Weight function triangle

Figure 12. The effect of the weight functions for extra moment densities on coexistence calculations, (a) The triangle and bell weight functions, for the case n— 1=3. (b) Coexistence curves for the triangle weight functions, for the same parent as in Fig. 9. Note that multiphase coexistence is generally detected for smaller n than in Fig. 9, where power-law weight functions were used for the extra moments. On the other hand, the predicted number of phases now no longer varies monotonically with n. (c) Dependence of log-error 5 on n for fixed y = 10 and three choices of weight functions for the extra moment densities Power law (solid line), triangle (dashed), bell (dot-dashed). The number of phases is marked next to the curves.

From reference (II), the weighting functions for linear triangle are given as following ... 

We now choose to approximate the weighting function by a symmetric triangle with height equal to unity and width equal to Then, the summation on the right-hand side of Equation (4.46) becomes equivalent to the area of this triangle ( ). This concludes the proof. 

Figure 3.8 Diffusion coefficients in LDPE at 22-25 °C, as a function of molecular weight. Filled circles Saleem et al. (1989), triangles Reynolds et al. (1990), open circles Moller and Gevert (1994), asterisks Moisan (1981).

Figure 1. Photoabsorption cross section for the dipole plasmon in axially deformed sodium clusters, normalized to the number of valence electrons N - The parameters of quadrupole and hexadecapole deformations are given in boxes. The experimental data [39] (triangles) are compared with SRPA results given as bars for RPA states and as the strength function (49) smoothed by the Lorentz weight with A = 0.25 eV. Contribntions to the strength function from p =0 and 1 dipole modes (the latter has twice larger strength) are exhibited by dashed curves. The bars are given in eVA. ...

Fig. 27 The effect of the molecular weight of PEI retinoate complexes on the surface tension of 0.15 m/L sodium chloride solutions as a function of time. The samples are PEI-600 retinoate (diamonds), PEI-2000 retinoate (circles), PEI-25000 retinoate (squares) and PEI-750000 retinoate (triangles). Reprinted with permission from [179]. Copyright 2000 American Chemical Society...

Fig.1. Dry thickness h0 of various PDMS layers irreversibly adsorbed on the silica surface of a silicon wafer, as a function of the scaling variable Nm 0m. N is the polymerisation index of the surface anchored chains and(])0 is the polymer volume fraction in the incubation bath. The triangles correspond to 4>0=1, i.e., to the maximum surface density of anchored chains for the particular molecular weight. The range of molecular weights used is 29.6 kg mol-1

Fig. 17 Comparison between the maximum adsorbed amount of polyampholyte with two high molecular weights and one low molecular weight as function of the block ratio Mn 65,000 g mol-1, squares Mn=101,000 g mol-1, triangles Mn 14,500 g mol-1, circles...

Fig. 3 Zero-shear viscosity at 169.5°C as a function of weight-averaged molecular weight for polystyrene melts. Open and closed circles and squares, linear triangles, H-polymers. (From Ref.. )...

Fig. 30. Weighted yield of 12C as a function of the initial mass for the Z = 0.02 (top), the Z = 0.008 (middle) and the Z = 0.004 models (bottom). We show results from [24] (black solid points), [162] (open magenta squares), [112] (solid green squares), [163] (open red circles), [164] (solid aqua triangles), and [79] (blue crosses)...

Figure 5 Relative diffusion coefficients (D/Dq) of water (triangles) and oil (circles) plotted as a function of a reduced temperature T— 7b)/C>g, where Tq is the balance temperature and g is the surfactant volume fraction, inthemainnticroemulsion channel of a Shinoda-cut diagram. Open S5mibols refer to the Cj2E5/water/cyclohexane/tetradecane (equal weights of cyclohexane and hexadecane) system (20) and filled s)mibols to the Cj2E5/water/tetradecane system (18). The data illustrate the s)mi-metric inversion of the micro structure aroimd Jq. (For a further discussion about the chosen re...

To calculate G x,u) for the 3-simplex lattice, we define restricted partition functions B and, for walks that cross the r-th order triangle, as shown in Fig. 13. is the sum of weight of all walks that enter an r-th order triangle of the 3-simplex from one corner and leave from another corner, but do not visit the third corner. Bi is the sum of weights of walks that enter and leave the r-order triangle, and also visit the third corner of the triangle. Then it is easy to see that these weights satisfy the recursion equations... 

FIGURE 4.18 Viscosity of aqueous solutions of HM polyacrylates as a function of SDS concentration (expressed in CMC). Polymers are (circles) PAA-500-1-C18 (triangles) PAA-500-3-C18 (squares) PAA-150-3-C18, where the first number refers to PAA molecular weight, the second to the percentage of hydrophobic groups, and the third to the number of carbons in the alkyl chains. Reprinted with permission from Iliopoulos et al. (1991). Copyright 1991 American Chemical Society. 

Fig. 19.24. Common logarithm of the pre-exponential factor of Go (circle) and the maximum crystal growt.h rate of Gmax (triangle) as a function of common logarithm of molecular weight for PESU [71]...

Fig. 19.29. Conunon logarithm of the growth rate as a function of common logarithm of molecular weight for PEO [77] at three different temperatures of 54.5°C circle), 56.9 C triangle) and 59.1 C (square)...

Figure 9. The influence of the molecular weight of the PLL backbone on the coefficient of friction was measured as a function of load by means of pin-on-disk tribometry. The test lubricants were aqueous buffer solutions containing PLL(350)-g[3.4]-PEG(5) (white triangles) and PLL(20)-g[3.4]-PEG(5) (black triangles). Mean values of three experiments standard deviation. Ball = steel (6mm in diameter), disc = glass, buffer solution lOmM HEPES (pH 7.4), polymer concentration = 0.25mg/ml, load = I0N,T = 25°C.

Fig. 65. Multicomponent system sodium oleate (NaO)-sodium taurocholate (NaTC)-l-monoolein (MO)-water as a function of pH. 37 C. Total lipid 2% by weight, total water 98% by weight. Triangles on right are given in mole % 100 moles NaO + NaTC + MO.

Fig. 13 (a) Tensile index and slrain-at-break of sheets made of fibres treated with PECs prepared from PAA of molecular weight 5000 and PAH of molecular weight 15,000 at a charge ratio (anion/ cation) of 0.8 as a function of PEC dose, (b) Tensile index of PEC-treated sheets (filled triangles) as a function of adsorbed amount of PAH [59] in comparison with the tensile index data for sheets made of fibres treated with PEMs after adsorption of layers 1, 3,5 and 7 (open squares) [57]... 

Characteristic width of the photon beat spectrum Itk of polystyrene solutions as a function of the wave vector AT. The width is normalized by the diffusion width />o A, and the wave vector is expressed through KR when R is the coil radius. The squares and the triangles correspond to two very different molecular weights, suggesting that there is indeed a universal scaling form. From M. Delsanti, Ph.D. Thesis, Orsay, 1978. 

Figure 38 Osmotic pressure of semidilute NaPSS solutions as a function of polymer and salt concentration. Filled circles are data of Takahashi etal. for molecular weight (MJ) =4.3 x 10 Da with no added salt and apparent salt concentration =3 x 10 M. Other symbols are data of Koene et al. for /I4 =6.5 X10 Da and salt concentrations =5 x 10 M (open circles), (%=5x 10" M (open squares), Cs=1 x 10 M (open diamonds), and c =1 X 10" M (filled triangles) for 114=4 x 10 Da and c =1 x 10 M (bottom-filled diamonds) and for 114=12 x 10 Da and c =1 x 10 M (top-filled diamonds). The solid curve corresponds to eqn [151] with A=0.16. Reproduced with permission from Dobrynin, A. V. Colby, R. H. Rubinstein, M. Macromolecules 9S6,28,1859-1871. Copyright 1995, American Chemical Society.

Figure 10 Apparent molecular weight Mapp as a function of the concentration c of giant surfactant micelles of sodium sulfbpropyl octadecyl maleate in brine, obtained by means of light scattering at a temperature of 323 K. Triangles experimental data of Berlepsch and coworkers [54], Line theory for self-assembled rigid rods fitted to the data [55]. The link free energy obtained from the fit amounts to G 26 Ab T, which is equivalent to about 65 kJ mor. The micelles have an estimated persistence length of 120 nm and a diameter of 6 nm. Also indicated by the dashed lines are the predictions of mean-field theory in the dilute and semidilute regimes [56].

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