The binary abacus is used to explain how computers manipulate numbers.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

( A cancellative binary operation is enough.

A simple form of AM, often used for digital communications, is on-off keying: a type of amplitude-shift keying in which binary data is represented by the presence or absence of a carrier.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

If there is a mixture of only two types of atoms, not counting impurities, such as a copper-nickel alloy, then it is called a binary alloy.

In this respect, all of the various forms of an alloy containing only two constituents, like iron and carbon, is called a binary system, while all of the alloy combinations possible with a ternary alloy, such as alloys of iron, carbon and chromium, is called a ternary system.

The observations show that the cloud is asymmetrical and matches the pattern of X-ray binaries ( binary star systems containing black holes or neutron stars ), mostly on one side of the galactic center.

In mathematics, the associative property is a property of some binary operations.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

* Absorbing element, in mathematics, an element that does not change when it is combined in a binary operation with some other element

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

In computer science, an AVL tree is a self-balancing binary search tree, and it was the first such data structure to be invented.

In other words AVL tree is a binary search tree where the height of the left subtree differs from the height of the right subtree by at most 1 level, if it exceeds 1 level then rebalancing occurs.

Lookup in an AVL tree is performed exactly like in any unbalanced binary search tree.

As with all binary trees, a node's in-order successor is the left-most child of its right subtree, and a node's in-order predecessor is the right-most child of its left subtree.

For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the list being searched, or in O ( log ( n )), colloquially " in logarithmic time ".

The redesigned AIM for Mac is a full universal binary Cocoa API application that supports both Tiger and Leopard – Mac OS X 10. 4. 8 ( and above ) or Mac OS X 10. 5. 3 ( and above ).

However, Alpha Centauri is a bright binary star, whose unresolved components to the naked eye are both fainter than Arcturus.

Hipparcos also suggested that Arcturus is a binary star, with the companion about twenty times dimmer than the primary and orbiting close enough to be at the very limits of our current ability to make it out.

For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log < sub > 2 </ sub > n + 1 time units are needed to return an answer.

Division of whole numbers can be thought of as a function ; if Z is the set of integers, N < sup >+</ sup > is the set of natural numbers ( except for zero ), and Q is the set of rational numbers, then division is a binary function from Z and N < sup >+</ sup > to Q.

A binary operation, ab, depends on the ordered pair ( a, b ) and so ( ab ) c ( where the parentheses here mean first operate on the ordered pair ( a, b ) and then operate on the result of that using the ordered pair (( ab ), c )) depends in general on the ordered pair (( a, b ), c ).

The enhanced assembler's source program was then assembled by its predecessor's executable ( A1 ) into binary or decimal code to give A2, and the cycle repeated ( now with those enhancements available ), until the entire instruction set was coded, branch addresses were automatically calculated, and other conveniences ( such as conditional assembly, macros, optimisations, etc.

This " knowledge " then leads to a tendency towards a binary opposition of the orient vs. the occident, where one is defined in opposition to the other, and they are unequal in value.

The amastigotes multiply by binary fission and differentiate into trypomastigotes, which are then released into the bloodstream.

If a planet orbits one member of a binary star system, then an uppercase letter for the star will be followed by a lowercase letter for the planet.

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

If G is a group and X is a set, then a ( left ) group action of G on X is a binary operator:

If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S then an isomorphism from X to Y is a bijective function such that:

If the list is stored as an ordered array, then binary search is almost always more efficient than linear search as with n > 8, say, unless there is some reason to suppose that most searches will be for the small elements near the start of the sorted list.

A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x, y ∈ N then x · y ∈ N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N = N *, where the superscript * is the Kleene star: the set is closed under composition or concatenation of its elements.

Energy input deforms the nucleus into a fat " cigar " shape, then a " peanut " shape, followed by binary fission as the two lobes exceed the short-range strong force attraction distance, then are pushed apart and away by their electrical charge.

One way to do this is to first perform a standard binary tree search for the element in question, and then use tree rotations in a specific fashion to bring the element to the top.

The dyadic monoid can be visualized as an infinite binary tree ; more generally, if the set S has p elements, then the monoid may be represented as a p-adic tree.

then we may take those and use them as bits in a binary number, i. e. with the encoding:

Since the prime factorisation of every number is unique, so then is every binary encoding of the square-free integers.

We can work the other way and start by choosing < as a transitive trichotomous binary relation ; then a total order ≤ can equivalently be defined in two ways:

But history has shown that these types of buyouts always end in the binary of an application being modified for the worse rather than the better.

It is possible to use a Compact Flash card for storing only information that isn't changed very often, such as music, videos and binary executables, while storing the small configuration files and other frequently modified data on a small hard drive or i-RAM.

This was commonly a BASIC language program named HELLO ( or some other name ) but DOS could be modified to run other types of programs at startup, such as an executable binary file.

Logs may be monitored using standard utilities such as tcpdump, which in OpenBSD has been extended especially for the purpose, or saved to disk in a modified tcpdump / pcap binary format using the pflogd daemon.

Any requirement for sending source modifications to anyone other than the recipient of the modified binary — in fact, any forced distribution at all, beyond giving source to those who receive a copy of the binary — would put the dissident in danger.

MESS is distributed under the MAME Licence, which allows for the redistribution of binary files and source code, either modified or unmodified, but disallows selling MESS or using it commercially.

" The MAME license requires source code be included with versions of MESS that are modified from the original source, while the MESS legal page states that when distributing binary files " you should also distribute the source code.

Lookup is not modified from a standard binary search tree, and has a worst-case time of O ( log n ).

PGPLOT may be redistributed to third parties and modified, but only in binary form.

The encoder uses 11 wires to pass the height information to the transponder in the form of a Gillham Code, a modified binary Gray code.

The word could then be modified directly using an additional 8 binary toggle switches to specify the data required.