By the 1070s, if not earlier in the reign of Máel Coluim mac Donnchada, it appears that the Scots again controlled Strathclyde.

By doing so, he came into conflict with High King Máel Sechnaill mac Domnaill whose power base was the Province of Meath.

By recognising Brian's authority over Leth Moga, that is, the Southern Half, which included the Provinces of Munster and Leinster ( and the Hiberno-Norse cities within them ), Máel Sechnaill was simply accepting the reality that confronted him and retained control over Leth Cuinn, that is, the Northern Half, which consisted of the Provinces of Meath, Connacht, and Ulster.

By his wife Eurydice, Amyntas had three sons, Alexander II, Perdiccas III and the youngest of whom was the famous Philip II of Macedon.

By Endeïs Aeacus had two sons, Telamon and Peleus ( father of Achilles ), and by Psamathe a son, Phocus, whom he preferred to the two others, both of whom contrived to kill Phocus during a contest, and then fled from their native island.

By 1525 Henry was infatuated with his mistress Anne Boleyn and dissatisfied that his marriage to Catherine had produced no surviving sons, leaving their daughter, the future Mary I of England, as heiress presumptive at a time when there was no established precedent for a woman on the throne.

By marrying Vespasia Polla he allied himself to the more prestigious patrician gens Vespasia, ensuring the elevation of his sons Titus Flavius Sabinus II and Vespasian to senatorial rank.

By 1909, Minnie Marx successfully managed to assemble her sons into a low-quality vaudeville singing group.

By his wife, the countess Katinka Kendeffy, whom he married in Paris in 1856, Count Andrássy left two sons, and one daughter, Ilona ( b. 1858 ).

By his first wife, Ermengarde of Hesbaye ( married ca 794-98 ), he had three sons and three daughters:

By a nymph, Pareia, he had four sons, Eurymedon, Nephalion, Chryses and Philolaus, who were killed by Heracles in revenge for the murder of the latter's two companions ; and by Dexithea, one of the Telchines, he had a son Euxanthius.

By his various wives and concubines Priam was the father of fifty sons and many daughters.

By this time his relationship with the 25-year-old writer Mira Mendelson ( 1915 – 1968 ) had finally led to his separation from his wife Lina, although they were never technically divorced: indeed Prokofiev had tried to persuade Lina and their sons to accompany him as evacuees out of Moscow, but Lina opted to stay.

By his first wife, the probably Spanish Aelia Flaccilla Augustus, he had two sons, Arcadius and Honorius and a daughter, Aelia Pulcheria ; Arcadius was his heir in the East and Honorius in the West.

By his wife Drahomíra, a Hevellian princess, Vratislaus had at least two sons, Wenceslaus and Boleslaus, both of whom succeeded him as Bohemian dukes.

By his wife Maria he had a family of nine sons and six daughters.

By an ordinance of Louis Philippe I of France of 13 August 1830, it was decided that the king's children ( and his sister ) would continue to bear the arms of Orléans, that Louis-Philippe's eldest son, as Prince Royal, would bear the title of duc d ' Orléans, that the younger sons would continue to have their existing titles, and that the sister and daughters of the king would only be styled " princesses d ' Orléans ", which meant the Orléans royalty did not take the name " of France ".

By Margaret, Edward had two sons, both of whom lived into adulthood, and a daughter who died as a child.

By 1675 George's eldest uncle had died without issue, but his remaining two uncles had married, putting George's inheritance in jeopardy as his uncles ' estates might pass to their own sons, should they have had any, instead of to George.

By the time of his death, his only surviving sons were James Brown Clay and John Morrison Clay, who inherited the estate and took portions for use.

By marrying Vespasia Polla he allied himself to the more prestigious patrician gens Vespasia, ensuring the elevation of his sons Titus Flavius Sabinus II and Vespasian to the senatorial rank.

By her he had ten children, four sons and three daughters surviving at the time of her death in 1818.

By his first wife, Jane Ferneley, he had three sons and three daughters.

By the Middle Ages it came to be a tradition to evade the inequality between daughters and sons via a legal fiction, in which the father claims that he is indebted to his daughter for a certain sum of money, and that this debt is due by him and his heirs, and then makes a formal agreement that this debt is to be paid upon his death, either in cash or as a proportion of his estate equal to half the share of one of the sons ; by this legal mechanism, the daughter would either gain a share in her father's estate, or a sum of money equal to its value.

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 Eustace had three sons:

By Asteria or Asterodia, Phocus had twin sons, Crisus and Panopeus.

By this time Woodruff had accurately measured Pike as a man of great personal pride, a man who would fly into a towering rage if his integrity were questioned, and who would be anxious to avenge himself.

By tradition, a red wine should be served at approximately room temperature -- if anything a little cooler -- and be aged enough for the tannin and acids to have worked out and the sediment have settled well.

By then, the stranger was thanking Haney profusely and had one arm around his shoulders as if he were an old friend.

By and large, Robert McEnroe's adaptation of Maurice Walsh's film, `` The Quiet Man '', provides the entertainment it set out to, and we have a lively musical show if not a superlative one.

By the conditions accepted in the treaty, the Avars were to take possession of Pannonia and the Lombards were promised military support in Italy should the need arise ; also, for a period of 200 years the Lombards were to maintain the right to reclaim their former territories if the plan to conquer Italy failed, thus leaving Alboin with an alternative open.

By definition a multilinear map is alternating if it vanishes whenever two of its arguments are equal.

By this definition, if previous versions ( N-1, N-2, etc.

By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property ; therefore, there is a smallest positive integer satisfying the property " not definable in under eleven words ".

* By whether the same key is used for both encryption and decryption ( symmetric key algorithms ), or if a different key is used for each ( asymmetric key algorithms ).

By not asking about the horses, Confucius demonstrates that the sage values human beings over property ; readers are led to reflect on whether their response would follow Confucius's and to pursue self-improvement if it would not have.

By definition a set S is countable if there exists an injective function

By contrast, if M ( t ) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M ( t ) is discontinuous.

By contrast, if we look at a point on the left and placed a small paddle wheel there, the larger " current " on its left side would cause the paddlewheel to rotate counterclockwise, which corresponds to a curl in the positive z direction.

By simply denying that an efficient first cause is problematic, being easily explained as the creative action of an omnipotent God, medieval theologians re-purposed and enhanced Aristotle's argument, as if the intention had been to prove God's existence.

These skillfully rendered paintings encouraged fan Glenn Bray to ask Barks if he could commission a painting of the ducks (" A Tall Ship and a Star to Steer Her By ", taken from the cover of Walt Disney's Comics and Stories # 108 by Barks ).

By repeatedly evaluating their course, and adjusting if they are moving in the wrong direction, bacteria can direct their motion to find favorable locations with high concentrations of attractants ( usually food ) and avoid repellents ( usually poisons ).

By treating the extra acceleration terms due to the rotation of the frame as if they were forces, subtracting them from the physical forces, it's possible to treat the second time derivative of position ( relative to the rotating frame ) as absolute acceleration.

By contrast, if the same historian had made the same claim five times in five different places ( and no other types of evidence were available ), the claim is much weaker because it originates from a single source.

By contrast, a person who had been acquitted of a lesser offence could not be tried for an aggravated form even if new evidence became available.

By contrast, if the bridge actually supported his weight then he might say he " thought " that the bridge was safe, and now after proving it to himself, he knows.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

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 Euclid's lemma p < sub > 1 </ sub > must divide one of the q < sub > j </ sub >; relabeling the q < sub > j </ sub > if necessary, say that p < sub > 1 </ sub > divides q < sub > 1 </ sub >.