On continuous preservation of norms and areas

Summary We give various properties, examples and equivalent conditions for mapsT of then-dimensional euclidean space into itself (n 2) satisfying the generalised orthogonality equation|Tx Ty| = |x y| for allx, y inR ⁿ , where stands for the usual dot product, and we prove that the only continuous maps verifying this condition are the orthogonal linear transformations.     

Crystal structure oftrans-[PtCl2(Me2PhP)(cis-2,5-dimethylpyrrolidine)]

The complextrans-[PtCl₂ (Me₂PhP)(cis-2,4-dimethylpyrrolidine)] has been characterized by X-ray crystallography. It crystallizes in the monoclinic space group P 2₂/c witha=8.762(3),b=16.034(5),c=12.600(4)Å,=93.43(3)° andZ=4. The structure was solved by heavy atom method and refined toR=0.052 for 3251 observed counter reflections. The coordination sphere of the Pt atom is square-planar with a geometry very similar to that of the studied complexes of the type trans-[PtCl₂(R₃P)L] whereL is either one of the possible isomers of theC,C-dimethylpyrrolidine or the unsubstituted pyrrolidine. In the present complex the puckered five-membered ring of the dimethylpyrrolidine ligand assumes an envelope conformation with the maximum puckering at the N atom.     

Lower bound of overall exponential sums for polynomialsϕ(x d )

LetS ⁽¹⁾ (n, Q) denote the maximum module of exponential sums for polynomials of degree n over the Galois fieldF _Q . In a previous paper the transition to the multiple exponential sums allowed us to obtain a good lower bound of the valueS ⁽¹⁾ (n, Q), which coincides with Weil's bound whenn = q ^(m-1)/2 + 1, whereq, m are odd andm 3. Here the same approach is used for the estimation of the valueS ^(d) (n, Q), which corresponds to polynomials(x ^d ) overF _Q , whered is any divisor ofq – 1.     

Separated flow over a strongly cambered conical wing at small angles of attack

The interference of supersonic flows on the concave surface of conical wings was experimentally investigated in [1] for various values of the camber and angles of attack. In order to establish the detailed structure of the interference flow the laminar flow past a wing model in the form of half the surface of a circular cone with vertex angle 2_k = 34° was numerically modeled within the framework of the quasiconical approximation for the three-dimensional Navier-Stokes equations [2]. Under this assumption, confirmed by analysis of the experimental data [1], it was found that the displacement of the external inviscid flow as a result of intense flow separation beyond the leading edges leads to flow patterns similar to those realized on V wing's with a break in the transverse contour [3]. At nonzero angles of attack weak secondary separation was detected beneath the flattened regions of primary separation located in the shaded parts of the concave surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 130–136, July–August, 1989.     

Complex homogeneous splines on the torus

We construct functions M_α which are piecewise homogeneous polynomials on the (d+1)-dimensional torus U^d+1. These functions possess complete symmetry with respect to the independent variables. The symmetry and homogeneous relations for these functions are exploited to obtain a recurrence relation and explicit representations. Furthermore, we show that , where ω=e^12x/k, 0≤j_t≤k−1, are linearly independent. By restricting M_α to U^d, we obtain the complex analogue of polynomial box splines on a (d+1)-direction mesh on U^d, which is a multivariate analogue of B-splines on the circle studied by I.J. Schoenberg^[8].     

Relaxation methods in control theory

Two different relaxed problems associated with a problem of optimal control theory, governed by an ODE, are considered: the first is obtained by Young's methods and the second by semicontinuity arguments. A formula which relates the two relaxed functionals to each other is given.     

Turing determinacy and the continuum hypothesis

From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of ₁ is constructible from a real.     

Algorithms for flows with parametric capacities

We consider the problem of finding maximal flows with respect to capacities which are linear functions of a parametert [0,T]. Since this problem is a special case of a parametric linear program the classichorizontal approach can be applied in which optimal solutions are computed for successive subintervals of [0,T]. We discuss an alternative algorithm which approximates in each iteration the optimal solution for allt [0,T]. Thisvertical algorithm is a labeling type algorithm where the flow variables are piecewise linear functions. Flow augmentations are done alongconditional flow augmenting paths which can be found by modified path algorithms. The vertical algorithm can be used to solve the parametric flow problem optimally as well as to compute a good approximation for allt if the computation of the optimal solution turns out to be too time consuming.Partially supported by NSF Grants ECS-8412926 and INT-8521433, and NATO Grant RG 85/0240.     

Entropy dissipation rate and convergence in kinetic equations

We give a lower bound of the entropy dissipation rate of Ka, Boltzmann and Fokker-Planck-Landau equations. We apply this estimate to the problem of the speed of convergence to equilibrium in large time for the Boltzmann equation.