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Thus this kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i. e. a tensor field, as opposed to a function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously.
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Some Related Sentences
Thus and kind
Thus, the Church was born and because of its intrinsic character was soon identified as a conservative institution, determined to resist the forces of change, to identify itself with the political rulers, and to maintain a kind of splendid isolation from the masses.
Thus, knowing the students ' level on a developmental sequence provides information on the kind and level of knowledge they can assimilate, which, in turn, can be used as a frame for organizing the subject matter to be taught at different school grades.
Thus, " iambic pentameter " is a meter comprising five feet per line, in which the predominant kind of foot is the " iamb ".
Thus it has been said that the House of Commons became a part of the government and it has been only a further step of this development that a new kind of prime minister should emerge.
Thus an RGB value does not define the same color across devices without some kind of color management.
Thus began one of the most famous apprenticeships in the musical theatre, as Hammerstein designed a kind of course for Sondheim on the construction of a musical.
Thus the American bison has the binomial Bison bison ; a name of this kind would not be allowed for a plant.
Thus, mark-up is a kind of pre-editing.
John Lemprière, in Bibliotheca Classica, notes that as the story was re-told in later versions it accumulated details from the stories of Noah and Moses: " Thus Apollodorus gives Deucalion a great chest as a means of safety ; Plutarch speaks of the pigeons by which he sought to find out whether the waters had retired ; and Lucian of the animals of every kind which he had taken with him & c ."
Thus, attempts to define electrochemical potential in this way cannot lead to the kind of universal parameter that thermodynamics ideally requires.
Thus, hanja are often used to clarify meaning, either on their own without the equivalent hangul spelling, or in parentheses after the hangul spelling as a kind of gloss.
Thus, it can change into various kinds of substances without remaining any kind of substance all the time.
Thus, virtual particles are often popularly described as coming in pairs, a particle and antiparticle, which can be of any kind.
“ The truth is that there is not enough of the right kind of freedom, the fundamental freedom to choose to be free or not to be free, according to one's preference .... Thus I demand, for each and every member of human society, freedom of association according to inclination and of activity according to aptitude.
Thus, if the information of the sub-matrices is available, the eigenvalues of the matrix can be described using that kind of information.
Thus a possible diagnostic outcome would be that all tosses yield the same outcome, and the probability of 5 of a kind is 6 % under the null hypothesis.
Thus the Barbary macaque, a kind of monkey, is popularly called the " Barbary ape " to indicate its lack of a tail.
Thus a particular kind of map of spaces is well modelled by a particular kind of morphism of groupoids.
… Thus, when we so use it in the sentence, < nowiki >'</ nowiki > This is good ,' we merely refer to this, and the addition of " is good " makes no difference whatever to our reference … it serves only as an emotive sign expressing our attitude to this, and perhaps evoking similar attitudes in other persons, or inciting them to actions of one kind or another.
Thus Johnston uses, " potentially hundreds of pitches per octave ," in way that is, " radical without being avant-garde ," and not for the creation of, " as-yet-unheard dissonances ," but in order to, " return ... to a kind of musical beauty ," he perceives as diminished in Western music since the adoption of equal-temperament ( Gann 1995, 1 ).
Thus the ratio that seemed to imply a sort of cost-benefit ratio for gun ownership was actually nothing of the kind because it did not take account of any benefits that corresponded to its costs.
Thus legal realism or " relationalism " has been favored in some common law jurisdictions, where the kind of legal codification associated with continental ( and Japanese ) law are virtually unknown.
In the Preface to the Meditations, Descartes asks the reader " not to pass judgment on the Meditations until they have been kind enough to read through all these objections and my replies to them .” Thus, this dialogue could be seen as an integral part of Descartes ' views expressed in the Meditations.
Thus and implicitly
Thus, he implicitly emphasized that it is God-made (" God has joined together "), " male and female ," lifelong (" let no one separate "), and monogamous (" a man ... his wife ").
Thus, for example, the statement, " It is true that two plus two equals four ", contains no more information than the statement " two plus two equals four ", because the phrase " it is true that ..." is always implicitly there.
Thus, implicitly, the Constitution creates a Cabinet that includes the principal officers of the various departments.
Thus, even those concerned only with interactions among young siblings implicitly or explicitly acknowledge that all relationships change over time and that any effects of birth order may be eliminated, reinforced, or altered by later experiences.
Thus, modifying the data through one name implicitly modifies the values associated to all aliased names, which may not be expected by the programmer.
Thus the court that she assembled about her in Ferrara corresponded to the tradition which the cultivation of science and art implicitly required, including scholars like Bernardo Tasso and Fulvio Pellegrini.
Thus and expresses
Thus, the outcome of a full partial fraction expansion expresses that function as a sum of fractions, where:
Thus, in addition to the inflammation within the thyroid, any cell that expresses the TSH receptor will likely experience lymphocytic infiltrates as well.
Thus, a + b expresses the sum of two scalars if a and b are scalars, or the sum of two arrays if they are arrays.
Thus can't ( or cannot ) is often used to express disbelief in the possibility of something, as must expresses belief in the certainty of something.
Thus, Dilleniaceae may be an ancient clade that expresses some phylogenetic relation between the higher Eudicots and the rather more primitive groups.
Thus, on his view the sentence " Samuel Clemens was witty " expresses exactly the same content as " Mark Twain was witty ", whether or not the competent user of these sentences recognizes it.
Thus this convention expresses deference to the authority of the court rather than an honour for the moderator as an individual.
Thus and fact
Thus throwing a rock at someone for the purpose of hitting him is a battery if the rock in fact strikes the person and is an assault if the rock misses.
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.
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, Annie's strong drive to take revenge on women scholars in general and on Miss de Vine in particular is perfectly comprehensible, and in fact does not in itself prove her to be mentally deranged.
Thus accounts such as that of the miraculous " Image Not Made by Hands ", and the weeping and moving " Mother of God of the Sign " of Novgorod are accepted as fact: " Church Tradition tells us, for example, of the existence of an Icon of the Savior during His lifetime ( the " Icon-Made-Without-Hands ") and of Icons of the Most-Holy Theotokos immediately after Him.
Thus the decision to involve the Inquisition, which therefore did not instigate the trial and in fact showed a reluctance throughout its duration.
Thus, the mergers were not done to see large efficiency gains, they were in fact done because that was the trend at the time.
Thus, for Rand, the fact that consciousness must itself possess identity implies the rejection of both universal skepticism based on the " limits " of consciousness and any claim to revelation, emotion or faith based belief.
Thus, Pakistan is in fact a multi-regional state and not a South Asian state actor only ; its history if analyzed in depth would prove the point.
Thus, while Bolivar knew that the nation of Panama was linked historically and culturally to South America, he was also conscious of the fact that the region was part of the Central American geography.
Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a twenty-one-year-old would have had only five birthdays, if he was born on a leap day.
Thus, ' truth ' involves both the quality of " faithfulness, fidelity, loyalty, sincerity, veracity ", and that of " agreement with fact or reality ", in Anglo-Saxon expressed by sōþ ( Modern English sooth ).
Thus, while Gandalf exerts a parental influence over Bilbo early on, it is Bilbo who gradually takes over leadership of the party, a fact the dwarves could not bear to acknowledge.
Thus, many of today's users are unaware of the fact that there are two types of DV AVI files.
Thus despite the fact no cigars have been brought into the hotel, each guest can smoke a cigar inside the property.
Thus a social fact is dependent on other social facts, but because each is only dependent on and reducible to other social facts, it is independent of and irreducible to psychological or biological factors.
Thus, the prefix ferro, meaning iron, was used to describe the property despite the fact that most ferroelectric materials do not contain iron.
Thus, Henri Rochefort wrote a piece in " L ' Intransigeant ", claiming Zola had committed suicide, having discovered Dreyfus to be in fact guilty.
Thus it would seem that Lewis Carroll did not intend care and hope from the repeating stanza to stand for two women, but was quite pleased with the interpretation after the fact.
Thus, despite appearing brilliant white to observers on Earth, Halley's Comet is in fact pitch black.
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 the Earth's " conservative potency " ( virtus conservativa ) provided the explanation for the fact that the Earth retains its atmosphere though travelling through space.
Despite his rejection of the Macchiaioli approach, Modigliani nonetheless found favour with his teacher, who referred to him as " Superman ", a pet name reflecting the fact that Modigliani was not only quite adept at his art, but also that he regularly quoted from Nietzsche's Thus Spoke Zarathustra.
Thus, under Kaldor – Hicks efficiency, a more efficient outcome can in fact leave some people worse off.
Thus, Internet access across the European Union is relatively open because of the laws forbidding discrimination based on nationality, but the fact of publication in, say, France, limits the target market to those who read French.
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