Manifolds with Loops

A non-contractible loop can be related to a non-contractible cycle in a graph [19] and so manifold tearing seeks to replace the standard graph used for spectral dimensionality reduction with one that contains no non-contractible cycles. This is 

One of the drawbacks of the manifold tearing procedure is that it does not work on spherical manifolds since they do not contain non-contractible loops. However, the adjustment of the parameter c does allow for two different types of tear to be produced—min and max tears. This allows for different numbers of non-contractible cycles to be broken and so allows for the method to be adapted to different types of manifold. 

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The concept of temporal variations in concentration at the flow-through detector explains why pronounced skewed peaks are often observed in flow analysis, especially with loop-based sample introduction. Taylor assumed that dispersion is symmetric in relation to an observer located at the dispersing zone [55,56], but in practice the recorded peaks are usually characterised by a rise time much shorter than the fall time (see also Fig. 1.3e). This skew effect is explained by the fact that the front and trailing portions of the flowing sample, which relate to the rise time and the fall time, respectively, have different residence times in the manifold and are therefore subjected to different extents of dispersion. 

Fig. 6 FI manifold with gas-diffusion separator nested in sample loop of the injection valve used for preconcentration of volatile species by time-bas sampling (sample loading sequence). AS, autosampler, T, heating thermostat (optional) CDS, gas-diffusion separator, V, injection valve Ri, reagent for generation of volatile species R2. acceptor reagent stream R3. derivatization reagent (optional) D, detector W, waste a, valve position in sample injection sequence. Crossed circles in valve represent blocked channels [20].

Lee, J.A., Verleysen, M. Nonlinear dimensionality reduction of data manifolds with essential loops. Neurocomputing 67, 29-53 (2005)... 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

Figure 6.8 Diagram of instrumental configuration of the LC/MS system used for characterization of crude fermentation extracts. The system consists of the following components (1) HPLC (2) loop injector (3) guard column (4) 5pm C18 HPLC column (4.6mm x 25cm) (5) zero dead volume tee (6) UV detector (7) fraction collector (8) triple quadrupole mass spectrometer equipped with ESI interface (9) ESI power supply and gas manifold and (10) syringe pump. (Reprinted with permission from Ackermann et al., 1996a. Copyright 1996 Elsevier.)...

The first of these four tables is Table 8.29. Table 8.29 lists a factor Fa that includes costs for a shell/tube steam surface condenser, steam vacuum jets with piping manifold, condensate pumps, and a condensate accumulator vessel. Please note here that plant-inhibited closed-loop cooling water is assumed to be available. 

Figure 5 Schematic of a complete multiplexed and integrated instrumental design with eight capillaries. Stars at I, U1, and U2 represent the multiplexed freeze/ thaw valves. The T-assembly is made up of eight pieces of commercial junctions stacked together. These connect to the manifold M1, the SEC (size-exclusion chromatography) purification columns, and the reaction loops. The cross-assembly is made of eight pieces of standard crosses packed together with built-in heaters. V8 is an eight-position motorized titanium valve with a center port. S1 is a two-position motorized PEEK valve. V6 is a six-position motorized PEEK valve. (Reprinted from Ref. 33 with permission.)...

Finding and characterizing the stationary states of systems with more than two DOFs is an unsolved problem. Isolated stationary points are the best known of these manifolds. In systems with two DOFs, in addition to isolated stationary points, it is also possible to find a closed loop of stationary points [77]. These are associated with parabolic resonances. More complicated manifolds wUl exist in systems having more than two DOFs. In the present discussion we wUl focus on the consequences of the existence of isolated points of stationary flow in phase space. 

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