Degenerate polynomial patches of degree 11 for almostGC 2 interpolation over triangles

We consider the problem of interpolating scattered data in ³ by analmost geometrically smoothGC ² surface, where almostGC ² meansGC ² except in a finite number of points (the vertices), where the surface isGC ¹. A local method is proposed, based on employing so-called degenerate triangular Bernstein-Bézier patches. We give an analysis of quintic patches forGC ¹ and patches of degree eleven for almostGC ² interpolation.     

A discrete method of optimal control based upon the cell state space concept

A discrete method of optimal control is proposed in this paper. The continuum state space of a system is discretized into a cell state space, and the cost function is discretized in a similar manner. Assuming intervalwise constant controls and using a finite set of admissible control levels (u) and a finite set of admissible time intervals (), the motion of the system under all possible interval controls (u, ) can then be expressed in terms of a family of cell-to-cell mappings. The proposed method extracts the optimal control results from these mappings by a systematic search, culminating in the construction of a discrete optimal control table.The possibility of expressing the optimal control results in the form of a control table seems to give this method a means to make systems real-time controllable.Dedicated to G. LeitmannThe material is based upon work supported by the National Science Foundation under Grant No. MEA-82-17471. The author is also indebted to Professor G. Leitmann for his many helpful comments.     

Scaling properties of ℤ k- 1 actions on the circle

We derive universal scaling properties for ^k–1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k–1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.     

Fractional occupation numbers and spin density functional calculations of degenerate systems

We implemented ab initio self‐consistent field (SCF) fractional occupation numbers (FON) calculation with Dunlap's interpolation scheme for the twisted ethylene, which is a prototype molecule of a σ–π biradical system. The calculational results are compared with those of complete‐active‐space (CAS) SCF and spin‐unrestricted Kohn–Sham (UKS) calculations on potential surfaces, occupation numbers of natural orbitals, and correlation entropies. It was found that the UKS methods gave similar results to CASSCF, while the FON solutions appeared in only the nearly complete degenerate region. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 317–323, 2003     

Topological degree of a family of convex functions and localization of nash equilibrium points

We define the degree of a family of convex functionsf _x, wherex , and prove that, if this degree is different from zero, one has at least one Nash equilibrium point in . This is a criterion of localization of such equilibrium points.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

From Weakly Coordinating to Non‐Coordinating Anions? A Simple Preparation of the Silver Salt of the Least Coordinating Anion and Its Application To Determine the Ground State Structure of the Ag(η2‐P4)2+ Cation

The unexpected but facile preparation of the silver salt of the least coordinating [(RO)₃Al‐F‐Al(OR)₃]^? anion (R=C(CF₃)₃) by reaction of Ag[Al(OR)₄] with one equivalent of PCl₃ is described. The mechanism of the formation of Ag[(RO)₃Al‐F‐Al(OR)₃] is explained based on the available experimental data as well as on quantum chemical calculations with the inclusion of entropy and COSMO solvation enthalpies. The crystal structures of (RO)₃Al←OC₄H₈, Cs⁺[(RO)₂(Me)Al‐F‐Al(Me)(OR)₂]^?, Ag(CH₂Cl₂)₃⁺[(RO)₃Al‐F‐Al(OR)₃]^? and Ag(η²‐P₄)₂⁺[(RO)₃Al‐F‐Al(OR)₃]^? are described. From the collected data it will be shown that the [(RO)₃Al‐F‐Al(OR)₃]^? anion is the least coordinating anion currently known. With respect to the fluoride ion affinity of two parent Lewis acids Al(OR)₃ of 685 kJ mol^?1, the ligand affinity (441 kJ mol^?1), the proton and copper decomposition reactions (?983 and ?297 kJ mol^?1) as well as HOMO level and HOMO–LUMO gap and in comparison with [Sb₄F₂₁]^?, [Sb(OTeF₅)₆]^?, [Al(OR)₄]^? as well as [B(R^F)₄]^? (R^F=CF₃ or C₆F₅) the [(RO)₃Al‐F‐Al(OR)₃]^? anion is among the best weakly coordinating anions (WCAs) according to each value. In contrast to most of the other cited anions, the [(RO)₃Al‐F‐Al(OR)₃] anion is available by a simple preparation in conventional inorganic laboratories. The least coordinating character of this anion was employed to clarify the question of the ground state geometry of the Ag(η²‐P₄)₂⁺ cation (D_2h, D₂ or D_2d?). In agreement with computational data and NMR spectra it could be shown that the rotation along the Ag‐(P‐P‐centroid) vector has no barrier and that the structure adopted in the solid state depends on packing effects which lead to an almost D_2h symmetric Ag(η²‐P₄)₂⁺ cation (0 to 10.6° torsion) for the more symmetrical [Al(OR)₄]^? anion, but to a D₂ symmetric Ag(η²‐P₄)₂⁺ cation with a 44° twist angle of the two AgP₂ planes for the less symmetrical [(RO)₃Al‐F‐Al(OR)₃]^? anion. This implies that silver back bonding, suggested by quantum chemical population analyses to be of importance, is only weak.     

Extended radial epiderivatives of non-convex vector-valued maps and parametric quasiconvex programming

In this paper, we consider extended vector-valued mappings defined on a normed linear space. Based on the recent semicontinuous regularizations related to hypographical and/or epigraphical profile mappings of the considered function introduced, we define semicontinuous radial epiderivatives. We, then, demonstrate that the properties of these epiderivatives amount to properties of hypographical and/or epigraphical profile mappings of the corresponding difference quotient of the underlying function, which simplify fairly well the proofs in the radial epiderivative formulaes. In particular, we stress the impact of semicontinuity, hence, we characterize with new arguments the radial epiderivatives in terms of the suprema and/or infima of the interiorly radial cone of the hypograph and/or epigraph of the considered function. Finally, we obtain optimality conditions for general non-convex constrained vector optimization problems. We apply thereafter the obtained pattern to a parametric quasiconvex programming problem for which we derive necessary and sufficient optimality conditions that are not sensitive to perturbation at the nominal level, yielding henceforth more – and strong at least under asymptotically regular constraints – information than the recent stability results obtained under additional conditions on the regularity of the normal cone to the adjusted sublevel sets of the underlying function.     

Phosphoryl‐Rich Flame‐Retardant Ions (FRIONs): Towards Safer Lithium‐Ion Batteries

The functionalized catecholate, tetraethyl (2,3‐dihydroxy‐1,4‐phenylene)bis(phosphonate) (H₂‐DPC), has been used to prepare a series of lithium salts Li[B(DPC)(oxalato)], Li[B(DPC)₂], Li[B(DPC)F₂], and Li[P(DPC)₃]. The phosphoryl‐rich character of these anions was designed to impart flame‐retardant properties for their use as potential flame‐retardant ions (FRIONs), additives, or replacements for other lithium salts for safer lithium‐ion batteries. The new materials were fully characterized, and the single‐crystal structures of Li[B(DPC)(oxalato)] and Li[P(DPC)₃] have been determined. Thermogravimetric analysis of the four lithium salts show that they are thermally stable up to around 200 °C. Pyrolysis combustion flow calorimetry reveals that these salts produce high char yields upon combustion.