Proof: By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s

By 2005, the university had established two 501 ( c )( 3 ) charitable organizations to provide scholarship assistance solely for minority students.

By convention, unknowns are denoted by letters at the end of the alphabet, x, y, z, w, …, while knowns are denoted by letters at the beginning, a, b, c, d, ….

By definition, equality follow a number of " equivalence relations ", including ( a ) reflexive ( i. e. ), ( b ) symmetric ( i. e. if then ) ( c ) transitive ( i. e. if and then ).

By definition, a and b can be written as multiples of c: a = mc and b = nc, where m and n are natural numbers.

By this process of " looping " or " somersaulting ", a hydra can move several inches ( c. 100 mm ) in a day.

By the time of the redaction of the Mishnah, Rosh Hashanah 1: 1 ( c. 200 CE ), jurists had identified four new-year dates:

By c. 1000, heofon was being used in reference to the Christianized " place where God dwells ", but originally, it had signified " sky, firmament " ( e. g. in Beowulf, c. 725 ).

By the 17th century, Osei Kofi Tutu I ( c. 1695 – 1717 ), with help of Okomfo Anokye, unified what became the Ashante into a confederation with the Golden Stool as a symbol of their unity and spirit.

Robin Lane Fox states " By the early fifth century, we know of the ownership of private icons of saints ; by c. 480-500, we can be sure that the inside of a saint's shrine would be adorned with images and votive portraits, a practice which had probably begun earlier ".

* By their critical temperature: they can be high temperature ( generally considered if they reach the superconducting state just cooling them with liquid nitrogen, that is, if T < sub > c </ sub > > 77 K ), or low temperature ( generally if they need other techniques to be cooled under their critical temperature ).

By the Magdalenian period ( c. 15000-9500 BC ), spear-throwers similar to the later atlatl were in use

By Thomas Gainsborough, c. 1765.

By the radiation pressure equation σT < sup > 4 </ sup >/ c ; the sun-facing photon pressure is 3. 61 µPa ( 3. 6 N / km², 2. 08 lbf / mi² ).

* c. 2410 BCE: By this time, kings in Sumer have ceased to be automatically high priests of the city deity.

By the early Middle Kingdom ( c. 2055 – 1650 BC ), non-royal Egyptians believed that they, too, could overcome death as Osiris had, by worshipping him and receiving the funerary rites that were partly based on his myth.

By the time of Herodotus ( c. 475 BC ), the Royal Road of the Persian Empire ran some 2, 857 km from the city of Susa on the Karun ( 250 km east of the Tigris ) to the port of Smyrna ( modern İzmir in Turkey ) on the Aegean Sea.

By the end of the Archaean c. 2500 Mya, plate tectonic activity may have been similar to that of the modern Earth.

By his first wife Anna Komnena Angelina ( b. c. 1176 ), married in 1199, Theodore had three daughters and two sons who died young:

By her he had at least one son, Constantine Doukas, co-emperor from c. 1075 to 1078 and from 1081 to 1087 / 8.

By the late 2nd century Chauci raiding was ongoing and more serious than before, continuing in the North Sea Channel until their last recorded raids c. 170 – 175.

By 300 AD, Uruk was mostly abandoned, and by c 700 AD it was completely abandoned.

By the fifteenth dynasty, they ruled Lower Egypt, and at the end of the seventeenth dynasty, they were expelled ( c. 1560 BC ).

By counting the number of stalls and urinals I attempted to form a loose estimate of how many men the hall would hold at one time.

By failing to do as he was told instantly -- to take out a permit or return the gun to his car -- he had played into Lord's hands.

By 1898, rustling losses had been driven down to the lowest level ever seen in Wyoming.

By now Curt was seeing clearly again.

By then Hez could see for himself, and so could the others.

By now Harmony could see that most of the adults in the train were winded and resting, or else siphoned off from the games by the challenging lure of the great cliff towering above them.

`` By God '', Waddell said, `` we don't want to upset the boy at this time of all times.

By prevailing over other claimants for the loyalties of men, the nation-state maintained an adequate measure of certainty and order within its territorial borders.

By subduing disparate lesser groups the nation has, to some degree at least, broadened the capacity for individual liberty.

`` By observing the conductor '', he says with a twinkle in his eyes, `` I learned how not to conduct ''.

By `` image '' is meant not only a visual presentation, but also remembered sensations of any of the five senses plus the feelings which are immediately conjoined therewith.

By the death of his father in 1888 he had come into possession of the family estate and had re-assumed its traditions.

By virtue of his self-reliance, his individualism and his freedom from external restraint, the private eye is a perfect embodiment of the middle class conception of liberty, which amounts to doing what you please and let the devil take the hindmost.

By upholding his own personal code of behavior, the private detective has placed himself in opposition to a society whose fabric is permeated with crime and corruption.

By monitoring, we mean some system of control over the types of information sent from the various centers.

By contrast, the energetic reaction of the leader to the full demands his decision imposes upon him strengthens the moral intuition and gives us the measure of the man.

By distinguishing superlunary ( celestial ) and sublunary ( terrestrial ) existence, and reinforcing this with the four-element physics of Empedocles, Aristotle came to speak of the stars as perfect bodies, which moved in only a perfect way, viz. in a perfect circle.

By her eighteenth birthday her bent for writing was so evident that Papa and Mamma gave her a Life Of Dickens as a spur to her aspiration.

By the time Lilian had been graduated from public school, her parents were doing quite well.

) By the time the streetcar pulled away, he had fallen in love with Paula.

By extending this help, we hope to make possible the enthusiastic enrollment of these nations under freedom's banner.

By now she was sure she was going to have a baby, deciding it would be born in India or Burma that November.

By this time she had learned that it was futile to argue with her young husband, yet the uncomfortable fact remained: the American Congregationalists were sending them as missionaries to the Far East and paying their salaries.

By early June they were a hundred miles off the coast of Ceylon, by which time all four missionaries were hardened seafarers.

By this time, as we shall see, the Tories were already planning to `` punish '' Steele for his political writing by expelling him from the House of Commons.