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, a gene for antibiotic resistance that evolves via natural selection may be shared.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Thus, natural beryllium bombarded either by alphas or gammas from a suitable radioisotope is a key component of most radioisotope-powered nuclear reaction neutron sources for the laboratory production of free neutrons.

Thus, any error in technology or judgment may have far-reaching consequences ( though the same can also happen through natural reproduction ).

Thus, what started as an effort to translate between languages evolved into an entire discipline devoted to understanding how to represent and process natural languages using computers.

Thus, to be valid, any law must conform to natural law and coercing people to conform to that law is morally acceptable.

Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.

Thus Esperanto achieved a stability of structure and grammar similar to that which natural languages enjoy by virtue of their native speakers and established bodies of literature.

Thus non-standard language varieties ( such as African American Vernacular English ) are considered to be natural while standard language varieties ( such as Standard American English ) which are more prescribed can be considered to be at least somewhat artificial or constructed.

Thus punctuated equilibrium contradicts some of Darwin's ideas regarding the specific mechanisms of evolution, but generally accords with Darwin's theory of evolution by natural selection.

Thus, a natural use of the turbocharger is with aircraft engines.

Thus, the moral value of one-celled organisms, as well as some multi-cellular organisms, and natural entities like a river, is only in the benefit they provide to sentient beings.

Thus, the effect of the accidental has to be understood in relation to the " natural " meaning of the note's staff position.

Thus, if in this example the composer actually wanted the note a semitone lower than G-natural, he might put first a natural sign to cancel the previous G-sharp, then the flat.

Thus a natural transformation is an infranatural transformation for which for every morphism.

Thus specific tissue effects of natural menopause cannot be attributed to loss of androgenic hormone production.

Thus T is uncountable: it cannot be placed in one-to-one correspondence with the set of natural numbers.

Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various " natural rings ".

Thus no formal system ( satisfying the hypotheses of the theorem ) that aims to characterize the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove.

Thus, they are natural guardians of treasures or sacred sites which cannot easily be moved out of harm's way.

Thus an event will not constitute a natural disaster if it occurs in an area without vulnerability, e. g. an earthquake in an uninhabited area .< ref > Luis Flores Ballesteros.

Thus he won her fairly and took her away to Troia, with the full consent of her natural protectors.

Thus his persistence values for some stem frequencies per meaning are: stem identical in 8 languages, 100% ; ;

Thus different sources give sometimes wildly different accounts of the number of languages within a family.

Thus, SGML is properly a meta-language, and many particular markup languages are derived from it.

Thus, in several languages of the Banks Islands, including Mwotlap, the simple m stands for, but an m with a macron ( m ̄) is a labial-velar nasal ; while the simple n stands for the common alveolar nasal, an n with macron ( n ̄) represents the velar nasal ; the vowel ē stands for a ( short ) higher by contrast with plain e ; likewise ō contrasts with plain o.

Thus, Friedrich Müller, in his ambitious classification ( 1876 – 88 ), separated the ' Negro ' and Bantu languages.

Thus Vietnamese, Tai – Kadai and Hmong – Mien ( Miao – Yao ), all languages with a similar tone system to Chinese, were classified under the Sino-Tibetan tree.

Thus similar optimization techniques can be used across languages.

Thus, L-systems are strict subsets of languages.

Thus, although all people perceive visible radiation the same way, in terms of a continuum of color, people who speak different languages slice up this continuum into discrete colors in different ways.

Thus, they are usually dealing with populations living in specific locales for generations without moving, but also with immigrant groups bringing their languages to new settlements.

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 in Dutch, Burggraaf is the rank above Baron, below Graaf ( i. e., Count ) in the kingdoms of the Netherlands and of Belgium ( by Belgian law, its equivalents in the other official languages are Burggraf in German and vicomte in French ).

Thus, the. NET Framework and Java, likewise any other programming languages under Windows, are ( or contain ) wrapper libraries.

Thus Old Latin, Classical Latin, Vulgar Latin, etc., are not considered different languages, but are all referenced under the name of Latin.

Thus to get an application to support multiple languages one would design the application to select the relevant language resource file at runtime.

Thus whistled tonal languages convey phonemic information solely through tone, length, and, to a lesser extent, stress, and most segmental phonemic distinctions of the spoken language are lost.

Thus if we set aside the notion of standardization, I believe it would be more appropriate to speak of 25 languages derived from Old Tibetan.

Thus, languages featuring destructive assignment rarely enforce referential transparency, which requires a procedure to return the same results for a given set of inputs at any point in time.

Thus it should be sufficient to test any class through its public interface or through its subclass interface, which some languages call the " protected " interface.

Thus, by 1800, the literary Ruthenian language had evolved into three modern literary languages.

Thus the Sudanic languages of East Africa and adjacent Afro-Asiatic families are part of the same area with inflectional past-marking that extends into Europe, whereas more westerly Nilo-Saharan languages often do not have past tense.