Some Qualitative Properties for the Total Variation Flow

We prove the existence of a finite extinction time for the solutions of the Dirichlet problem for the total variation flow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time. The asymptotic profile of the solutions of the Dirichlet problem is also studied. It is shown that the profiles are nonzero solutions of an eigenvalue-type problem that seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations.     

Total Scalar Curvature and L 2 Harmonic 1-Forms on a Minimal Hypersurface in Euclidean Space

Let M ⁿ , n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R ⁿ⁺¹. We show that if the total scalar curvature on M is less than the n/2 power of 1/C _s , where C _s is the Sobolev constant for M, then there are no L ² harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.     

Explicit Elastic Curves

The equilibria of thin rods are given by curves which are critical points of the modified total squared curvature. The critical curves are known as elastic curves. It is shown how all the elastic curves are given explicitly in terms of elliptic functions as soon as a certain set of three parameters is known. Every regular curve can be parametrized to have a constant speed but the parametrization is rarely known explicitly. Remarkably, all the elastic curves are here explicitly parametrized to have a constant speed. Curves with fixed distinct endpoints as well as closed curves are admitted. The tangent direction may be constrained at one, both, or neither of the endpoints. There are three major strands of formulas corresponding to: fixed length L, variable length without tension, and variable length with tension (let > 0 and add a term L to the total squared curvature). In the most complicated cases the three parameters are given as solutions to a non-linear system of three equations. In the least complicated case everything is given explicitly in terms of elliptic functions. If the length is variable and there is no tension, at least one of the parameters is completely determined (the elliptic modulus m = 1/2). Using the same set of parameters explicit formulas are given for: the length when it is variable, the total squared curvature, and the tangent angle along the elastic curve. A number of examples are presented which illustrate the full range of constraints.     

Extracting Algorithms from Intuitionistic Proofs

This paper presents a new method - which does not rely on the cut-elimination theorem - for characterizing the provably total functions of certain intuitionistic subsystems of arithmetic. The new method hinges on a realizability argument within an infinitary language. We illustrate the method for the intuitionistic counterpart of Buss's theory S, and we briefly sketch it for the other levels of bounded arithmetic and for the theory IΣ₁.     

The Fary-Milnor theorem in Hadamard manifolds

The Fary-Milnor theorem is generalized: Let be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If has total curvature less than or equal to , then is the boundary of an embedded disk. The example of a trefoil knot which moves back and forth abritrarily close to a geodesic segment shows that the bound is sharp in any such space. The original theorem was for closed curves in Euclidean 3-space and the proof by integral geometry did not apply to spaces of variable curvature. Now, instead, a combinatorial proof has been devised.

     

A "chiral aldehyde" equivalent as a building block towards biologically active targets

Chiral gamma-aryloxybutenolides, readily accessible through dynamic kinetic asymmetric transformation (DYKAT) of racemic acyloxybutenolides, were utilized as "chiral aldehyde" building blocks for intermolecular cycloadditions and Michael reactions. Unprecedented selectivity in trimethylenemethane cycloadditions with this building block allowed an efficient synthesis of a novel metabotropic glutamate receptor 1 antagonist in development by the Bayer corporation. These studies further inspired work that culminated in the total synthesis of (+)-brefeldin A, a natural product with a range of significant biological properties. All of the stereochemistry in this target molecule was derived from two palladium-catalyzed asymmetric allylic alkylation reactions. The trans-alkenes were synthesized by a Julia olefination and a ruthenium-catalyzed trans-hydrosilylation-protodesilylation protocol. The route to (+)-brefeldin A lends itself to analogue syntheses and was completed in 18 steps in 6 % overall yield.     

A Formal Total Synthesis of Eleutherobin Using the Ring‐Closing Metathesis (RCM) Reaction of a Densely Functionalized Diene as the Key Step: Investigation of the Unusual Kinetically Controlled RCM Stereochemistry

Asymmetric oxyallylation reactions and ring‐closing metathesis have been used to synthesize compound 3 , a key advanced intermediate used in the total synthesis of eleutherobin reported by Danishefsky and co‐workers. The aldehyde 6 , which is readily prepared from commercially available R‐(?)‐carvone in six steps in 30 % overall yield on multigram quantities, was converted into the diene 5 utilizing two stereoselective titanium‐mediated Hafner–Duthaler oxyallylation reactions. The reactions gave the desired products ( 8 and 12 ) in high yields (73 and 83 %, respectively) as single diastereoisomers, with the allylic alcohol already protected as the p‐methoxyphenyl (PMP) ether, which previous work has demonstrated actually aids ring‐closing metathesis compared to other protective groups and the corresponding free alcohol. Cyclization under forcing conditions, using Grubbs' second‐generation catalyst 13 , gave the ten‐membered carbocycle (E)‐ 14 in 64 % yield. This result is in sharp contrast to similar, but less functionalized, dienes, which have all undergone cyclization to give the Z stereoisomers exclusively. A detailed investigation of this unusual cyclization stereochemistry by computational methods has shown that the E isomer of the ten‐membered carbocycle is indeed less thermodynamically stable than the corresponding Z isomer. In fact, the selectivity is believed to be due to the dense functionality around the ruthenacyclobutane intermediate that favors the trans‐ruthenacycle, which ultimately leads to the less stable E isomer of the ten‐membered carbocycle under kinetic control. During the final synthetic manipulations the double bond of enedione (E)‐ 16 isomerized to the more thermodynamically stable enedione (Z)‐ 4 , giving access to the advanced key‐intermediate 3 , which was spectroscopically and analytically identical to the data reported by Danishefsky and co‐workers, and thereby completing the formal synthesis of eleutherobin.     

天然产物(－)-Patriscabrol和(＋)-Isopatriscabrol的全合成研究

以反丁烯二酰氯和薄荷醇为起始原料, 经十四步反应, 得到光活的(－)-patriscabrol和(＋)-isopatriscabrol, 总收率约为4.6%. 化合物的结构经¹H NMR, MS, NOE差谱, HMBC, HMQC确定.     

A General Method for the Synthesis of Bastaranes and Isobastaranes: First Total Synthesis of Bastadins 5, 10, 12, 16, 20, and 21

A general strategy for the synthesis of twenty naturally occurring bastadins (all but bastadin 3) is presented. A key retrosynthetic disconnection of the two amide bonds, common in all target molecules, bisects the macrocyclic core into two diaryl ether fragments, an α,ω‐diamine (western part) and an α,ω‐dicarboxylic acid (eastern part). Efficient preparation of the synthetically challenging o‐mono or dibromo‐substituted diaryl ether linkages was achieved employing the diaryl iodonium salt method. Regarding the western part, variations of the aliphatic chain were more efficiently secured by the preparation of two different α,ω‐aminonitrile moieties. Cobalt boride mediated reduction of the nitrile functionality established the required diamines and, at the same time, provided the necessary variation of the aromatic‐ring bromination pattern. Regarding the eastern part, two different dicarboxyl precursors had to be prepared in order to accommodate bromination‐pattern variations. Coupling and subsequent macrolactamization of different combinations of these key intermediates may lead at will to any member of this family of marine natural products. Four bastaranes (bastadins 5, 10, 12 and 16) and two isobastaranes (bastadins 20 and 21) were synthesized as a demonstration of the flexibility and efficiency of the approach presented.     

Inclusion of exact exchange for self-interaction corrected H3 density functional potential energy surface

The effect of the inclusion of the exact exchange into self-interaction corrected generalized gradient approximation density functional theory (GGA-DFT) for the simplest hydrogen abstraction reaction, H + H₂ → H₃ → H₂ + H, is presented using a triple-zeta augmented 6-311++G(d,3pd) basis set. The introduction of the self-interaction correction has a considerably larger effect on molecular geometry and vibrational frequencies than the inclusion of the exact exchange. We investigate the influence of the self-interaction error on the shape of the potential energy surface around the transition state of the hydrogen abstraction reaction. The decomposition of the self-interaction error into correlation and exchange parts shows that the exchange self-interaction error is the main component of the energy barrier error. The best agreements with the experimental barrier height were achieved by self-interaction corrected B3LYP, B-LYP and B3PW functionals with errors of 1.5, 2.9 and 3.0 kcal/mol, respectively. Received: 13 August 1997 / Accepted: 14 November 1997