Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

Thus, the two immiscible phases ( hydrophilic vs. hydrophobic ) will change so that their corresponding interfacial area will be minimal.

Thus a phonemic split has occurred in London English, exemplified by the minimal pair wholly vs. holy.

Thus, it is exactly the unique minimal convex set containing X.

Thus, sets of complementary wavelets are useful in wavelet based compression / decompression algorithms where it is desirable to recover the original information with minimal loss.

Thus, minimal time is spent providing prehospital care ( spine immobilization ; " ABCs ", i. e. ensure airway, breathing and circulation ; external bleeding control ; endotracheal intubation ) and the victim is transported as fast as possible to a trauma centre.

Thus, a candidate key is a minimal set of attributes that functionally determine all of the attributes in a relation.

Thus, many speakers do not differentiate between the vowel sounds ( as in " dress ") and ( as in < trap >), except in cases where a minimal pair such as < bed >/< bad > exists in the vocabulary of the speaker.

Thus there must have been exacerbating factors that made the effect as bad as it was over and above the minimal set-up factors.

Thus, Japanese Combined Fleet commander, Admiral Isoroku Yamamoto, authorized the use of faster warships at night to make the deliveries when the threat of detection was much less and aerial attack minimal.

Thus the Logic presents the categories of thought as they are in themselves ; they are the minimal conditions for thinking anything at all, the conceptions that run in the background of all our thinking.

Thus, although tabbed windows are adequate in environments where there is a minimal necessity for tabs ( around ten tabs or less ), this scheme does not scale, and alternate methods may be required to address this issue.

Thus, any successful attack against the larger program will gain minimal access, even though the pair of programs will be capable of performing privileged operations.

Thus, betrayal was considered as the most rational thing to do, i. e. minimaxing the losses ( getting the possible loss to be the minimal possible ).

Thus, the language included distinct sounds and minimal pairs which were not separate in writing.

Thus, the extent to which language processing occurs within an anatomical module is considered to be minimal.

Thus, his style of minimal composition is truly European in its orientation and has been little influenced by North American minimal composers, whose works draw more often on rock, jazz, and world music.

Thus the only concentration of troops in Conakry appeared to be the armoured battalion, with a modest number of Soviet medium tanks manufactured in the late 1940s, as well as Soviet APCs, and elements of the engineer battalion.

Thus such a gain medium must have a gain bandwidth sufficiently broad to amplify those frequencies.

Thus the Luftwaffe's reliance on tactical mid-range, twin engined medium bombers and short range dive-bombers was a rational option under these circumstances.

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, if a wave is traveling through several different media, then the optical path length of each medium can be added to find the total optical path length.

Thus, an optical fiber or a copper cable is a transmission medium.

Thus, in the Aristotelian view, a projectile moving through the air would owe its continuing motion to eddies or vibrations in the surrounding medium, a phenomenon known as antiperistasis.

Thus, many users of medium speed diesel engines such as railroad locomotives use V16 powerplants, including most Electro-Motive Diesel and GE Transportation Systems locomotives.

Thus it tends to be the medium of exchange function that constrains what can be used as a form of financial capital.

Thus the medium of exchange function has become wholly a part of the marketplace and its signals, and is utterly integrated with the unit of account function, so that, given the integrity of the public key system on which these are based, they become to that degree inseparable.

Thus, the chansons were primarily an oral medium.

Thus, adoption of sandstone as the medium for carvings in this temple has made it more advanced in its architectural design vis-à-vis other temples in its vicinity, which were mostly wood-based.

Thus, in the scenario above, a task with medium priority ran before a task with high priority, effectively giving us a priority inversion.

Thus, ODR is a different medium to resolve disputes, from beginning to end, respecting due process principles.

Thus, they chose to hide the data entry medium while not in use to make a more compact device.

Thus Rational argument, integral to print typography, is militated against by the medium of television for the aforesaid reason.

Thus it became the first international university and the first university in Malaysia to use English as the medium of instruction and administration.

" Thus, the advent of services such as Television on demand, Pay-per-view and even premium cable subscription services such as HBO and Showtime open up the opportunity for niche content to reach the right audiences, in an otherwise mass medium.

Thus if a beam of light with intensity I is incident on the surface, a beam of intensity RI is reflected, and a beam with intensity TI is transmitted into the medium.

Thus, based on visual evidence from the show, the battlecrabs appear to come in various sizes, answering the need for medium to very large capital ships with just one basic model.

Thus " M11 / 39 " means the 11 ton medium tank of 1939.

Thus it is possible that an individual ’ s experience with using a certain lean medium, will prompt that individual to use it for equivocal tasks

Thus, through the medium of musical subcultures ( industrial metal, gothic metal and neo folk music scenes ) and the creation of autonomous villages for völkisch communities, they hope to disseminate their subversive ideas throughout society in order to achieve cultural hegemony.

Thus this laser does not need a concave lens or refocusing lenses and beam quality improves along the gain medium.