field.

Thus, a tax on the sale of property would be considered an indirect tax, whereas the tax on simply owning the property itself would be a direct tax.

Expected to make use of devices with which listeners could press a button to indicate whether they liked or disliked a particular piece of music, Adorno bristled with distaste and astonishment: “ I reflected that culture was simply the condition that precluded a mentality that tried to measure it .” Thus Adorno suggested using individual interviews to determine listener reactions and, only three months after meeting Lasarzfeld, completed a 160-page memorandum on the Project ’ s topic, “ Music in Radio .” Adorno was primarily interested in how the musical material was affected by its distribution through the medium of radio and thought it imperative to understand how music was affected by its becoming part of daily life.

Thus the Whig Party leader in Illinois, Abraham Lincoln, simply abandoned politics after 1849.

Thus the heater did not function as an electrode, but simply served to heat the cathode sufficiently for it to emit electrons by thermionic emission.

Thus, where the money supply expanded, people would not simply wish to hold the extra money in idle money balances ; i. e., if they were in equilibrium before the increase, they were already holding money balances to suit their requirements, and thus after the increase they would have money balances surplus to their requirements.

Thus, a 0-ary operation ( or nullary operation ) can be represented simply as an element of A, or a constant, often denoted by a letter like a.

Thus, by the time of the Roman Empire, a sophist was simply a teacher of rhetoric and a popular public speaker.

Thus it is usually called Euler's phi function or simply the phi function.

Thus, in order to simply flip through the possible values for a 128-bit symmetric key ( ignoring doing the actual computing to check it ) would theoretically require 2 < sup > 128 </ sup > − 1 bit flips on a conventional processor.

Thus the operation of the oracle is simply written as:

Thus, there is no possibility of simply copying each list, let alone of using a template, as can be done, if different languages share one project with or without separate namespaces.

Thus, a legacy switch system can be easily " converted " to allow wavelength multiplexed transport over a fiber simply by judicious choice of transceiver wavelengths, combined with an inexpensive passive optical multiplexing device.

Thus, a sphere and a disk are each simply connected, while a torus is not.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus, the broadcast domain is the entire inter-connected layer two network, and the segments connected to each switch / bridge port are each a collision domain.

Thus, while D-Cinema is a defined standard, though one that is still partly being framed by SMPTE as of 2007, E-Cinema may be anything, ranging from a DVD player connected to a consumer projector to something that approaches the quality of D-Cinema without conforming to some of the standards.

Thus, according to Wittgenstein, mental states are intimately connected to a subject's environment, especially their linguistic environment, and conceivability or imaginability arguments that claim otherwise are misguided.

Thus for a large or very distributed company that requires multiple PBXs, users can receive the same services across the network and be unaware of the switch that their telephone is connected to.

Thus one writer alleges that Johnson's campaign manager, future Texas governor John B. Connally, was connected with 202 ballots in Precinct 13 in Jim Wells County that had curiously been cast in alphabetical order and just at the close of polling, with all of the people whose names appeared on the ballots being dead on election day.

By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient can be formed unless on a connected component of D. Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers.

: Thus, in Bergsland & Vogt ( 1962 ), the authors make an impressive demonstration, on the basis of actual language data verifiable by extra-linguistic sources, that the " rate of change " for Icelandic constituted around 4 % per millennium, whereas for closely connected Riksmal ( Literary Norwegian ) it would amount to as much as 20 %.

Thus, manifolds, Lie groups, and graphs are all called connected if they are connected as topological spaces, and their components are the topological components.

Thus, a category is connected if it is, intuitively, all one piece.

Thus, when discussing simply connected topological spaces, it is far more common to speak of simple connectivity than simple connectedness.

Thus, any autoimmune response was perceived to be abnormal and postulated to be connected with human disease.

Thus, the high-speed alternator would produce a steady radio signal when connected to an aerial.

Thus, Gawain is connected in multiple myths to the Celtic concept of Other-world.

Thus, the war histories of Fort Branch and Plymouth are intimately connected.

Thus, " where a court of equity has all the parties before it, it will adjudicate upon all of the rights of the parties connected with the subject matter of the action, so as to avoid a multiplicity of suits.

Thus although Fischer and Kern connected with the Berlin chapter of the O. C.

Thus, from birth, Charles Stewart Parnell possessed an extraordinary number of links to many elements of society ; he was linked to the old Irish Parliamentary tradition via his great-grandfather and grandfather, to the American War of Independence via his grandfather, to the War of 1812 ( where his grandfather had been awarded a gold medal by the United States Congress for gallantry ); he belonged to the disestablished Church of Ireland ( its members mostly unionists ) though in later years he was to drop away from formal church attendance ; and he was connected with the aristocracy through the Powerscourts.

Thus, " maund " is connected to the Latin mendicare, and French mendier, to beg.

Thus, with multiple layers of encryption, it is effectively impossible to know who is connected to any particular service or pseudonym.

Thus, the 0-homology group measures the number of path connected components of the space.

Thus each client builds up a list of all the torrents shared by the peers it connected to in the current session ( or it can even maintain the list between sessions if instructed ).