Uniform consistency of automatic and location-adaptive delta-sequence estimators

Summary The class of delta-sequence estimators for a probability density includes the kernel, histogram and orthogonal series types, because each can be characterized as a collection of averages of some function that is indexed by a smoothing parameter. There are two important extensions of this class. The first allows a random smoothing parameter, for example that specified by a cross-validation method. The second allows the smoothing parameter to be a function of location, for example an estimator based on nearest-neighbor distance. In this paper a general method is presented which establishes uniform consistency for all of these estimators.Research partially supported by AFOSR Grant No. S-49620-82-C-0144, and by NSF Grant DMS-850-3347Research supported by NSF Grant DMS-8400602     

On the nature of the perfluoroalkyl (Rf) effect

In this paper, we assess the ability of trifluoro-methyl groups to thermodynamically stabilize unsaturated and strained organic species relative to saturated analogues. Theoretical calculations with the STO-3G basis set are utilized in this analysis. The resulting theoretical data, as well as experimental calorimetric data, indicate little stabilization and perhaps some destabilization resulting from the replacement of H by CF₃ (CF₃ is unambiguously less stabilizing than methyl). This substituent effect is readily explained by dominance in CF₃-substituted compounds of an electron-withdrawing inductive effect which destabilizes the unsaturated linkages. We thus conclude that the striking thermal stability of perfluoroalkylated strained rings is completely kinetic in nature. The origins of the kinetic effect are explored.     

New uniqueness proofs for the (5, 8, 24), (5, 6, 12) and related Steiner systems

New elementary proofs of the uniqueness of certain Steiner systems using coding theory are presented. In the process some of the codes involved are shown to be unique.The uniqueness proof for the (5, 8, 24) Steiner system is due to John Conway. The blocks of the system are used to generate a length 24 binary code. Any two such codes are then shown to be equivalent up to a permutation of the coordinates. This code turns out to be the extended Golay code.In the uniqueness proof for the (4, 7, 23) system, the blocks generate a length 23 code which is extended to a length 24 code. The minimum weight vectors of this larger code hold a (5, 8, 24) Steiner system. This result together with the previous one completes the proof. At this point it is also possible to conclude that the codes involved are unique and hence equivalent to the binary perfect Golay code and its extension.Continuing with the uniqueness result for the (3, 6, 22) Steiner system, the blocks generate a length 22 code which is extended to the same length 24 code by the addition of two coordinates and one additional vector. This extension ultimately requires the computation of the coset weight distribution of the length 22 code, a result heretofore unknown. The complete coset weight distribution for a specific (22, 11, 6) self-dual code is computed using the CAMAC computer system.The (5, 6, 12) and (4, 5, 11) Steiner systems are treated differently. It is shown that each system is completely determined by the choice of six blocks which may be assumed to lie in any such design. These six blocks in fact form a basis for length 12 (and 11) ternary codes corresponding to the two systems and may be generated by an algorithm independent of the designs. This algorithm is presented and the minimum weight vectors of the resulting codes, the perfect ternary Golay code and its extension, are calculated by the CAMAC system.     

An explicit retraction of symplectic flag manifolds onto complex flag manifolds

We construct an explicit deformation retraction of the manifold of symplectic flags onto the manifold of complex flags. The main tool is the polar decomposition of symplectic matrices. We also give a new definition of symplectic Stiefel manifold and prove that it has the same homotopy type as the complex Stiefel manifold.     

Interactive High-Dimensional Data Visualization

Abstract

We propose a rudimentary taxonomy of interactive data visualization based on a triad of data analytic tasks: finding Gestalt, posing queries, and making comparisons. These tasks are supported by three classes of interactive view manipulations: focusing, linking, and arranging views. This discussion extends earlier work on the principles of focusing and linking and sets them on a firmer base. Next, we give a high-level introduction to a particular system for multivariate data visualization—XGobi. This introduction is not comprehensive but emphasizes XGobi tools that are examples of focusing, linking, and arranging views; namely, high-dimensional projections, linked scatterplot brushing, and matrices of conditional plots. Finally, in a series of case studies in data visualization, we show the powers and limitations of particular focusing, linking, and arranging tools. The discussion is dominated by high-dimensional projections that form an extremely well-developed part of XGobi. Of particular interest are the illustration of asymptotic normality of high-dimensional projections (a theorem of Diaconis and Freedman), the use of high-dimensional cubes for visualizing factorial experiments, and a method for interactively generating matrices of conditional plots with high-dimensional projections. Although there is a unifying theme to this article, each section—in particular the case studies—can be read separately.     

Synthesis and host-guest studies of chiral N-linked peptidoresorc[4]arenes

Four cone resorc[4]arene octamethyl ethers (10, 11, ent-10, and ent-11) tetrafunctionalized at the feet with valyl-leucine [LL- (6); DD- (ent-6)] and leucyl-valine [LL- (9); DD- (ent-9)] methyl esters have been synthesized. These compounds, obtained by conjugation of macrocycle tetracarboxylic acid chlorides with the appropriate terminal amino groups of the above dipeptides, are N-linked peptidoresorc[4]arenes. We found that these macrocycles (M) are capable of recognizing the homologue dipeptides as guests (G), both in solution and in the gas phase, by forming relatively stable host-guest complexes ([M.G]), resistant to chromatographic purification but not to heating. Complexation phenomena between M and G in solution were investigated by NMR methods, including NMR DOSY experiments, for the detection of translational diffusion. Heteroassociation constants of 2030 and 186 M(-1) were obtained by the Foster-Fyfe method for the complexes [10.6] and [10.ent-6], respectively, the latter being comparable to the self-association constant of dipeptide itself. Conversely, the structural features of the proton-bound complexes [M.H.Gn]+ (n = 1, 2), generated in the gas phase by electrospray ionization mass spectrometry (ESI-MS), were investigated by collision-induced dissociation (CID) experiments. In both cases, the four N-linked peptidoresorc[4]arenes were shown to act as synthetic receptors and to recognize the homologue dipeptide by means of hydrogen bonds.