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The 4-dimensional spacetime ( c t ,x , y, z ) in which we find ourselves is called a Minkowski space, and a point in such a space is termed an event. The equation s2 C2t2 - x 2 - y 2 - 2 2 = 0 (1.1) defines the light cone (Fig. 1.1). Events that have s2 > 0 are called timelike; events that have s2 < 0 are called spacelike; events on the light cone ( s 2 = 0) are connected by signals propagating at the speed of light from the origin and are called lightlike. Timelike events can be causally connected to the origin with signals for which v < c, but spacelike events cannot be connected to the origin except by signals for which v > c . The special theory of relativity is based on two postulates: (1)the velocity of light in a vacuum is constant in all inertial systems (an inertial system is a system of reference in which Newton s first law of motion holds), and (2) the laws of physics are invariant under transformations between inertial systems (covariance). The transformations that respect these postulates are called the Lorentz transformations. Before discussing them we take a short notational detour.

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The value of this hook is 3; the product of hooks for the diagram is d = l ~ 3 ~ 4 ~ 1 ~ 2 = 2 4 . 3. The dimension of the representation is given by the ratio of the numerator n and denominator d, n Dim = -. (5.84) d Let us use these rules to work out a few simple examples. Consider the direct product of the fundamental representation of S U ( N ) with itself [eq. (5.79)]. What are the dimensions of the two representations that result In a notation inspired by the preceding rules, D i m ( m )=

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While the frameset tag (<frameset>) is responsible for de ning the layout of the entire page (in terms of number of frames and their size), the frame tag (<frame>) is responsible for de ning properties of each frame. The frame tag has the following minimal syntax:

CSS @import rule), or inline in individual elements. These styles represent the way the author intends the document to look. User styles Styles that the end user speci es should be used for the document. These styles are selected by the end user from local style sheets and can be used to modify a document s default look. The styles can be changed in Microsoft Internet Explorer, for example, within the Accessibility section of Internet Options, as shown in Figure 28-3. User agent styles Styles that a user agent uses by default when no other styles are speci ed for a particular element or document. These styles are usually very simple in nature black text on a white background, slightly larger fonts for headings, and so on.

Mixing time techniques all work on the principle of adding material to the vessel which has different properties from the bulk. Measurements are then made (usually in a controlled volume within the vessel) that show the presence of the added material. The decay of material property uctuations is used to measure the mixing time for the system. Many of the devices used for measurement of the added liquid are intrusive, and the experimenter must try to minimize the number of probes installed in the vessel in order to: Minimize the perturbation of the bulk ows by the probes and the probe supports. Reduce the amount of associated hardware required. Minimize the amount of data that need to be collected, stored, and processed. However, it is important that the probes in the vessel be in the correct positions to provide as much relevant mixing information as possible. There will be certain points in the mixing vessel that will be less intensely agitated and where mixing rates are lower than in the bulk of the tank. These points will be the last points to become mixed and will control and limit the mixing rate for the entire mixing vessel. So if probes are used, one or more should be positioned in these poorly mixed regions to obtain the correct mixing time. Other probe locations should be chosen to indicate whether the bulk of the tank is well mixed. Extra probes could be distributed axially to provide information on ow compartmentalization, particularly in multiple-impeller systems.