No previous antitrust case, he said, had involved interests of such magnitude or presented such complex problems of relief.

We now observe that the case in which **zg is a Af curve on a quadric is impossible if the complex of singular lines consists exclusively of the lines which meet Aj.

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.

A more 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 ).

In the complex case, one can also check that the bilinear form is linear over i in one argument, and conjugate linear in the other.

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 ).

The case was extremely complex, requiring the court to review diplomatic exchanges dating back over 100 years.

In the case of infinite sets, the behavior is more complex.

In this case signals will naturally flow through the bus in physical or logical order, eliminating the need for complex scheduling.

If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).

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 contrast, patient testimonials, case reports, and even expert opinion ( however some critics have argued that expert opinion " does not belong in the rankings of the quality of empirical evidence because it does not represent a form of empirical evidence " and continue that " expert opinion would seem to be a separate, complex type of knowledge that would not fit into hierarchies otherwise limited to empirical evidence alone.

Expert systems are designed to solve complex problems by reasoning about knowledge, like an expert, and not by following the procedure of a developer as is the case in conventional programming.

However, there could still be an even more complex flash-back construction in American films in the case of W. S. Van Dyke ' sThe Lady of the Dugout ( 1918 ).

In general, the surrounding graph must be systematically recolored to turn the ring's coloring into a good one, as was done in the case above where there were 4 neighbors ; for a general configuration with a larger ring, this requires more complex techniques.

Successful vaults depend on the speed of the run, the length of the hurdle, the power the gymnast generates from the legs and shoulder girdle, the kinesthetic awareness in the air, and the speed of rotation in the case of more difficult and complex vaults.

The most important case is the field of complex numbers.

The state variable S may be a 32-or 64-bit unsigned integer ; in that case, S0 can be 0, and G ( S, n ) can be just S mod n. The best choice of F is a complex issue and depends on the nature of the data.

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.

And while every writer must be dealt with as a special case, the interested student will want to ask himself a number of questions about each.

Later experiments are often designed to test a hypothesis that a treatment effect has an important magnitude ; in this case, the number of experimental units is chosen so that the experiment is within budget and has adequate power, among other goals.

In many cases such a selection can be made without invoking the axiom of choice ; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin.

In that case it is equivalent to saying that if we have several ( a finite number of ) boxes, each containing at least one item, then we can choose exactly one item from each box.

Sometimes, Roman numerals are still used for enumeration of lists ( as an alternative to alphabetical enumeration ), for sequential volumes, to differentiate monarchs or family members with the same first names, and ( in lower case ) to number pages in prefatory material in books.

In the case of multiple alleles at a diploid locus, the number of possible genotypes ( G ) with a number of alleles ( a ) is given by the expression:

Formerly, some companies changed their ANAC number every month for secrecy ; this is still the case with a few numbers.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

In general, this is the case where the Mach number in part or all of the flow exceeds 0. 3.

The cheek teeth were short-crowned ( brachyodont ), with the tubercles more-or-less completely fused into transverse ridges, or cross-crests ( lophodont type ), and the total number of teeth was in one case the typical 44, but in another was fewer.

After stipulating a number of conditions to Hroðgar in case of his death ( including the taking in of his kinsmen and the inheritance by Unferth of Beowulf's estate ), Beowulf dives into the lake.

BRP is similar to other generic systems such as GURPS, Hero System or Savage Worlds in that it uses a simple resolution method which can be broadly applied, in this case an attempt to roll under a certain number with percentile dice.

SAT is also easier if the number of literals in a clause is limited to 2, in which case the problem is called 2SAT.

Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: " Von Neumann and I have anticipated " the question whether probabilities " might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities ( cf.

This approach is considered more cost-effective than multiprocessing, as only a small number of components within a CPU is replicated to support MT as opposed to the entire CPU in the case of MP.

The number of neutrons in a nucleus usually has very little effect on an element's chemical properties ( except in the case of hydrogen and deuterium ).

In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which make 0 a natural number .</ ref >

Even in this case, however, it is still possible to list all the elements, because the set is finite ; it has a specific number of elements.

Braille, like Baudot, uses a number symbol and a shift symbol, which may be repeated for shift lock, to fit numbers and upper case into the 31 codes that 6 bits offer.

This is usually the case when a large number of frames need to be animated.

It was fully declined for gender, case and number.

If a user added more than one drive to a system the user had to open the case and cut a trace in the circuit board to permanently change the drive's number, or hand-wire an external switch to allow it to be changed externally.

In the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument.

The condition number may also be infinite, in which case the algorithm will not reliably find a solution to the problem, not even a weak approximation of it ( and not even its order of magnitude ) with any reasonable and provable accuracy.