For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

The two methods produce the same result ; string concatenation is associative ( but not commutative ).

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

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.

If A itself is commutative ( as a ring ) then it is called a commutative R-algebra.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism of commutative rings.

The subcategory of commutative R-algebras can be characterized as the coslice category R / CRing where CRing is the category of commutative rings.

* Any commutative ring R is an algebra over itself, or any subring of R.

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

In fact, this is the free commutative R-algebra on the set

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

More compactly, an abelian group is a commutative group.

A group in which the group operation is not commutative is called a " non-abelian group " or " non-commutative group ".

This is not so obvious because xor is commutative and associative.

Each diagram is accompanied by a `` dog chart '', a list of the levers that show which other levers any particular lever will lock if pulled.

Fig. 1 is a simplified block diagram of the ultraviolet microscopy system developed at the Medical Electronics Center of Rockefeller Institute.

The ratio of the measured antenna temperature change during a drift scan across the moon to the average brightness temperature of the moon over the antenna beam ( assuming that the brightness temperature of the sky is negligible ) was found, by graphical integration of the antenna directivity diagram, to be 0.85.

A mathematical block diagram for the leveling system is shown in Fig. 7-2.

The exact number and placement of Endosymbiotic theory | endosymbiotic events is currently unknown, so this diagram can be taken only as a general guide It represents the most parsimonious way of explaining the three types of endosymbiotic origins of plastids.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables ( see more at finite state machine, state transition table and control table ), as flowcharts ( see more at state diagram ), or as a form of rudimentary machine code or assembly code called " sets of quadruples " ( see more at Turing machine ).

An adiabat is a curve of constant entropy on the P-V diagram.

The following diagram is a P-V diagram with a superposition of adiabats and isotherms:

Hence the speedup is limited up to 20 ×, as the diagram illustrates.

An Abbe diagram is produced by plotting the Abbe number V < sub > d </ sub > of a material versus its refractive index n < sub > d </ sub >.

* In a discharging battery or galvanic cell ( diagram at right ) the anode is the negative terminal since that is where the current flows into " the device " ( i. e. the battery cell ).

A monochrome signal is identical to a color one, with the exception that the elements shown in color in the diagram ( the color burst, and the chrominance signal ) are not present.

Bridges can also be classified by their lineage, which is shown as the vertical axis on the diagram.

The chessboard is placed with a light square at the right-hand end of the rank nearest to each player, and the pieces are set out as shown in the diagram, with each queen on its own color.

The outcome of a cladistic analysis is a cladogram – a tree-shaped diagram ( dendrogram ) that represents a phylogenetic hypothesis, a hypothesis on evolutionary relationships.

If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

In this diagram, abcd is a cylindrical vessel, cd is a movable piston, and A and B are constant – temperature bodies.

In the diagram, the simian clade is nested within the primate clade.

In cladograms, the clade the authors are most interested in is usually shown in greater detail and as far away from the root – the base – as possible, commonly in the top left corner ( if the root is at the bottom ) or, like in the diagram above, at the bottom ( if the root is to the left or right ).

Extensive use is usually made of a diagrammatic style known as the block diagram.

Clayton is with him, takin him out of the valley.

His wife had said to him: `` Nellie is in love with Clayton Roy.

`` Exterminatin' cow thieves is just a business proposition with me '', he'd blandly announce.

It is also possible, but equally doubtful, that he actually shot down the hundreds of men with which his legend credits him.

Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.

One of my virtues or vices is a sort of three-dimensional imagination complete with sound effects and glorious living color.

this is not so, for education offers all kinds of dividends, including how to pull the wool over a husband's eyes while you are having an affair with his wife.

It is nothing you can put your fingers on but the air suddenly fills with a high charge of electricity.

It is Eromonga -- look hard, you can see with your naked eye the wooden scaffolding on the cliff ''.

I clapped the big man with the bleached hair on his shoulder and said heartily, hoping it would make an impression on the women: `` This one is the maku Frayne.

That place is crawling with Bill Doolin and his gang ''.

The woman eyed the youth with the avidity a coin collector might display toward a rare doubloon which is not yet in his collection.

`` What is with this vow jazz ''??

-- liberal considers that the need for a national economy with controls that will assure his conception of social justice is so great that individual and local liberties as well as democratic processes may have to yield before it.

Why, in the first place, call himself a liberal if he is against laissez-faire and favors an authoritarian central government with womb-to-tomb controls over everybody??

In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.

Reduced to its simplest terms, it is an assumption of a collective duty to compensate for the inability of individuals to cope with the rigors of the era.

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

Only one rule prevailed in my conversations with these men: The more highly placed they are -- that is, the more they know -- the more concerned they have become.

He was, and is, with the RAND Corporation, a nonprofit pool of thinkers financed by the U.S. Air Force.

They include the Navy's Atlantic Command at Norfolk, Virginia, which is in contact with the Polaris subs ; ;

In point of fact, this is a beige box with a bright red door, about one and a half feet square and hung from the wall about six feet from the door to Wisman's right.

It has nothing of the proud stride of the trained runner about it, it is not a lope, it is not done with style or verve.