Fourth dimension

Hopfinger et al. [53, 54] have constructed 3D-QSAR models with the 4D-QSAR analysis formahsm. This formalism allows both conformational flexibility and freedom of alignment by ensemble averaging, i.e., the fourth dimension is the dimension of ensemble sampling. The 4D-QSAR analysis can be seen as the evolution of Molecular Shape Analysis [55, 56]. [Pg.429]

These are three of the four quantum numbers familiar from general chemistry. The spin quantum number s arises when relativity is included in the problem, introducing a fourth dimension. [Pg.171]

If the distances satisfy the triangle inequalities, they are embeddable in some dimension. One possible solution is therefore to try to start refinement in four dimensions and use the allowed deviation into the fourth dimension as an additional annealing parameter [43,54]. The advantages of refinement in higher dimensions are similar to those of soft atoms discussed below. [Pg.260]

Although time as a physical or philosophical concept is an extremely subtle quantity, in chemical kinetics we adopt a fairly primitive notion of time as a linear fourth dimension (the first three being spatial dimensions) whose initial value (t = 0) can be set by the experimenter (for example, by mixing two reactant solutions) and whose extent is accurately measurable in standard units. The time dimension persists as a variable until the experimenter stops observing the reaction, or until... [Pg.1]

Normally, solids are crystalline, i.e. they have a three-dimensional periodic order with three-dimensional translational symmetry. However, this is not always so. Aperiodic crystals do have a long-distance order, but no three-dimensional translational symmetry. In a formal (mathematical) way, they can be treated with lattices having translational symmetry in four- or five-dimensional space , the so-called superspace their symmetry corresponds to a four- or five-dimensional superspace group. The additional dimensions are not dimensions in real space, but have to be taken in a similar way to the fourth dimension in space-time. In space-time the position of an object is specified by its spatial coordinates x, y, z the coordinate of the fourth dimension is the time at which the object is located at the site x, y, z. [Pg.25]

Steiner, Rudolph. The fourth dimension sacred geometry, alchemy, and mathematics listeners notes of lectures on higher-dimensional space and of questions and answers on mathematical topics introduction by David Booth. Great Barrington (MA) Anthroposophic P, 2001. xxii, 238 p. ISBN 0-88010-472-4... [Pg.535]

Moffitt, John F. Hermeticism and the art of the Fourth Dimension a review essay. Cauda Pavonis 3, no. 2 (Fall 1984) 4-6. [Pg.628]

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. [Pg.67]

To obtain pictures of the orbital ip = R0< >, we would need to combine a plot of R with that of 0, which requires a fourth dimension. There are two common ways to overcome this problem. One is to plot contour values of ip for a plane through the three-dimensional distribution as shown in Figures 3.8a,c another is to plot the surface of one particular contour in three dimensions, as shown in Figures 3.8b,d. The shapes of these surfaces are referred to as the shape of the orbital. However, plots of the angular function 0 (Figure 3.7) are often used to describe the shape of the orbital ip = RQ because they are simple to draw. This is satisfactory for s orbitals, which have a spherical shape, but it is only a rough approximation to the true shape of p orbitals, which do not consist of two spheres but rather two squashed spheres or doughnut shapes. [Pg.61]

We can conveniently think of p as a gas with a nonuniform density, which is more compressed and therefore more dense in some regions, and less compressed or less dense in other regions. Since the electron density p(x, y, z) of a molecule varies in three dimensions, we need a fourth dimension to represent it completely. Nevertheless we can get a good idea of the behavior of p by plotting constant electron density envelopes. [Pg.136]

Pomes, R. Eisenmesser, E. Post, C.B. Roux, B., Calculating excess chemical potentials using dynamic simulations in the fourth dimension, J. Chem. Phys. 1999, 111, 3387-3395... [Pg.457]

Thus, the magnetization is transferred from the amide proton to the attached nitrogen and then simultaneously to the intra- and interresidual 13C spins and sequential 13C spin. The 13C chemical shift is labelled during /, and 13C frequency during t2. The desired coherence is transferred back to the amide proton in the identical but reverse coherence transfer pathway. The 15N chemical shift is frequency labelled during t3, and implemented into the 13C 15N back-INEPT step. The sensitivity of the HNCOmCA-TROSY experiment is excellent and nearly similar to HNCA-TROSY except for the inherent sensitivity loss by a factor of /2, arising from additional quadrature detection needed for 13C frequency discrimination in the fourth dimension. The excellent sensitivity is due to a very efficient coherence transfer pathway,... [Pg.264]

Some broader contexts and meanings of these esoteric astral bodies were subsequently enunciated by C. W. Leadbeater in his book called Man Visible and Invisible (1902). Passages from this important Theosophical publication are worthy of citation because, besides reiterating standard notions about the wholly occultist astral bodies and haloes—the kind discussed by Revel in the book Duchamp is now known to have owned—they additionally document the role played within the Esoteric Tradition of important emblematic themes employed by Duchamp and long recognized by various Duchamp scholars. In short, the writings of Besant and Leadbeater prove that Theosophists often discussed both symbolic chemistry and the fourth dimension. Especially significant in this context is the fact that their pub-... [Pg.104]

The chapter following deals with Man s Vehicles. After listing their exotic nomenclature— the names used in Theosophical literature for the higher planes are derived from Sanskrit, for in Western philosophy we have as yet [1902] no terms for these worlds —Leadbeater enumerates the colorful, painterly effects of the fourth plane, or fourth dimension. Again, he relates such esoteric and fourth-dimensional phenomena directly to the spiritual operations of Occult Chemistry ... [Pg.106]

As man learns to function in these higher types of matter, he finds that the limitations of the lower life are transcended, and fall away one by one. He finds himself in a world of many dimensions, instead of one of three only and that fact alone opens up a whole series of entirely new possibilities in various directions. The study of these additional dimensions is one of the most fascinating that can be imagined. Short of really gaining the sight of the other planes, there is no method by which so clear a conception of astral life can be obtained as by the realization of the fourth dimension. ... [Pg.107]

Once plausibly unmasked as the Alchemist of the Avant-Garde, we may now in turn proceed to reveal the essentially esoteric underpinnings of Duchamp s much discussed, truly artful manipulations of Ghance and the Fourth Dimension. [Pg.264]

Everything happens as if Time itself represented a fourth dimension of space it is as if fourth-dimensional space—resulting from the combination... [Pg.269]

The transition in Duchamp s mind was between a statement like Bosse s, representing standard canons of Renaissance spatial illusionism, and what was to come much later, the Fourth Dimension. For Duchamp, the latter represented a step leading beyond mundane perspective practice, such as is briefly exemplified in his Note 6 Use transparent glass and a mirror for fourth-dimensional perspective. The idea was treated in greater detail in his Note 5 ... [Pg.274]

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