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.

In addition to real life counterparts of the Abyssinian maid, Milton's Paradise Lost describes Abyssinian kings keeping their children guarded at Mount Amara and a false paradise, which is echoed in Kubla Khan.

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:

Modifications would need to be made to allow such, as there is only enough room for a single track on the bridge ; it would need to be twinned or a passing loop would need to be installed on Lemieux Island, or other similar upgrade ( s ) such as a four rail gauntlet track like the one still in use on the Bordeaux Railway Bridge between Montreal and Laval, in addition to the passing loop, would need to be made.

Multiplication, subtraction, and addition can then be accomplished as they appear in the equation by starting at the left end of the equation and working toward the right.

In addition, dirt particles can be held onto a soiled surface by sticky substances or by the surface tension of liquids, including liquid greases.

and natural waters, in addition, often contain impurities such as calcium salts which can react with soils to make them more difficult to remove.

The effect of drugs that act on the iodide-concentrating mechanism can be counteracted by addition of relatively large amounts of iodine to the diet.

The foaming can be accomplished at some future time at a different location by the addition of the correct proportion of catalyst in solution.

Just before game time, Robinson's pretty wife, Connie informed him that an addition to the family can be expected late next summer.

AAC can be a permanent addition to a person's communication or a temporary aid.

Acid-base conjugate pairs differ by one proton, and can be interconverted by the addition or removal of a proton ( protonation and deprotonation, respectively ).

In addition, the exterior of an adobe wall can be covered with glass to increase heat collection.

In addition, the researchers also mention that the shifting robot can even display an individual's hair style and skin color if a photo of their face is projected onto the 3D Mask.

Such modifications can also determine the localization of the protein, e. g., the addition of long hydrophobic groups can cause a protein to bind to a phospholipid membrane.

A stream anchor, which is usually heavier than a kedge anchor, can be used for kedging or warping in addition to temporary mooring and restraining stern movement in tidal conditions or in waters where vessel movement needs to be restricted, such as rivers and channels.

For example, if the numbers involved in a computation may be arbitrarily large, the time required by a single addition can no longer be assumed to be constant.

If the amount in the sample is below an instrument's range of measurement, the method of addition can be used.

Standard addition can be applied to most analytical techniques and is used instead of a calibration curve to solve the matrix effect problem.

The University can rarely provide all the necessary training in first hand experience that an apprenticeship can, and therefore in addition to graduate level training the profession also tends towards encouraging conservation students to spend time as an intern.

In addition the fixed cell structure means that ATM can be readily switched by hardware without the inherent delays introduced by software switched and routed frames.

According to the special theory of relativity, the aberration looks as a transform of the celestial sphere due to different frames of reference and is virtually a special case of velocity addition, namely one of the light beam and frames ' relative velocity v. The formula from relativistic aberration can be simplified to

While acrylic retarders can slow drying time to several hours, it remains a relatively fast-drying medium, and the addition of too much acrylic retarder can prevent the paint from ever drying properly.

In addition, they can develop considerable amount of shame over their inadequacy to liberate their parents from alcoholism.

On some occasions a patient with a brain AVM may become aware of the noise, which can compromise hearing and interfere with sleep in addition to causing psychological distress.

In addition, cutting operations can smear the lead globules over the surface.