A Key Intermediate in Copper‐Mediated Arene Trifluoromethylation, [nBu4N][Cu(Ar)(CF3)3]: Synthesis,Characterization, and C(sp2)−CF3 Reductive Elimination

The synthesis, characterization, and C(sp²)?CF₃ reductive elimination of stable aryl[tris(trifluoromethyl)]cuprate(III) complexes [nBu₄N][Cu(Ar)(CF₃)₃] are described. Mechanistic investigations, including kinetic studies, studies of the effect of temperature, solvent, and the para substituent of the aryl group, as well as DFT calculations, suggest that the C(sp²)?CF₃ reductive elimination proceeds through a concerted carbon–carbon bond‐forming pathway.     

A carbohydrate-based approach for the total synthesis of strictifolione

A chiral pool approach starting with d-glucose, using the Yamaguchi protocol and a Z-selective HWE reaction followed by lactonization, has been applied to execute the total synthesis of strictifolione.     

Sequential Reflexive Logics with Noncontingency Operator

Hilbert systems L and sequential calculi [L ] for the versions of logics L= T,S4,B,S5, and Grz stated in a language with the single modal noncontingency operator A=A¬ A are constructed. It is proved that cut is not eliminable in the calculi [L ], but we can restrict ourselves to analytic cut preserving the subformula property. Thus the calculi [T ], [S4 ], [S5 ] ([Grz ], respectively) satisfy the (weak, respectively) subformula property; for [B ₂ ], this question remains open. For the noncontingency logics in question, the Craig interpolation property is established.     

On the number of zero-patterns of a sequence of polynomials

Let be a sequence of polynomials of degree in variables over a field . The zero-pattern of at is the set of those ( ) for which . Let denote the number of zero-patterns of as ranges over . We prove that for and

for . For , these bounds are optimal within a factor of . The bound () improves the bound proved by J. Heintz (1983) using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz's but slightly weaker than () follow from results of J. Milnor (1964), H.E.  Warren (1968), and others; their proofs use techniques from real algebraic geometry. In contrast, our half-page proof is a simple application of the elementary ``linear algebra bound'.

Heintz applied his bound to estimate the complexity of his quantifier elimination algorithm for algebraically closed fields. We give several additional applications. The first two establish the existence of certain combinatorial objects. Our first application, motivated by the ``branching program' model in the theory of computing, asserts that over any field , most graphs with vertices have projective dimension (the implied constant is absolute). This result was previously known over the reals (Pudlák-Rödl). The second application concerns a lower bound in the span program model for computing Boolean functions. The third application, motivated by a paper by N. Alon, gives nearly tight Ramsey bounds for matrices whose entries are defined by zero-patterns of a sequence of polynomials. We conclude the paper with a number of open problems.

     

图的min-max型最优消去顺序问题

文[1]从算法复杂性的估计中提出一个图的最优标号(排序)问题-顶点的最优消去问题.本文将给出若干基本的理论结果,其中包含NP-完全性、上下界、与其它目论参数的关系及特殊图结果等.     

A procedure for parameterizing a constraint in linear programming

A post-optimal procedure for parameterizing a constraint in linear programming is proposed. In the derivation of the procedure, the technique of pivotal operations (Jordan eliminations) is applied. The procedure is compared to another by Orchard-Hays [2], and a numerical example of the procedure is provided.     

Solution of bordered singular systems in numerical continuation and bifurcation

In numerical continuation and bifurcation problems linear systems with coefficient matrices in the block form arise naturally. Here

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and n may be large but m is small. A usually has a special structure (banded, block banded, sparse,…) and B, C, D are dense, so that it is advisable to use a specialized solver for A and to solve with M by some block method. Unfortunately, A is often also a nearly singular matrix (in fact, made nonsingular only by roundoff and truncation errors). On the other hand, M is usually nonsingular but can be ill-conditioned and in certain situations will degenerate to singularity as well. We describe numerical tests for this problem using the mixed block elimination method of Govaerts and Pryce (1993) for solving bordered linear systems with possibly nearly singular blocks A. To this end, we compute by Newton's method a triple-point bifurcation point in a parameterized reaction—diffusion equation (the Brusselator). The numerical tests show that the linear systems are solved in a stable way, in spite of the use of a black-box solver (SGBTRS from LAPACK) for a nearly singular matrix.