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Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that a < sup > s </ sup > ≡ a < sup > t </ sup > ( mod n ).
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Even and without
Even to be `` from hope and fear set free '' is at least better than to have lost the first without having got rid of the second.
Even in normal experience one gets impressions without knowing exactly how -- of atmosphere, of one another's personalities, moods, intentions.
Even the two-year-old feels miffed if the family has a prayer-time without her.
Even nominally disordered ( amorphous ) boron contains regular boron icosahedra which are, however, bonded randomly to each other without long-range order.
Even now, advanced radiosity calculations may be reserved for calculating the ambiance of the room, from the light reflecting off walls, floor and ceiling, without examining the contribution that complex objects make to the radiosity — or complex objects may be replaced in the radiosity calculation with simpler objects of similar size and texture.
Even in languages without such a distinction, such as Xhosa, experiments have shown that when the click release is removed from a recording, the resulting sound is judged to be uvular, not velar.
Even more important: a difference generally implies positive terms between which the difference is set up ; but in language there are only differences without positive terms.
Even older rocks, such as the Acasta gneiss of the Slave craton in northwestern Canada, the oldest known rock in the world have been metamorphosed to the point where their origin is undiscernable without laboratory analysis.
Even qualified international or out of state doctors are not allowed to practice without acceptance to by the local medical guild ( Medical board ).
Even without considering the effects of fatigue, the longer the handle, the harder it is to guide the head of the hammer to its target at full speed.
Even without specific restrictions on powder and primers, they may be covered under other laws governing explosive materials.
; Poor fuel economy: Even compared to other heavy passenger vehicles, Hummers without the Diesel engine options have very poor fuel economy.
Even in Wycliffe's time the " Lollards " had reached wide circles in England and preached " God's law, without which no one could be justified.
Even Grimm himself did not at first intend to include all the languages in his Grammar, but he soon found that Old High German postulated Gothic, and that the later stages of German could not be understood without the help of other West Germanic varieties including English, and that the rich literature of Scandinavia could not be ignored either.
Even when we make all due allowance for the prejudices of critics whose only possible enthusiasm went out to ' the pointed and fine propriety of Poe ,' we can hardly believe that the exquisite art which is among the most valued on our possessions could encounter so much garrulous abuse without the criminal intervention of personal malignancy.
Even without doping the vibrational motions ( the phonons ) are delocalized over the crystal, explaining the heat conduction.
Even without having any superhuman powers, he has on several occasions matched Marvel superhumans such as Spider-Man and the Thing.
Even more unusual variants include a ruleset in which the rules are hidden from players ' view, and a game which, instead of allowing voting on rules, splits into two sub-games, one with the rule, and one without it.
Even without third-party developers, by 1983 over one million Odyssey² units were sold in the US alone.
Even without external triggers, regions of the cloud can reach conditions where they become unstable against collapse.
Even countries without those diplomatic relations participate in international organizations of which the Holy See is a full member.
Even in diploid organisms, many somatic cells are polyploid due to a process called endoreduplication where duplication of the genome occurs without mitosis ( cell division ).
Even this was achieved without engine changes.
Even without this consideration, refusing such a request would arguably create a constitutional crisis, as it is considered a fairly strong constitutional convention that the head of state always grants a parliamentary dissolution.
Even though it was independent from the surface for a very short duration, the Rouquayrol-Denayrouze apparatus reached worldwide celebrity after having been mentioned by Jules Verne in his adventure book Twenty Thousand Leagues Under the Sea ; but Jules Verne wildly exaggerated its dive duration without external air supply.
Even and knowledge
Even the knowledge that she was losing another boy, as a mother always does when a marriage is made, did not prevent her from having the first carefree, dreamless sleep that she had known since they dropped down the canyon and into Bear Valley, way, way back there when they were crossing those other mountains.
Even then, as you go into the house oppressed by the knowledge that something is cooking and that your house has passed under this unaccountable, official control, could you go on forgetting that you still had that ridiculous hat on your head and you were still carrying that childish horn in your hand??
Even though slaves outnumbered the Portuguese colonists, the lack of weapons, the colonial law, the disagreement between slaves coming from different African cultures and their complete lack of knowledge about the land and its surroundings would usually discourage the idea of a rebellion.
Even though Eva Sapir was an important influence, Sapir received his lust for knowledge and interest in scholarship, aesthetics and music from his father.
Even today, knowledge in the field advances so rapidly that many of the etymologies in contemporary dictionaries are outdated.
Even though in general, ordinary relations between directors and shareholders in a business corporation are not of such a fiduciary nature as to make it the duty of a director to disclose to a shareholder the general knowledge which he may possess regarding the value of the shares of the company before he purchases any from a shareholder, yet there are cases where, by reason of the special facts, such duty exists.
# Even if something can be known about it, knowledge about it can't be communicated to others.
Even in the field of science, the practices of which were once viewed as being merely the objective testing and reporting of knowledge, scientists must persuade their audience to accept their findings by sufficiently demonstrating that their study or experiment was conducted reliably and resulted in sufficient evidence to support their conclusions.
Even the geographical knowledge needed in transferring the war from one part of the world to another was considerable.
Even as a child he gained knowledge of two foreign languages.
Even with our imperfect knowledge of Byzantine genealogy, no less than eleven emperors may be traced among his ancestors.
# Even if something could be known about it, knowledge about it can't be communicated to others.
Even if it is far more probable that Jordanes just introduced his own philosophical knowledge in the text, a lot of modern Romanian authors considers that Deceneus is a priest who reforms the Getae cult, changing Zalmoxis ' cult into a popular religion and imposing strict religious rules such as the restriction of wine consumption.
Even the astronomer Yi Xing's isn't comparable to his value ( who was beginning to utilize foreign knowledge ).
Even now Cumberland was still unsure of the situation facing him, " I cannot come at any certain knowledge of the enemy's numbers, but I have concurring information that the body on this side the Schelde does not exceed 31 battalions and 32 squadrons.
Even in the oldest times the chief qualifications demanded of the hazzan, in addition to knowledge of Biblical and liturgical literature as well as the prayer motifs ( known as " steiger "), were a pleasant voice and an artistic delivery ; for the sake of these, many faults were willingly overlooked.
Even at the height of the Cold War, Francis ' music was well received in Iron Curtain countries, and some of her recordings were made available on state-owned record labels such as Melodiya in the former Soviet Union and on Jugoton in former Yugoslavia, although it was common knowledge that rock n ' roll was highly looked down upon in Eastern bloc countries.
Even here Xenocrates's preference for symbolic modes of sensualising or denoting appears: he connected the above three stages of knowledge with the three Fates: Atropos, Clotho, and Lachesis.
Even the " reading knowledge " required for a PhD degree is comparable only to what second-year language students read and only very few researchers who are native English speakers can read and assess information written in languages other than English.
Even with this knowledge, Riley initially refused medical treatment from the government out of paranoia, though Buffy eventually convinced him to do so.
Even so, it is possible for scraps of knowledge or experiences gained from their time with the Great Race to remain in dreams.
Even though human knowledge seems to abstract various properties or parts of God, such extrapolations are merely existent in human conceptions of God.
Even his wit and knowledge of the world were spoiled, and his affected gaiety was touched with sadness, by the odour of falsehood which escaped through every pore of his body.
Even in his book Meditations on the Peaks: Mountain Climbing as Metaphor for the Spiritual Quest Evola discussed mountaineering as a possible approach or support on the way of initiatic ascesis in which heroic action is combined with specialized knowledge and training culminating in an initiation — the climbing of the mountain.
Even though photography has such effect, there are limits to photographic knowledge of the world.
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