In addition to the authentication and acknowledgment procedures which precede and follow the sending of the go messages, again in special codes, each message also contains an `` internal authenticator '', another specific signal to convince the recipient that he is getting the real thing.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

Since this holds true when performing addition on any real numbers, we say that " addition of real numbers is an associative operation.

* In arithmetic, addition and multiplication of real numbers are associative ; i. e.,

In addition, phase transitions in colloidal suspensions can be studied in real time using optical techniques, and are analogous to phase transitions in liquids.

In addition, if the defensive team gains possession, but then moves backwards into the endzone and is stopped, a one point safety will be awarded to the offense, although, unlike a real safety, the offense kicks off, opposed to the team charged with the safety.

In addition, the Cauchy distribution is the only univariate distribution which is closed under linear fractional transformations with real coefficients.

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

In addition, as part of the first commercial implementation of DPNSS ( in the Government Telephone Network or GTN in 1983 ), BT insisted that the core of the network be made from PBXs of different manufacture to prove the interoperability in real life.

The real numbers are a ring, having both addition and multiplication.

Note that ƒ cannot be extended to a homomorphism of rings ( from the complex numbers to the real numbers ), since it does not preserve addition:

For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication.

In the complex case of this theorem, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ U, the vector be also in U and.

But the real numbers under addition also form a group.

By working with logarithms, multiplication of positive real numbers is replaced by addition of logs.

In addition, DOS programs would not be allowed to access or modify data and code segments that did not belong to them as they could in real mode.

In addition, hypotheses and popular rumors exist about Stalin's real father.

Thus, the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function:

In addition, he said, he has answered more than 400 messages of congratulations which led him to the comment that he himself had decided he wouldn't send another congratulatory message for the rest of his life.

In addition to this standard variety, in everyday life most Austrians speak one of a number of Upper German dialects.

In addition to the scarcely documented and confirmed biography of Cuyp ’ s life, and even more so than his amalgamated style from his three main influences, there are yet other factors that have led to the misattribution and confusion over Aelbert Cuyp ’ s works for hundreds of years.

Amputations are traumatic experiences that reduce the quality of life for patients in addition to being expensive.

In addition to de facto renunciation through apostasy, heresy, or schism, the Roman Catholic Church envisaged from 1983 to 2009 the possibility of formal defection from the Church through a decision manifested personally, consciously and freely, and in writing, to the competent church authority, who was then to judge whether it was genuinely a case of " true separation from the constitutive elements of the life of the Church ... ( by ) an act of apostasy, heresy or schism.

In addition to souvenirs of Monet and other objects of his life, the house contains his collection of Japanese woodcut prints.

Documents in Maya hieroglyphs stated chocolate was used for ceremonial purposes, in addition to everyday life.

Xavier makes the connection Milbury and X-Men villain, Mister Sinister, who has apparently long been manipulating Charles ' life in addition to other X-Men.

In addition to exercising absolute political power, Domitian believed the Emperor's role encompassed every aspect of daily life, guiding the Roman people as a cultural and moral authority.

By the end of her life Elizabeth was also reputed to speak Welsh, Cornish, Scottish and Irish in addition to English.

In addition to serving in political office, Elias supported many civic, religious, and educational causes during his life.

The story thus symbolically reflects the way in which life is brought forth through the addition of water to the land, and once it grows, water is required to bring plants to fruit.

In addition to reducing weight, implementing a wireless solution has the potential to reduce costs throughout an aircraft's life cycle.

In addition to focusing on knowledge, skills and computer literacy, the centres also carried out life enrichment education, which includes basic workers ’ rights and the dangers of HIV / AIDS.

In addition to reversing the wrongs done by Adam, Irenaeus thinks of Christ as " recapitulating " or " summing up " human life.

Marine life is a vast resource, providing food, medicine, and raw materials, in addition to helping to support recreation and tourism all over the world.

In addition, in matrilineal culture, the mother's brother was the leading male figure in a male child's life, as he mentored the child within the mother's clan.

In addition to the two main teachings of chanting and the Gohonzon, Nichiren Buddhism expounds the doctrine of the Ten Worlds of life, The Ten Factors of existence, the principle of The Three Thousand Realms in a single moment of life and the teachings of The Three Proofs for verification of the validity of teachings.

He covers the value of human life and quality of life ethics in addition to abortion and other controversial ethical questions.

In addition to maintaining normal levels of oxygen in the atmosphere, photosynthesis is the source of energy for nearly all life on earth, either directly, through primary production, or indirectly, as the ultimate source of the energy in their food, the exceptions being chemoautotrophs that live in rocks or around deep sea hydrothermal vents.

The " Lombard " appeared in 1999, ( AKA: Bronze Keyboard ) a thinner, lighter, and faster ( 333 or 400 MHz ) PowerBook with a longer battery life and had both USB and SCSI built in and was a New World ROM Mac, and then the " Pismo " in 2000, which replaced the single SCSI port with two FireWire ports, updated the PowerBook line to AGP graphics, a 100 MHz bus speed, and DVD-ROM optical drives standard, in addition to dropping the " G3 " from the PowerBook name.

In addition, the likenesses of the characters in the comics do not bear any resemblance to their prior movie counterparts, except for Ash, who still maintains the likeness of Bruce Campbell.

In one questionnaire study of university students, in addition to men overall justifying some types of aggression more than women, USA respondents justified defensive physical aggression more readily than Japanese or Spanish respondents, whereas Japanese students preferred direct verbal aggression ( but not indirect ) more than their American and Spanish counterparts .< ref > Within American culture, southern men were shown in a study on university students to be more affected and to respond more aggressively than northerners when randomly insulted after being bumped into, which was theoretically related to a traditional culture of honor in the Southern United States.

In addition, the internet makes it possible for workers in developing countries to provide in-person services and compete directly with their counterparts in other nations.

In addition to the two main protagonists, several of the fictional characters depicted in the film also have real-life counterparts.

It consists of six years, after 8 years ( including kindergarten ) of primary school, in which pupils study the same subjects as their German counterparts, with the addition of compulsory Ancient Greek, Latin and Klassieke Culturele Vorming, Classical Cultural Education, history of the Ancient Greek and Roman culture and literature.

The Xeon CPUs generally have more cache than their desktop counterparts in addition to multiprocessing capabilities.

In addition to weighing less, composite sticks can be manufactured with more consistent flexibility properties than their wooden counterparts.

In addition to the increased torque that these composite sticks possess, the sticks do not warp or absorb moisture like their wooden counterparts.

In addition, the students enjoy more freedom in the combination of subjects between Year 1-4 as compared to their non-IP counterparts.

In addition, compared to their South American counterparts, male white-headed capuchins are relatively more alert to rival males than to predators, reducing the predator detection benefits that the Central American squirrel monkey receives from associating with the white-headed capuchin compared to its South American counterparts.

In addition, Luftwaffe pilots generally flew many more individual sorties ( sometimes up to 1000 ) than their Allied counterparts.

The Committee members are generally the same as those of their British counterparts, with the addition of special representatives for such things as sports, housing, morale, etc.

The Society are said to be evil dopplegangers of the heroes of Earth-2, and make their first solo appearance in Countdown Presents The Search for Ray Palmer: Crime Society # 1 written by Sean McKeever and illustrated by Jamal Igle, In addition to the five known members, this version of the CSA includes evil versions of Green Arrow, Wildcat, Black Canary, Hawkwoman, Stargirl, and the Spectre Later issues introduce Annataz Arataz ( the evil counterpart of Zatanna ), and counterparts of Supergirl ( Kara Zor-El ), Wonder Girl ( Donna Troy ), and Booster Gold.

In addition to majors in the pure disciplines of biology, chemistry, mathematics and their education counterparts, Union has programs in place for professional and health science careers.

In addition to their modified proportions, super deformed characters typically lack the detail of their normal counterparts.

In addition, pair-fed rats lost only 47 % as much weight as their RYGB counterparts.

In addition, all Genesis locomotives have four-stroke engines instead of the two-stroke engines previously used in EMD counterparts.

Super Street Fighter II Turbo allows players to play as versions of characters from the original Super Street Fighter II in addition to their regular counterparts in the game by inputting a code for each character.

In addition, the Italian aircraft were often based at better airfields than their British and Commonwealth counterparts.

Japanese models appeared in late 1984, sporting the same specifications as their American counterparts, except for the addition of a downsized body shape and a modern C-shape maple neck with 22 medium-jumbo frets.

* TS-600 series — These models were mainly identical to their 400 series counterparts but offered the 6 Meters band as an addition.

This also meant that most bilateral agreements the UK government had negotiated with overseas counterparts contained no provisions for a second British scheduled airline in addition to the incumbent UK flag carrier.

In addition, Underground 2 is the first game in the Need for Speed series to offer three SUVs as racing vehicles, which may be modified more extensively than their compact counterparts.