Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus every part and plane of the picture keeps changing place in relative depth with every other part and plane ; ;

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

Thus, if every historian were to claim that there was a solar eclipse in the year 1600, then though we might at first naively regard that as in violation of natural laws, we'd come to accept it as a fact.

Thus, if a publicly traded company specialises in the acquisition of privately held companies, from a per-share perspective there is a gain with every acquisition that falls within these guidelines.

Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

Thus in Austinian terms a moral code can objectively determine what people ought to do, the law can embody whatever norms the legislature decrees to achieve social utility, but every individual remains free to choose what to do.

Thus, every chlorine is in the middle of a cube formed by caesium atoms and every caesium is in the middle of a cube formed by chlorine.

Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i. e. acting on every point in it.

Thus, the molecular formula of ethanol may be written in a paper in conformational, three-dimensional, full two-dimensional ( indicating every bond with no three-dimensional directions ), compressed two-dimensional ( CH < sub > 3 </ sub >– CH < sub > 2 </ sub >– OH ), separating the functional group from another part of the molecule ( C < sub > 2 </ sub > H < sub > 5 </ sub > OH ), or by its atomic constituents ( C < sub > 2 </ sub > H < sub > 6 </ sub > O ), according to what is discussed.

Thus, every non-Han dynasty saw itself as the legitimate holder of the " Mandate of Heaven ", which legitimized the dynastic cycle regardless of social or ethnic background as it was moral integrity and benevolent leadership that had been validating the " Mandate of Heaven.

Thus every proposition has three components: the two terms, and the " copula " that connects or separates them.

Thus every module direct summand of R is generated by an idempotent.

Thus we have in the first five centuries such epithets applied to her as " in every respect holy ", " in all things unstained ", " super-innocent " and " singularly holy "; she is compared to Eve before the fall, as ancestress of a redeemed people ; she is " the earth before it was accursed.

Thus it is true in every case above except case 2, because this is the only case when is true but is not.

Thus, every MUMPS command or function name could be abbreviated from one to three letters in length, e. g. Quit ( exit program ) as Q, $ P

Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

Thus the number of preorders is the sum of the number of partial orders on every partition.

( Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between − and winding.

Thus every even perfect number is a pernicious number.

Thus DA means that a cocking action separate from the trigger pull is unnecessary ; and every trigger pull will result in a complete cycle.

Thus, only a small number of computers ( the servers ) need to be rigorously secured, rather than securing every single client computer.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

Thus, besides the training provided to youth in school, the existence of the school program can have supplementary benefits to industry which make it an asset to industrial development efforts.

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 flight test program was set to integrate the AARGM onto Tornado ECR's weapon system.

Thus, the Dhrystone score counts only the number of program iteration completions per second, allowing individual machines to perform this calculation in a machine-specific way.

Thus, the software program automatically interpolates the two values, creating a smooth transition.

Thus it made sense to not map the memory between programs for IPC, but instead migrate the program code being used into the local space of the program.

Thus future teachers at Lehigh often take a five year program earning both a Bachelors Degree in a specialized field and a Masters Degree in Education.

Thus, an interpreter might load a portable bytecode program, decode the bytecodes to generate platform-dependent threaded code, then execute threaded code without further reference to the bytecodes.

Thus, the Steinway Artists program was born.

Thus biography can form one part of the larger study of the cultural significance, underlying program, or agenda of a work ; a study which gained increasing importance in the 1980s and early 1990s.

Thus, a program will achieve greater performance if it uses memory while it is cached in the upper levels of the memory hierarchy and avoids bringing other data into the upper levels of the hierarchy that will displace data that will be used shortly in the future.

Thus the structure of the resulting program closely mirrors that of the grammar it recognizes.

Thus he developed and strongly supported " Vietnamization ", a program intended to expand, equip, and train South Vietnam's forces and assign to them an ever-increasing combat role, at the same time steadily reducing the number of U. S. combat troops.

Thus the key technique here — which characterises a program as a chatbot rather than as a serious natural language processing system — is the production of responses that are sufficiently vague and non-specific that they can be understood as " intelligent " in a wide range of conversational contexts.

Thus the two histories present the basic program of the Enlightenment in crystallized form.

Thus " Fulbrighters " were grant recipients under the USIA educational and cultural exchange program.

Thus, a good maintenance program might lubricate the bearings frequently but never clean them.

Thus its use will spread rapidly, long before a program developed using the MIT approach has a chance to be developed and deployed.

Thus, a program " Foo " has all of its specific files and libraries in < tt >/ Programs / Foo </ tt >.

Thus, it is very unlikely that it will be possible to program a reasonably fast algorithm for playing the Go endgame flawlessly, let alone the whole Go game.

Thus as sensitivity increases, a screening program will cost more, or be able to screen a smaller number of women.

Thus, King's estate is able to require a license fee for redistribution of the speech, whether in a television program, a history book, a dramatic re-enactment, or otherwise.