Clearly, any line, l, of any bundle having one of these points of tangency, T, as vertex will be transformed into the entire pencil having the image of the second intersection of L and Q as vertex and lying in the plane determined by the image point and the generator of Af which is tangent to **zg at T.

Finally, the image of a general bundle of lines is a congruence whose order is the order of the congruence of invariant lines, namely Af and whose class is the order of the image congruence of a general plane field of lines, namely Af.

Now consider the transformation of the lines of a bundle with vertex, P, on **zg which is effected by the involution as a whole.

From the preceding remarks, it is clear that such a bundle is transformed into itself in an involutorial fashion.

The gala is the Thrift Shop's annual bundle party and, as all Thrift Shop friends know, that means the admission is a bundle of used clothing in good condition, contributions of household equipment, bric-a-brac and such to stock the shelves at the shop's headquarters at 1213 Walnut St..

In salamandrids, the male deposits a bundle of sperm, the spermatophore, and the female picks it up and inserts it into her cloaca where the sperm is stored until the eggs are laid.

This book is a bundle of four previous books ( Ensucklopedia, Huh Huh for Hollywood, The Butt-Files, and Chicken Soup for the Butt ) which are no longer in print separately.

Bundle theory, originated by the 18th century Scottish philosopher David Hume, is the ontological theory about objecthood in which an object consists only of a collection ( bundle ) of properties, relations or tropes.

According to bundle theory, an object consists of its properties and nothing more: thus neither can there be an object without properties nor can one even conceive of such an object ; for example, bundle theory claims that thinking of an apple compels one also to think of its color, its shape, the fact that it is a kind of fruit, its cells, its taste, or at least one other of its properties.

The difficulty in conceiving of or describing an object without also conceiving of or describing its properties is a common justification for bundle theory, especially among current philosophers in the Anglo-American tradition.

The apple is said to be a bundle of properties including redness, being four inches ( 100 mm ) wide, and juiciness.

Redness and juiciness, for example, may be found together on top of the table because they are part of a bundle of properties located on the table, one of which is the " looks like an apple " property.

According to this understanding, the self can not be reduced to a bundle because there is nothing that answers to the concept of a self.

According to the standard interpretation of Hume on personal identity, he was a Bundle Theorist, who held that the self is nothing but a bundle of experiences (" perceptions ") linked by the relations of causation and resemblance ; or, more accurately, that the empirically warranted idea of the self is just the idea of such a bundle.

According to this view, Hume is not arguing for a bundle theory, which is a form of reductionism, but rather for an eliminative view of the self.

In modern geometry, the extra fifth dimension can be understood to be the circle group U ( 1 ), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U ( 1 ).

* Every Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )

Each group of wires is wound in a helix so that when the wire is flexed, the part of a bundle that is stretched moves around the helix to a part that is compressed to allow the wire to have less stress.

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint union < ref group = note name =" disjoint "> The disjoint union assures that for any two points x < sub > 1 </ sub > and x < sub > 2 </ sub > of manifold M the tangent spaces T < sub > 1 </ sub > and T < sub > 2 </ sub > have no common vector.

A bundle map from the base space itself ( with the identity mapping as projection ) to E is called a section of E. Fiber bundles can be generalized in a number of ways, the most common of which is requiring that the transition between the local trivial patches should lie in a certain topological group, known as the structure group, acting on the fiber F.

If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL ( n, C ).

Let P → M be a principal bundle over a manifold M with structure Lie group G and a principal connection ω.

If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle ( i. e. it has horizontal and vertical components ), then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle.

The classical Yang – Mills action on a principal bundle with structure group G, base M, connection A, and curvature ( Yang – Mills field tensor ) F is

Such solutions usually exist, although their precise character depends on the dimension and topology of the base space M, the principal bundle P, and the gauge group G.

* Any principal bundle with structure group G gives a groupoid, namely over M, where G acts on the pairs componentwise.

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with

Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.

A principal G-bundle, where G denotes any topological group, is a fiber bundle π: P → X together with a continuous right action P × G → P such that G preserves the fibers of P and acts freely and transitively on them.

* Export-Package: Expresses which Java packages contained in a bundle will be made available to the outside world.

* Import-Package: Indicates which Java packages will be required from the outside world to fulfill the dependencies needed in a bundle.

Each bundle is a tightly coupled, dynamically loadable collection of classes, jars, and configuration files that explicitly declare their external dependencies ( if any ).

The key feature was the language's support for a " second chance " method in all classes ; if a method call on an object failed because the object didn't support it ( normally not allowed in most languages due to strong typing ), the runtime would then bundle the message into a compact format and pass it back into the object's method.

The d20 system also introduced the concept of prestige classes which bundle sets of mechanics, character development and requirements into a package which can be " leveled " like an ordinary class.

Any vector bundle V over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle ; these universal Chern classes in turn can be explicitly written down in terms of Schubert cycles.

That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of H² ( X ; Z ), which associates to a line bundle its first Chern class.

Whitney sum formula: If is another complex vector bundle, then the Chern classes of the direct sum are given by

If we work on an oriented manifold of dimension 2n, then any product of Chern classes of total degree 2n can be paired with the orientation homology class ( or " integrated over the manifold ") to give an integer, a Chern number of the vector bundle.

The Chern classes of M are thus defined to be the Chern classes of its tangent bundle.

On the other hand there was now a framework which produced families of classes, whenever there was a vector bundle involved.

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

The vanishing of the Pontryagin classes and Stiefel-Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial.