Locating the lowercase letters in columns 6 and 7 caused the characters to differ in bit pattern from the upper case by a single bit, which simplified case-insensitive character matching and the construction of keyboards and printers.

Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired ( as is often the case in category theory ).

Extremely large objects are normally shipped in sections and assembled onsite, but shipping an assembled unit reduced costs and avoided reliance on construction labor at the delivery site ( which in this case was still recovering from Hurricane Katrina ).

With their careful construction — and in the case of Easy Street and The Immigrant, their social commentary — these films are considered by Chaplin scholars to be among his finest work.

A typical application is furnished by the Arzelà – Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

The same type of construction works in the general case of congruence equations.

Before beginning the construction, the vertex figure ABCD is obtained by cutting each connected edge at ( in this case ) its midpoint.

In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve first-and second-order equations, while doubling a cube requires the solution of a third-order equation.

The case helped halt the construction of a power plant on Storm King Mountain in New York State.

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 ).

Placing the modifying noun in the genitive case is one way to indicate that two nouns are related in a genitive construction.

Modern English typically does not morphologically mark nouns for a genitive case in order to indicate a genitive construction ; instead, it uses either the

Early Italian instruments, on the other hand, were so light in construction that they were treated rather like a violin: kept for storage in a protective outer case, and played after taking it out of its case and placing it on a table.

# We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L < sub > g </ sub >( h ) = gh.

( Diophantus also resorts to what could be called a special case of a secant construction.

In categorical terms, this is a special case of a ( co ) filtered limit construction.

In 1888, during the construction of the new building, workers discovered the dismembered torso of a female ; the case, known as the " Whitehall Mystery ", has never been solved.

In the case of extremely mountainous terrain, railway engineers opted for the more economical narrow gauge construction.

Ideon's article also mentioned the fact that Meri himself had shown concern for the condition of the then Soviet-era construction ( in one memorable case Meri, having arrived from Japan, led the group of journalists that were expecting him, to the airport's toilets to do the interview there, in order to point out the shoddy condition of the facilities ).

(" Screen " in this case refers to electrical " screening " or shielding, not physical construction: all " grid " electrodes in between the cathode and plate are " screens " of some sort rather than solid electrodes since they must allow for the passage of electrons directly from the cathode to the plate ).

One such application, ccsNet, was tested in a case study funded by the Federal Transportation Administration ( FTA ) in which the efficacy of configuration management was measured through comparing the approximately 80 % complete construction of the Los Angeles County Metropolitan Transit Agency ( LACMTA ) 1st and 2nd segments of the Red Line, a $ 5. 3 billion rail construction project.

The Jones matrix in each case may therefore be written with the convenient construction

Most Lithuanian nouns can take the illative ending, indicating that from the descriptive point of view the illative still can be treated as a case in Lithuanian, although since the beginning of the 20th century it isn't included in the lists of standard Lithuanian cases in most grammars and textbooks and the prepositional construction į + accusative is more frequently used today to denote direction.

The coordinates of the vector are equal to the projections of the vector ( yellow ) onto the x-component basis vector ( green )-using the dot product ( a special case of an inner product, see below ).

A special case is the inner product of a vector with itself, which is square of its norm ( magnitude ):

The analogy in the case of rotating bucket is that the element of water surface will " slide " up or down the surface unless the normal to the surface aligns with the vector resultant formed by the vector addition F < sub > g </ sub > + F < sub > Cfgl </ sub >.

In the case where the vector field v and ∇ are interchanged:

( Note: ∇< sup > 2 </ sup > F represents, in this case, the vector formed by the individual Laplacian of each component of the vector in question )

where vertical bars |...| denote the vector magnitude, which in the case of r ( t ) is simply the radius R of the path.

In light of the physical interpretation, a vector field with constant zero divergence is called incompressible or solenoidal – in this case, no net flow can occur across any closed surface.

The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution if and only if the determinant is nonzero, while in the second case that same condition means that the transformation has an inverse operation.

For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:

The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids.

If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory ; this special case has very different properties.

In this case, the group is also called a permutation group ( especially if the set is finite or not a vector space ) or transformation group ( especially if the set is a vector space and the group acts like linear transformations of the set ).

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.

Two quarks can have their spins aligned, in which case the two spin vectors add to make a vector of length S = 1 and three spin projections ( S < sub > z </ sub > = + 1, S < sub > z </ sub > = 0, and S < sub > z </ sub > = − 1 ), called the spin-1 triplet.

For computing in either case, the system must be initialized, for example into the all-zeros string,, corresponding to the vector ( 1, 0, 0, 0, 0, 0, 0, 0 ).

There is no figure which has as full symmetry group the circle group, but for a vector field it may apply ( see the 3D case below ).

In each case, lines of the bundles are transformed by involutions within the pencils they determine with the multiple secant.

( the theorem can actually be further generalized to the case where the Lagrangian depends on up to the n < sup > th </ sup > derivative using jet bundles )

The preservation, use, and reconfiguration of mummy bundles and skeletal remains, like the later Inca, may suggest that this is the case.

In the last case, slow-moving arboreal mammals such as sloths, lorises and arboreal anteaters possess retia of the highly developed type known as vascular bundles.

As in the case of vector bundles, a principal connection ω on P defines, for each curve γ in M, a mapping

case is generalized to gauge natural bundles in the form of a new geometric concept

In the case when there are just two non-zero bundles in the complex this implies that the symbol is an isomorphism off the zero section, so an elliptic complex with 2 terms is essentially the same as an elliptic operator between two vector bundles.

Conversely the index theorem for an elliptic complex can easily be reduced to the case of an elliptic operator: the two vector bundles are given by the sums of the even or odd terms of the complex, and the elliptic operator is the sum of the operators of the elliptic complex and their adjoints, restricted to the sum of the even bundles.

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 ).

In the case of the complex line bundle, we are looking in fact also for circle bundles.

The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product.

In December 2006, along with Larry Jones and his group Feed the Children, ABS participated in the " largest food distribution effort " in the U. S .. More than 10, 000 bundles of food were handed out in one day ( consisting of a case of chicken, juices, fruits and vegetable ) in Harlem, outside the Abyssinian Baptist Church on 138th Street.

In the first case three bundles of tow are burnt before the newly elected pontiff, who sits on the sedia gestatoria, while a master of ceremonies says: " Pater Sancte, sic transit gloria mundi " ( Holy Father, so passes the glory of the world ).

Furthermore, in this case the structure group of the normal bundles is the circle group ; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from to the circle, which in turn equals the first integral cohomology group.

It is no longer true that the preimages π < sup > − 1 </ sup >( x ) must all look alike, unlike fiber bundles where the fibers must all be isomorphic ( in the case of vector bundles ) and homeomorphic.

The definition in that case is usually said with greater care ( using invertible sheaves or holomorphic line bundles ); see below.

Justification of the expected properties comes from the line bundle case, where V is a Whitney sum of line bundles.

In general a mechanism for reducing to that case comes from the splitting principle for vector bundles.

A hedonic regression equation treats these attributes ( or bundles of attributes ) separately, and estimates prices ( in the case of an additive model ) or elasticity ( in the case of a log model ) for each of them.

To define the Todd class td ( E ) where E is a complex vector bundle on a topological space X, it is usually possible to limit the definition to the case of a Whitney sum of line bundles, by means of a general device of characteristic class theory, the use of Chern roots ( aka, the splitting principle ).