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.

In the first case the fixed elements within each pencil are the multiple secant and the line joining the vertex, P, to the intersection of **zg and the plane of the pencil which does not lie on the multiple secant.

Later he lies freely in order to gain the confidences of other characters, either inventing his own reason for being interested in the case or a family excuse for pursuing a line of questioning.

It is an example of the widespread motif of Greek ( or Indo-European more generally ) names expressing " battle-prowess ", in this case the ability to withstand or push back an enemy battle line.

A case can be made that discriminatory lines came to be drawn more sharply under Athenian democracy than before or elsewhere, in particular in relation to woman and slaves, as well as in the line between citizens and non-citizens.

For the simple case where the direction of the light beam along the line SE is perpendicular to the motion of the observer then only light emitted at an angle equal to the arc cosine of the ratio of the observer's speed to the speed of light will reach the telescope.

In the first case the line is an oblique asymptote of ƒ ( x ) when x tends to +∞, and in the second case the line is an oblique asymptote of ƒ ( x ) when x tends to −∞

ASCII included the distinction of upper and lower case alphabets and a set of control characters to facilitate the transmission of written language as well as printing device functions, such as page advance and line feed, and the physical or logical control of data flow over the transmission media.

Following the rapid change from a loose formation to a rigid line of battle both fleets raised their colours ; each British ship added additional Union Flags in its rigging in case its main flag was shot away.

In case the line is busy, prospective recruits are asked to meditate on the five magic words for a few minutes, and then try calling again.

In either case, to print a line, precisely timed hammers strike against the back of the paper at the exact moment that the correct character to be printed is passing in front of the paper.

There have been repeated case studies regarding the installation of a high speed line between the cities of Valparaíso and Santiago, some even considering maglev trains, but no serious action has ever been taken on the matter.

Proof sheets were the means by which syndicates provided newspapers with black-and-white line art for the reproduction of strips ( which they arranged to have colored in the case of Sunday strips ).

Odds can also be placed on a don't come-bet just like a pass line bet ; in this case the dealer ( not the player ) places the odds bet on top of the bet in the box, because of limited space, slightly offset to signify that it is an odds bet and not part of the original don't come bet.

* Following a single or rouge, play resumes with the non-scoring team scrimmaging from its own 35-yard line, unless the single is awarded on a missed field goal, in which case the non-scoring team scrimmages from either the 35-yard line or the yard line from which the field goal was attempted, whichever is greater.

As in the two-dimensional case, each axis becomes a number line.

The simplest case is when y is a linear function of x, meaning that the graph of y divided by x is a straight line.

Polling ( often called scanning in this case ) is done by activating each x line in sequence and detecting which y lines then have a signal, thus which keys are pressed.

For a set of points on the same line there is no Delaunay triangulation ( the notion of triangulation is degenerate for this case ).

The case b ≠ 0 with A = 0 is just a straight line in the direction of b:

A radial elliptic trajectory is a non-trivial special case of an elliptic orbit, where the ellipse is a line segment.

The issue central to the case was whether Microsoft was allowed to bundle its flagship Internet Explorer ( IE ) web browser software with its Microsoft Windows operating system.

Histones dimerise about their long α2 helices in an anti-parallel orientation, and, in the case of H3 and H4, two such dimers form a 4-helix bundle stabilised by extensive H3-H3 ’ interaction.

In the case of a locomotive-type boiler, a boiler barrel extends from the firebox and the hot gases pass through a bundle of fire tubes inside the barrel which greatly increase the heating surface compared to a single tube and further improve heat transfer.

* a lens system transmitting the image from the objective lens to the viewer, typically a relay lens system in the case of rigid endoscopes or a bundle of fiberoptics in the case of a fiberscope

Informally, the tangent bundle of a manifold ( in this case a circle ) is obtained by considering all the tangent spaces ( top ), and joining them together in a smooth and non-overlapping manner ( bottom ).

The theorem of the previous section is the special case of when L is a point bundle.

A tensor density is the special case where L is the bundle of densities on a manifold, namely the determinant bundle of the cotangent bundle.

The simplest example is the case that the family of vector spaces is constant, i. e., there is a fixed vector space V such that V ( x ) = V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X × V over X.

Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle ( respectively ).

There is also an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case.

An important special case occurs when V is a line bundle.

Thus a context in which the Hodge dual is very often seen is the exterior algebra of the cotangent bundle ( i. e. the space of differential forms on a manifold ) where it can be used to construct the codifferential from the exterior derivative, and thus the Laplace-de Rham operator, which leads to the Hodge decomposition of differential forms in the case of compact Riemannian manifolds.

Fibrations do not necessarily have the local Cartesian product structure that defines the more restricted fiber bundle case, but something weaker that still allows " sideways " movement from fiber to fiber.

In the case for holomorphic vector bundle E over a smooth compact complex manifold V, the statement is in the form:

After Serre we recognise l ( K − D ) as the dimension of H < sup > 1 </ sup >( D ), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H < sup > 1 </ sup >( D ) and H < sup > 0 </ sup >( KD *), and we are reading off dimensions ( notation: K is the canonical line bundle, D * is the dual line bundle, and juxtaposition is the tensor product of line bundles ).

This is like the orientable double cover on a differential manifold: indeed that's a special case in which the line bundle is the determinant bundle ( top exterior power ) of the tangent bundle.