Focus fixed point

An ion beam causes secondary electrons to be ejected from a metal surface. These secondaries can be measured as an electric current directly through a Faraday cup or indirectly after amplification, as with an electron multiplier or a scintillation device. These ion collectors are located at a fixed point in a mass spectrometer, and all ions are focused on that point — hence the name, point ion collector. In all cases, the resultant flow of an electric current is used to drive some form of recorder or is passed to an information storage device (data system). 

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

A parabola is the set of points that are equidistant from a given fixed point (the focus) and from a given fixed line (the directrix) in the plane. The key-feature of a parabola is that it is quadrilateral in one of its coordinates and linear in the other. 

For all CE instruments that use on-line detection at a fixed point along the capillary, CIEF must include a means of transporting the focused zones past the... 

Figure 3. The stable and unstable manifolds of the critical points A and B on the TCM for Z = 2. (a) The collinear eZe configuration (a = ti). The critical points A and B on the TCM are hyperbolic fixed points, (b) The Wannier ridge configuration (x = ti/2). The critical points A and B on the TCM are a stable focus and an unstable focus, respectively.

Next, we focus on a portion of the film that is small in lateral extent compared with the SAW wavelength. As the wave passes a fixed point, the lower surface of the film oscillates in response to the sinusoidal SAW surface displacement. If the film is acoustically thick (/7 1), the upper portions of the film tend to lag behind the driven substrate/film interface, inducing strains across the thickness of the film. This inertial deformation of the film results in nonuniform displacement across the film. 

A conic section or cotdc is the locus of a point which moves so that its distance from a fixed point (called the focus) is in a constant ratio (called the eccentricity) to the distance from a fixed straight line (called the directrix). 

Figure 26. Skeleton bifurcation diagram in the t/-p parameter plane for the model equation (16). Shown are Hopf and saddle-node bifurcations (SUN = saddle-unstable-node bifurcation) as well as the border of the focus-node transition (dashed line) mixed-mode wave forms exist close to the dark region (which marks the region where a fixed point is a ShQ nikov saddle focus). The phase portraits sketch the Unear stability of the fixed point(s). (Reprinted with permission from M. T. M. Koper and P. Gaspard, J. Chem. Phys. 96, 7797, 1992. Copyright 1992, American Institute of Physics.)...

To obtain an asymptotic solution in the Da 1 limit we focus on the region around the stationary front and introduce a new rescaled coordinate with the origin at the center of the front a o(Da) (which in general will not coincide with the coordinate origin, determined by the strain flow fixed point) ... 

Pig. 1.7. Normalized fluctuations of the inter-spike interval versus the noise intensity for the FitzHugh-Nagmno model. Black curve reproduces the result shown in 40). Parameters 6o = 1.05, e = 0.01, 7 = 1,4, o = 1/3 in Eqs. 1.31. The fixed point is a stable focus. Same results with iip = 1.2 shown by the red curve, where the fixed point is a stable node. 

The scaled particle theory of fluids developed by Reiss, Lebowitz, Helfand and Frisch > " need concern itself [in the case of hard spheres by virtue of Eq. (26)] only with calculating g a). To accomplish this we focus our attention on a spherical cavity of radius at least r centered about a fixed point in the fluid. A cavity is defined as a region of space devoid of molecular (hard sphere) centers (see Fig. 8). Such a cavity can be formed spontaneously in our fluid as a result of a local density fluctuation. 

Figure 26 Generation of a torus attractor via two Hopf bifurcations. The first Hopf bifurcation converts a stable fixed point (a focus) into an unstable focus. A stable limit cycle generally originates at this bifurcation point. A second Hopf bifurcation occurs, rendering the limit cycle unstable, and giving rise to a stable torus. Each Hopf bifurcation results in one additional frequency of oscillation in the system.

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

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