### Table 1. Separable potentials for the Schrodinger equation.

1997

"... In PAGE 5: ...pecifying in a proper way functions V1; V2 yields further possibilities for SV in Eq.(1). The list of the corresponding potentials is presented below in Table 1. Note that the potentials V (x1; x2) given in Table1 are not inequivalent in a usual sense. They are distinguished by the form and number of coordinate systems providing separability of the corresponding Schrodinger equations.... In PAGE 5: ...II. Coordinate systems providing separability of Eq.(1) In this section we will obtain full lists of inequivalent coordinate systems providing sep- arability of Eq.(1) with the potentials listed in Table1 . To this end we have to perform the following steps: 1.... In PAGE 5: ...rability of Eq.(1) with the potentials listed in Table 1. To this end we have to perform the following steps: 1. To insert the potentials listed in Table1 into the last equation of the system (8) and to split equalities obtained with respect to x1; x2 (this gives us systems of nonlinear ordinary di erential equations for the functions A(t), B(t), W (t), W1(t), W2(t)).... In PAGE 10: ...(1) with the potential (13) is invariant with respect to the groups (19) and (21), the above coordinate system is equivalent to the following one: x1 = cosh !1 cos !2 1; x2 = sinh !1 sin !2: Summarizing we conclude that the Schrodinger equation having the potential (13) admits SV in four inequivalent coordinate systems. The remaining cases 1{15 and 17 from Table1 are handled in a similar way, the results obtained being listed below. 1.... In PAGE 20: ...of the factor Q(t; x1; x2) I: Q(t; x1; x2) = exp ( ?1 4 A0 A x2 1 + B0 B x2 2 ? 1 2 W 0 1 A x1 + W 0 2 B x2 ) ; II: Q(t; x1; x2) = expfR(t; x1; x2)g; III: Q(t; x1; x2) = expfR(t; x1; x2)g; IV: Q(t; x1; x2) = expfR(t; x1; x2)g; where R(t; x1; x2) is given by (10). Thus, Table1 gives the full list of the potentials V (x1; x2) such that the heat equation (2) admits SV within the framework of our approach. The coordinate systems providing separability of the heat equations are given by the corresponding formulae from Sec.... In PAGE 22: ...in the form ai = ?2 @f @xi f?1; i = 1; 2 (27) with a su ciently smooth function f = f(x1; x2). Given the conditions (27), the change of the dependent variable u(t; x1; x2) = U(t; x1; x2)f(x1; x2) transforms the Fokker-Plank equation (26) to become Ut + Ux1x1 + Ux2x2 ? V (x1; x2)U = 0; (28) where we have denoted V (x1; x2) = (fx1x1 + fx2x2)f?1: (29) Table1 gives the full list of potentials V such that Eq.(26) admits SV.... ..."

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### TABLE 7.2-2 ACCURACY OF THE TRAPEZOIDAL INTEGRATION RULES

### Table 2b. Maximum Errors of the C1 Cubic Spline Method with Adaptive Mesh for Nonlinear Biharmonic Equations (1.6) Matrix Sizes 150 150 615 615 2395 2395

1997

"... In PAGE 23: ... The numerical so- lutions we thus obtained from our computer programs are compared against the exact solution. In Table2... In PAGE 24: ...points equally-spaced over the L-shape domain, of the numerical approximations against the exact solutions, where the L-shape domain is quadrangulated as shown in Figure 1 and is re ned two times and = 0:01. Table2 a. Maximum Errors of the C1 Cubic Spline Method for nonlinear Biharmonic Equations (1.... In PAGE 24: ... Maximum Errors of the C1 Cubic Spline Method for nonlinear Biharmonic Equations (1.6) Matrix Sizes 150 150 527 527 1971 1971 x3y3 3:805 10?2 2:709 10?3 3:254 10?4 x4 + y4 1:999 10?2 1:199 10?3 1:809 10?4 sin(x + 3y) 5:587 10?2 3:343 10?3 1:706 10?4 exp(2x + y) 4:795 10?1 6:569 10?2 3:758 10?3 1=(1 + x + y) 1:572 10?3 1:577 10?4 1:291 10?5 r5=2 sin(5 =2)(1 + x) 2:051 10?2 5:429 10?4 5:826 10?5 r3=2 sin(3 =2)(1 + x) 1:208 10?2 4:319 10?3 1:503 10?3 It can be also seen from Table2 a that the convergence of the numerical solu- tions is close to a fourth order convergence when the solution is su ciently smooth. For the last two functions, the convergence rate is very slow.... In PAGE 24: ... We apply a local re- nement prior to the global re nements to speedup the convergences. Refer to the Table2 b. In general, one should re ne the quadrangulation locally at all corners.... ..."

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### Table 1. Evidence of non-linear relationships in length-affinity data for several MHC class II alleles

"... In PAGE 3: ... Parametric fits were then made based on polynomials with one, two, or three fitted parame- ters (linear, quadratic, and cubic, respectively). Analysis of vari- ance from these fits showed that for these MHC class II alleles the nature of the relationship was most likely nonlinear ( Table1 ). A quadratic or cubic fit resulted in a significant reduction in sum of squares in all three cases at the 0.... ..."

### Table 2: Computation of the smallest eigenvalue for a model Schr odinger operator in dimension 30.

2002

"... In PAGE 6: ... We then examine ear- lier iterates that are accurate to within , for various values of . In Table2 we vary and present the number of iterations needed to obtain that precision, the separation rank of Fk, and the run time in seconds on a Sun Ultra10 with 360MHz processor. Although we did not enforce antisymmetry in this example, it is possible to guide the iteration to a fermionic, rather than bosonic, eigenspace.... ..."

Cited by 13

### Table 6.I: Schrodinger apos;s Equation for di erent observers 48

### Table 1. Cubic Samples

"... In PAGE 4: ... From this combined catalogue we selected a number of subsamples with di erent magnitude limits and mor- phological type in order to investigate separately voids de ned by all galaxies and by elliptical galaxies (only de Vaucouleurs type T 0). A summary of the observational samples used is given in Table1 (for further explanations of Table 1 cf.... In PAGE 4: ... From this combined catalogue we selected a number of subsamples with di erent magnitude limits and mor- phological type in order to investigate separately voids de ned by all galaxies and by elliptical galaxies (only de Vaucouleurs type T 0). A summary of the observational samples used is given in Table 1 (for further explanations of Table1 cf.... In PAGE 15: ...2. Results Mean void diameters of galaxy samples for all cube sizes used are given in Table1 , separately for voids de ned by all galaxies and by elliptical galaxies, as well as by poor clusters of galaxies. Nearby galaxy samples contain both faint and bright galaxies, thus for nearby samples it is possible to calculate void diameters using several absolute magnitude limits.... In PAGE 17: ... Error bars indicate the rms error of void diameters. 90 and 120 h?1 Mpc) in Table1 . The relative rms scatter (calculated as v over hDvi from Table 1) of void diameters for clusters amounts to 32 % whereas the mean relative scatter of galaxy de ned voids for all galaxy samples in Table 1 is 27 %.... In PAGE 17: ... 90 and 120 h?1 Mpc) in Table 1. The relative rms scatter (calculated as v over hDvi from Table1 ) of void diameters for clusters amounts to 32 % whereas the mean relative scatter of galaxy de ned voids for all galaxy samples in Table 1 is 27 %. The overall range of mean void diameters de ned by all galaxies ranges from 9 to 25 h?1 Mpc, and from 13 to 36 h?1 Mpc for voids de ned by elliptical galaxies.... In PAGE 17: ... 90 and 120 h?1 Mpc) in Table 1. The relative rms scatter (calculated as v over hDvi from Table 1) of void diameters for clusters amounts to 32 % whereas the mean relative scatter of galaxy de ned voids for all galaxy samples in Table1 is 27 %. The overall range of mean void diameters de ned by all galaxies ranges from 9 to 25 h?1 Mpc, and from 13 to 36 h?1 Mpc for voids de ned by elliptical galaxies.... In PAGE 19: ...etermined in section 5.3). The spread is largest in bright galaxy and cluster samples (cf. Table1 ) where mean diam- eters of voids are close to the diameters of voids in Poisson samples with the same number density). We therefore con- clude that statistical properties of the identi ed voids are di erent from \voids quot; found in Poissonian samples.... ..."

### Table 3: Cubic elds

### Table 2: Cubic Polynomials

2005