Generalizations of Tucker-Fan-Shashkin lemmas

Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.Comment: 10 pages, 2 figure

Electron-topological, energetic and π-electron delocalization analysis of ketoenamine-enolimine tautomeric equilibrium

The ketoenamine-enolimine tautometic equilibrium has been studied by the analysis of aromaticity and electron-topological parameters. The influence of substituents on the energy of the transition state and of the tautomeric forms has been investigated for different positions of chelate chain. The quantum theory of atoms in molecules method (QTAIM) has been applied to study changes in the electron-topological parameters of the molecule with respect to the tautomeric equilibrium in intramolecular hydrogen bond. Dependencies of the HOMA aromaticity index and electron density at the critical points defining aromaticity and electronic state of the chelate chain on the transition state (TS), OH and HN tautomeric forms have been obtained

Gas-phase reactions of HONO with HNO and NH3: an ab initio MO/TST study

The reduction of MONO by HNO and NH3 has been investigated by means, of ab initio molecular orbital and transition-state theory (TST) calculations. The main reaction channels for the HNO + trans-MONO (cis-HONO) reactions are those proceeding via five-member ring transition states, leading to the production of NO and H2O. In the temperature range 300-1000 K, TST calculations predict an A factor of 2.25 x 10(10) cm(3) mol(-1) s(-1) (or 3.63 x 10(10) cm(3) mol(-1) s(-1)) and an apparent activation energy of 20.9 kcal/mol (or 21.9 kcal/mol) for the HNO + trans-HONO (or cis-HONO) reaction. In the NH3 + HONO system, the reaction NH3 + cis/trans-HONO --> H2NNO + H2O with barrier heights centering around 34 kcal/mol can occur at high temperatures. The reversible H-atom exchange reaction NH3 + cis-HONO double left right arrow H2NH-O(H)NO double left right arrow NH2H + cis-HONO occurs readily. The calculated rate constant for the reaction at 300 K is 1.06 x 10(6) cm(3) mol(-1) s(-1), in reasonable agreement with the experimental value of 2.2 x 10(6) cm(3) mol(-1) s(-1)

A panorama of positivity. I: dimension free

This survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included