Minkowski-type and Alexandrov-Type Theorems for Polyhedral Herissons

Classical H. Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H. Minkowski uniqueness theorem due to A.D. Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be described as polyhedra with injective spherical image.     

Guaranteed-accuracy approximation of reachable sets for a linear dynamic system subject to impulse actions

A method is proposed for approximating the reachable set of a dynamic system with a state space dimension no higher than six-eight considered on a finite time interval. The system is governed by linear differential equations with piecewise constant coefficients and impulse actions specified at prescribed times. The method is based on guaranteed-accuracy polyhedral approximations of reachable sets at researcher-specified times. Every approximation is constructed using the preceding one. A procedure is described for choosing parameters of the method that ensure the required accuracy with close-to-minimal time costs.     

Proof of the Mixing Theorem for Statistical Systems in Classical Physics

In a general measure-theoretic context, it is proved that the action of a stochastic map on a pair of probability distributions can be characterized by the criterion of decreasing mixing distance.     

Optimal node disjoint paths on partial 2-trees: A linear algorithm and polyhedral results

We present anO(p · n) algorithm for the problem of finding disjoint simple paths of minimum total length betweenp given pairs of terminals on oriented partial 2-trees withn nodes and positive or negative arc lengths. The algorithm is inO(n) if all terminals are distinct nodes. We characterize the convex hull of the feasible solution set for the casep=2.We gratefully acknowledge the referee's many helpful suggestions to improve the presentation of this paper.     

Synthetic studies of proanthocyanidins. Part 3: Stereoselective 3,4-cis catechin and catechin condensation by TMSOTf-catalyzed intramolecular coupling method

TMSOTf-catalyzed intramolecular condensation for catechin and epicatechin units are described. A potential electrophile and a nucleophile were connected with diester linkers and TMSOTf-catalyzed condensation was examined. In comparison with intermolecular catechin and catechin condensation, the intramolecular condensation required high reaction temperature and reversed 3,4-cis product was obtained. The condensed product was transformed into the natural 3,4-cis (+)-catechin-(4β→8)-(+)-catechin dimer.     

Representations of unbounded optimization problems as integer programs

Any optimization problem in a finite structure can be represented as an integer or mixed-integer program in integral quantities.We show that, when an optimization problem on an unbounded structure has such a representation, it is very close to a linear programming problem, in the specific sense described in the following results. We also show that, if an optimization problem has such a representation, no more thann+2 equality constraints need be used, wheren is the number of variables of the problem.We obtain a necessary and sufficient condition for a functionf:SZ, withS Z ⁿ , to have a rational model in Meyer's sense, and show that Ibaraki models are a proper subset of Meyer models.This research was supported by NSF Grant No. GP-37510X1 and ONR Contract No. N00014-75-C0621, NR047-048.     

Three-dimensional configurations of organic/inorganic hybrid nanostructural blocks (I). A quantum mechanical investigation for ladder-like structure of vinylsilsesquioxane

Silsesquioxanes (SSO) or polyhedral oligomericsilsesquioxanes (POSS) are generally prepared frommolecular precursors using the hydrolytic condensa-tion of trialkoxysilane, RSi(OR')3. They are organic/inorganic hybrid nanostructural blocks with theircomplete general formula Tn(T = RSiO1.5, n = 1,2, …), and the incomplete generic formula is Tn-(OH)x(OR')y[x, y = 0, 1, 2, …, T= RSiO1.5 ?(x+y)/2n][1,2].Each VSSO, possessing a certain structural formulaand molecular weight, may h…     

Crown compounds for anions. Sandwich complex of cyclic trimeric perfluoro-o-phenylenemercury with [B12H11SCN]2– anion

The reaction of cyclic trimeric perfluoro-o-phenylenemercury (o-C₆F₄Hg)₃ (1) with the polyhedral [B₁₂H₁₁SCN]^2– anion in THF at 20 °C affords the {[(o-C₆F₄Hg)₃](B₁₂H₁₁SCN)}^2– (4) and {[(o-C₆F₄Hg)₃]₂(B₁₂H₁₁SCN)}^2– (5) complexes. Complex 5 was isolated as the tetrabutylammonium salt. X-ray diffraction analysis showed that this complex has a bent-sandwich structure in which the [B₁₂H₁₁SCN]^2– anion is located between the planes of two molecules 1 and is coordinated to both these molecules through B—H—Hg bridges and S—Hg bonds. The stability constants of complexes 4 and 5 in THF (20 °C), which were determined from the IR spectroscopic data, are 16 L mol^–1 and 992 L² mol^–2, respectively.     

Synthesis of Cobalt Bis(Dicarbollide)—Curcumin Conjugates for Potential Use in Boron Neutron Capture Therapy

A series of novel cobalt bis(dicarbollide)—curcumin conjugates were synthesized. Two conjugates were obtained through the nucleophilic ring-opening reaction of the 1,4-dioxane and tetrahydropyran derivatives of cobalt bis(dicarbollide) with the OH group of curcumin, and using two equiv. of the oxonium derivatives, two other conjugates containing two cobalt bis(dicarbollide) units per molecule were obtained. In contrast to curcumin, the conjugates obtained were found to be non-cytotoxic against both tumor and normal cell lines. The analysis of the intracellular accumulation of the conjugates by flow cytometry showed that all cobalt bis(dicarbollide)—curcumin conjugates entered HCT116 colorectal carcinoma cells in a time-dependent manner. New non-cytotoxic conjugates contain a large amount of boron atoms in the biomolecule and can potentially be used for further biological research into boron neutron capture therapy (BNCT).     

Synthesis of peptide dendrimers with polyhedral oligomeric silsesquioxane cores via click chemistry

Inorganic polyhedral oligomeric silsesquioxane (POSS) was used as the core for the synthesis of poly(Llysine) peptide dendrimer via copper-catalyzed azide-alkyne click chemistry. The inorganic/organic composite dendrimer was characterized by MS, 1H NMR, FTIR, GPC and DLS.