In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

A character χ is a linear functional on A which is at the same time multiplicative, χ ( ab )

A functional group with a hydroxyl and cyanide bonded to the same carbon is called cyanohydrin.

where the functional form of in the Lagrangian description is not the same as the form of in the Eulerian description.

This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in the same way with his interests in physics and logic.

Conversely, in functional code, the output value of a function depends only on the arguments that are input to the function, so calling a function f twice with the same value for an argument x will produce the same result f ( x ) both times.

Depending on the particular formalism adopted for the calculus, it may be seen as a simple application of a " functional substitution " rule of inference, as in Gödel's paper, or it may be proved by considering the formal proof of, replacing in it all occurrences of Q by some other formula with the same free variables, and noting that all logical axioms in the formal proof remain logical axioms after the substitution, and all rules of inference still apply in the same way.

Each data system bus ( aka string ) was composed of the same functional elements, consisting of multiplexers ( MUX ), high-level modules ( HLM ), low-level modules ( LLM ), power converters ( PC ), bulk memory ( BUM ), data management subsystem bulk memory ( DBUM ), timing chains ( TC ), phase locked loops ( PLL ), Golay coders ( GC ), hardware command decoders ( HCD ) and critical controllers ( CRC ).

The Copenhagen Stroke Study found that patients that presented with ipsilateral pushing used 3. 6 weeks more to reach the same functional outcome level on the Barthel Index, than did patients without ipsilateral pushing.

The weapons are also almost exact functional copies of those from Doom, with the same key bindings.

While the real values of the specific economic items generally stay the same in terms of relatively stable foreign currencies, in hyperinflationary conditions the general price level within a specific economy increases rapidly as the functional or internal currency, as opposed to a foreign currency.

Harpsichords vary in size and shape, but all have the same basic functional arrangement.

In his above-mentioned account, he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis ; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system.

But Halliday ’ s conception of grammar – or ‘ lexicogrammar ’ ( a term he coined to argue that lexis and grammar are part of the same phenomenon ) – is based on a more general theory of language as a social semiotic resource, or a ‘ meaning potential ’ ( see systemic functional linguistics ).

A functional group is a molecular module, and the reactivity of that functional group is assumed, within limits, to be the same in a variety of molecules.

The Network Layer provides the functional and procedural means of transferring variable length data sequences from a source host on one network to a destination host on a different network, while maintaining the quality of service requested by the Transport Layer ( in contrast to the data link layer which connects hosts within the same network ).

For non-strict functional languages, functions and language constructs exist to achieve the same result, but they are not necessarily called control flow statements.

* The second factor has been the use of English for " decorative " or " design " rather than functional purposes ; i. e., for Japanese consumption, not for English speakers per se, and as a way of appearing " smart, sophisticated and modern ," in much the same way as Japanese and similar writing script is used in western fashion.

* Suppose & B is equivalent to & D. If we acquire new information A and then acquire further new information B, and update all probabilities each time, the updated probabilities will be the same as if we had first acquired new information C and then acquired further new information D. In view of the fact that multiplication of probabilities can be taken to be ordinary multiplication of real numbers, this becomes a functional equation

For example, the color red in a line of clothing may not be functional ( and thus part of protectable trade dress ) whereas the same color on a stop sign would be functional because the color red serves the function of putting drivers on alert ( and thus would not be part of a protectable trade dress ).

The content is not the same, however: rather than individual security, it is the security and continuing existence of an `` ideological group '' -- those in the `` free world '' -- that is basic.

Mr. Claude is a specialist in torso development and he has long favored the now-famous Weider Push-Pull Super-Set technique in which one exercise of the Super-Set is a pressing or `` pushing '' movement which accents one sector of a muscle group in a specific way, followed by a `` pulling '' exercise which works the opposing sector of the same muscle group.

Helmut Roloff, playing with a group of musicians from the Bayreuth Ensemble, gives a sturdy reading, in much the same vein as that of the last-mentioned pianists.

We quickly ran into the same trouble that plagued Bill Ruger in his first experiments: Three or four bullets would be placed well in a six-inch bull at 100 yards and then, unaccountably, one could stray far out of the group.

The improvement was most noticeable in the greater consistency among reaction cells prepared as a group on the same manifold.

Specific differences exist in the action of different drugs belonging to the same group as, for instance, between reserpine and chlorpromazine.

The parent of a group filing consolidated returns might be treated as the same corporation following a reorganization defined in section 368(a)(1), but as a different corporation for this purpose after a tax-free acquisition by another corporation which had not, for example, elected to file consolidated returns with its own subsidiaries.

Noting the complaints of inventors and members of the patent bar, he admitted that some of the strictures `` were fairly well founded '', but he added that under existing rules the courts could not consolidate testimony in a group of suits involving separate infringements of the same patent.

However, in the same three-month period, toll-road bonds, as a group, have bucked this trend.

Within a people, different means to accomplish the same goal can be adopted by subsets of the larger group.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

This behaviour under CPT is the same as the statement that the particle and its antiparticle lie in the same irreducible representation of the Poincaré group.

Around the same time, in France, Charles-Louis Havas extended the services of his news agency, Havas to include advertisement brokerage, making it the first French group to organize.

A backplane ( or " backplane system ") is a group of electrical connectors in parallel with each other, so that each pin of each connector is linked to the same relative pin of all the other connectors forming a computer bus.

On the same day, a separate rebel group Rally of Democratic Forces had captured Biltine.

As the same game is played repeatedly among a group of players, precedents build up about how a particular infraction of the rules should be handled.

* A " professional community " is a group of people with the same or related occupations.

Collectivism is a basic cultural element that exists as the reverse of individualism in human nature ( in the same way high context culture exists as the reverse of low context culture ), and stresses the priority of group goals over individual goals and the importance of cohesion within social groups ( such as an " in-group ", in what specific context it is defined ).

Note the position of the hydroxyl group ( red or green ) on the anomeric carbon relative to the CH < sub > 2 </ sub > OH group bound to carbon 5: they are either on the opposite sides ( α ), or the same side ( β ).

Some years after their arrival, the Nephites met with a similar group, the Mulekites who left the Middle East during the same period.

If you don't leave this country within 3 days, your life will be taken the same as Powell's was.

Women themselves have come to look upon matters in the same light as the outside world, and scarcely find any wrong in submitting to the importunities of a stronger will, even when their affections are withheld.

I am certain that Carl Sandburg will not fall into the same sad philosophy.

By the same test predispositions destructive of human personality exercise their most sinister impact, with the result that men of good will are often trapped and nullified.

The student of ideas and their place in history will always be concerned with the patterns of transition, which are at the same time patterns of transformation, whereby ideas pass from one area of activity to another.

An ordinary hot bath or shower will do the same.

It will determine the amount of money to be spent by the college and will certify this amount to the board of freeholders, which `` shall appropriate in the same manner as other appropriations are made by it the amount so certified and the amount shall be assessed, levied and collected in the same manner as moneys appropriated for other purposes ''.

If he had had a son, he would tell him, `` Gather ye rosebuds while ye may This same flower that smiles today tomorrow will be dying ''.

We all seek the same thing through different ways -- an end to this long night of two thousand years of darkness and unspeakable abuses which will continue to plague us until the Star of David flies over Zion ''.

Gradually others will move up to the same level

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The building will contain 430,000 square feet, approximately the same as our present plant.

Mr. Brown will at the same time undertake expansion of VecTrol's custom design program for electronic control users with a greatly increased engineering staff.

Mrs. Long still feels the same unique spirit of Westminster which she stated the present Juniors will experience today but probably will not appreciate in full for a number of years.

Since the same method of shaping and fitting the dogs and notches is used throughout, we will only describe the construction of one locking bar.

The result will be the same regardless of the order used.

The compression ratio arrived at with the formula will be the same regardless of whether cubic inches or cubic centimeters are used.

Frontage on a body of clear, clean water will be vastly different from the same amount of frontage on polluted water.

Since charges are relatively the same, reserving a car before you leave for Europe will assure you of having one on tap when you want it.

The same model car might be available in six or eight countries, yet not two countries will have the same rate either for the daily rate or rate per kilometer driven.