Eduard A. Nigsch

Melting point prediction employing k-nearest neighbor algorithms and genetic parameter optimization.

We have applied the k-nearest neighbor (kNN) modeling technique to the prediction of melting points. A data set of 4119 diverse organic molecules (data set 1) and an additional set of 277 drugs (data set 2) were used to compare performance in different regions of chemical space, and we investigated the influence of the number of nearest neighbors using different types of molecular descriptors. To compute the prediction on the basis of the melting temperatures of the nearest neighbors, we used four different methods (arithmetic and geometric average, inverse distance weighting, and exponential weighting), of which the exponential weighting scheme yielded the best results. We assessed our model via a 25-fold Monte Carlo cross-validation (with approximately 30% of the total data as a test set) and optimized it using a genetic algorithm. Predictions for drugs based on drugs (separate training and test sets each taken from data set 2) were found to be considerably better [root-mean-squared error (RMSE)=46.3 degrees C, r2=0.30] than those based on nondrugs (prediction of data set 2 based on the training set from data set 1, RMSE=50.3 degrees C, r2=0.20). The optimized model yields an average RMSE as low as 46.2 degrees C (r2=0.49) for data set 1, and an average RMSE of 42.2 degrees C (r2=0.42) for data set 2. It is shown that the kNN method inherently introduces a systematic error in melting point prediction. Much of the remaining error can be attributed to the lack of information about interactions in the liquid state, which are not well-captured by molecular descriptors.

A New Approach to Diffeomorphism Invariant Algebras of Generalized Functions

We develop the diffeomorphism invariant Colombeau-type algebra of nonlinear generalized functions in a modern and compact way. Using a unifying formalism for the local setting and on manifolds, the construction becomes simpler and more accessible than previously in the literature.

On a nonlinear Peetre's theorem in full Colombeau algebras

We adapt a nonlinear version of Peetres theorem on local operators in order to investigate representatives of nonlinear generalized functions occurring in the theory of full Colombeau algebras.

Nonlinear generalized sections and vector bundle homomorphisms in Colombeau spaces of generalized functions

We define and characterize spaces of manifold-valued generalized functions and generalized vector bundle homomorphisms in the setting of the full diffeomorphism-invariant vector-valued Colombeau algebra. Furthermore, we establish point value characterizations for these spaces.

On regularization of vector distributions on manifolds

Abstract One can represent Schwartz distributions with values in a vector bundle E by smooth sections of E with distributional coefficients. Moreover, any linear continuous operator which maps E-valued distributions to smooth sections of another vector bundle F can be represented by sections of the external tensor product E * ⊠ F

Nonlinear tensor distributions on Riemannian manifolds

{E^{*}\boxtimes F}

Colombeau algebras without asymptotics

with coefficients in the space ℒ ⁢ ( 𝒟 ′ , C ∞ )

Approximation properties of local smoothing kernels

{\mathcal{L}(\mathcal{D}^{\prime},C^{\infty})}

The functional analytic foundation of Colombeau algebras

of operators from scalar distributions to scalar smooth functions. We establish these isomorphisms topologically, i.e., in the category of locally convex modules, using category theoretic formalism in conjunction with L. Schwartz’ notion of ε-product.

Convolvability and regularization of distributions

We construct an algebra of nonlinear generalized tensor fields on manifolds in the sense of J.-F. Colombeau, i.e., containing distributional tensor fields as a linear subspace and smooth tensor fields as a faithful subalgebra. The use of a background connection on the manifold allows for a simplified construction based on the existing scalar theory of full diffeomorphism invariant Colombeau algebras on manifolds, still having a canonical embedding of tensor distributions. In the particular case of the Levi-Civita connection on Riemannian manifolds one obtains that this embedding commutes with pullback along homotheties and Lie derivatives along Killing vector fields only.