Thus technical efficiency is achieved at the expense of actual experience.

Thus, the actual name that Chiang received at birth was Jiang Ruiyuan ().

Thus conclusions about actual agriculture in the Letters are as likely to evoke Latium as Attica.

Thus, this supposedly distinctive feature was easy to use, but had nothing to do with actual phylogenic relationship.

Thus, if n < sub > r </ sub > is the average number of times a new civilization reappears on the same planet where a previous civilization once has appeared and ended, then the total number of civilizations on such a planet would be ( 1 + n < sub > r </ sub >), which is the actual reappearance factor added to the equation.

Thus, churches actually had money-changing tables at which each coin would be examined separately and a token of actual worth given to the layperson so that he or she could be seen by the other parishioners as putting money in the basket during that part of the service.

Thus, all three models for paramagnetic Hb-O < sub > 2 </ sub > may contribute to some small degree ( by resonance ) to the actual electronic configuration of Hb-O < sub > 2 </ sub >.

Thus, actual address space utilization rates will be small in IPv6, but network management and routing efficiency is improved by the large subnet space and hierarchical route aggregation.

Thus, we might as well treat that eigenspace as the actual Hilbert space of the system.

Thus the odds of losing symbols appearing on the payline became disproportionate to their actual frequency on the physical reel.

Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics.

Thus the principality ruled politically by a prince-bishop could wholly or largely overlap with his diocesan jurisdiction, but not necessarily ; several lost their actual see ( the city itself ), which could obtain the status of free imperial city.

Thus, actual discussion and information exchanges can happen.

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 actual flowers of the fig are unseen unless the fig is cut open.

Thus, they found the length of 1 ° of meridian to be 111. 8 km ( 115. 1 km ), which differs from the actual value by 850 metres.

: 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 Posidonius ' measure of 240, 000 stadia translates to 24, 000 miles, not much short of the actual circumference of 24, 901 miles.

Thus Confucius, whose actual name was Kǒng Qiū ( 孔丘 ), was given the zì Zhòngní ( 仲尼 ), where the first character zhòng indicates that he was the second son in his family.

Thus, in practice quality was usually very poor, though the actual localization and translation were closer to a professional level than those found in modern fansubs.

Thus, the attempt to maintain a given surface temperature in an environment of faster heat loss results in both the perception of lower temperatures and an actual greater heat loss increasing the risk of adverse effects such as frostbite, hypothermia, and death.

Thus, frequencies close to the Nyquist frequency may be distorted in the sampling and reconstruction process, so the bandwidth should be kept below the Nyquist frequency by some margin ( frequency headroom ) that depends on the actual filters used.

Thus the Supreme Court's actual decision never hinged on the equal protection claims.

Thus, if popular fashion says that a ‘ blouse ’ is ideal for a certain situation or ensemble, this idea is immediately naturalized and accepted as truth, even though the actual sign could just as easily be interchangeable with ‘ skirt ’, ‘ vest ’ or any number of combinations.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines ( e. g. Fuzzy Sets and Systems ), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct / refute / refound are taken as intuitively given.

Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same.

Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game.

Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.

Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed.

Thus Babylonian mathematics remained stale in character and content, with very little progress or innovation, for nearly two millennia.

Thus Grosseteste concluded, following very much in what Boethius had argued, that mathematics was the highest of all sciences, and the basis for all others, since every natural science ultimately depended on mathematics.

Thus, while Fessenden was only a teenager, he was teaching mathematics to the young children at the school while simultaneously studying with the older students at Bishop's University.

Thus Hilbert is insisting that mathematics is not an arbitrary game with arbitrary rules ; rather it must agree with how our thinking, and then our speaking and writing, proceeds :".

Thus, providing students of archaeology with a solid background in quantitative sciences such as mathematics, statistics and computer sciences seems today more important than ever.

Thus, inverse problems are one of the most important, and well-studied mathematical problems in science and mathematics.

Thus in 1900 he enrolled to École Normale Supérieure to study mathematics.

Thus the root of mathematics and scientific pursuits in Pythagoreanism is also based on a spiritual desire to free oneself from the cycle of birth and death.

Thus the order of arguments in some of the constructions below is exactly the opposite from those in many mathematics textbooks.

Thus humans do not invent mathematics, but rather discover and experience it, and any other intelligent beings in the universe would presumably do the same.

Thus the rationalists took mathematics as their model for knowledge, and the empiricists took the physical sciences.

Thus Kant arrives at the conclusion that all pure mathematics is synthetic though a priori ; the number 7 is seven and the number 5 is five and the number 12 is twelve and the same principle applies to other numerals ; in other words, they are universal and necessary.

Thus Bardas founded the Magnaura School with seats for philosophy, grammar, astronomy and mathematics, supported scholars like Leo the Mathematician and promoted the missionary activities of Cyril and Methodius to Greater Moravia.

In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k-th power for 1 ≤ k ≤ P. Thus, a magic cube is bimagic when it is 2-multimagic, and trimagic when it is 3-multimagic, tetramagic when it is 4-multimagic.