C & O president Walter J. Tuohy was summoned back for cross-examination by New York Central attorneys before examiner John Bradford who is hearing the complex case.

In the case of the complex numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugation, but there are infinitely ( uncountably ) many " wild " automorphisms ( assuming the axiom of choice ).

At extremely rare intervals the thermometer has fallen below zero (- 18 ° C ), as was the case in the remarkable cold wave of the 12th-13 February 1899, when an absolute minimum of-17 ° F (- 29 ° C ) was registered at Valley Head.

In C, the array element indices are 0-9 inclusive in this case.

a − λ1 is not invertible ( because the spectrum of a is not empty ) hence a = λ1: this algebra A is naturally isomorphic to C ( the complex case of the Gelfand-Mazur theorem ).

Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B. C.

In the case of SWNT, covalent functionalization will break some C = C double bonds, leaving " holes " in the structure on the nanotube and, thus, modifying both its mechanical and electrical properties.

Peeling vegetables can also substantially reduce the vitamin C content, especially in the case of potatoes where most vitamin C is in the skin.

A bijective *- homomorphism π is called a C *- isomorphism, in which case A and B are said to be isomorphic.

Nevertheless, the case for support of the contras continued to be made in Washington, D. C. by both the Reagan administration and the Heritage Foundation, which argued that support for the contras would counter Soviet influence in Nicaragua.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

In this case, we might compute T = A *( B * D ) only once and simply blend T * C to produce F, a single operation.

As was the case with LORAN C, its primary use was for ship navigation in coastal waters.

To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant of the 3 × 3 matrix B / 2, D / 2 ; B / 2, C, E / 2 ; D / 2, E / 2, F: that is, ∆

Pearson v. Chung, the case of a Washington, D. C. judge, Roy Pearson, who sued a dry cleaning business for $ 67 million ( later lowered to $ 54 million ), has been cited as an example of frivolous litigation.

Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ).

Ligands may also bind elsewhere, however, as is the case for bulkier ligands ( e. g., proteins or large peptides ), which instead interact with the extracellular loops, or, as illustrated by the class C metabotropic glutamate receptors ( mGluRs ), the N-terminal tail.

This is true in the case of phospholipase C beta, which possesses GAP activity within its C-terminal region.

Since annealing brass requires heating it to about 660 F ( 350 C ), the heating must be done in such a way as to heat the neck to that temperature, while preventing the base of the case from being heated and losing its hardness.

The most important case is when such an algebra is a C *- algebra.

This is not a *- ring structure ( unless the characteristic is 2, in which case it's identical to the original *), as ( so * is not a ring homomorphism ), neither is it antimultiplicative, but it satisfies the other axioms ( linear, involution ) and hence is quite similar.

The Gelfand representation or Gelfand isomorphism for a commutative C *- algebra with unit is an isometric *- isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of A with the weak * topology.

*- ian, similar to above case but with the addition of the plural suffix "- an ", common among Persians and Armenians.

Results on direct integrals can be viewed as generalizations of results about finite dimensional C *- algebras of matrices ; in this case the results are easy to prove directly.

*- Casting numbers on the case and extension housing.

The most common arguments in favour of a relationship between Indo-European and Uralic are based on seemingly common elements of morphology, such as the pronominal roots (* m-for first person ; * t-for second person ; * i-for third person ), case markings ( accusative *- m ; ablative / partitive *- ta ), interrogative / relative pronouns (* kʷ-' who ?, which?

Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case.

Thus, for example, the studies of " hypercomplex numbers ", such as considered by the Quaternion Society, were put onto an axiomatic footing as branches of ring theory ( in this case, with the specific meaning of associative algebras over the field of complex numbers.

SU ( n / n )/ U ( 1 ) A special case of the superunitary Lie algebras where we remove one U ( 1 ) generator to make the algebra simple.

Also note that Heyting algebras can be viewed as Lindenbaum algebras of intuitionistic logic, which makes them a special case of the above example.

In the case of algebras that are not posets, one uses different substructures instead of filters.

As is the case for all Hopf algebras, U < sub > q </ sub >( G ) has an adjoint representation on itself as a module, with the action being given by where.

As is the case for all Hopf algebras, the tensor product of two modules is another module.

If it is assumed that and ( this is the case, for instance, for von Neumann algebras ), then the above equality gives

In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the orthogonal Clifford algebra.

introduced algebraic groups and Lie algebras of type E < sub > 8 </ sub > over other fields: for example, in the case of finite fields they lead to an infinite family of finite simple groups of Lie type.

Boolean algebras are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices ( with extra structure ).

The case of the eigenvalue of a single operator corresponds to the algebra and a map of algebras is determined by which scalar it maps the generator T to.

In finite dimensions, the simple formally real Jordan algebras come in four infinite families, together with one exceptional case:

As a C * algebra, the Calkin algebra is remarkable because it is not isomorphic to an algebra of operators on a separable Hilbert space ; instead, a larger Hilbert space has to be chosen ( the GNS theorem says that every C * algebra is isomorphic to an algebra of operators on a Hilbert space ; for many other simple C * algebras, there are explicit descriptions of such Hilbert spaces, but for the Calkin algebra, this is not the case ).

Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form 〈 S, ·, +, ', 0, 1, < sup > C </ sup >〉, where 〈 S, ·, +, ', 0, 1 〉 is again a Boolean algebra and < sup > C </ sup > satisfies the above identities for the closure operator.

A special case is the class of trivial interior algebras which are the single element interior algebras characterized by the identity 0

In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representation by considering fields of sets instead of whole power sets.

It follows that the theory of varieties is of limited use in the study of finite algebras, where one must often apply techniques particular to the finite case.

Not every author assumes that all algebras on a pseudovariety are finite ; if this is the case, one sometimes talks of a variety of finite algebras.